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The area bounded by the X-axis and y=x^(2)-4 from -5 to 0 2014-04-15
From Lexie:
Determine the area that is bounded by the following curve and the x-axis on the interval below. (Round your answer to three decimal places)

y=x^(2)-4, -5 ≤ x ≤ 0

The answer is 32.333 but I have no idea how to get there.

Answered by Penny Nom.
A tangent of the curve (x/a)^n+(y/b)^n =2 2014-04-15
From sudhir:
the equation of tangent of the curve (x/a)^n+(y/b)^n =2. at(a,b) is
Answered by Penny Nom.
The popcorn box problem 2013-11-07
From Dave:
We know that calculus can be used to maximise the volume of the tray created when cutting squares from 4-corners of a sheet of card and then folding up.

What I want is to find the sizes of card that lead to integer solutions for the size of the cut-out, the paper size must also be integer. EG 14,32 cutout 3 maximises volume as does 13,48 cutout 3.

I have done this in Excel but would like a general solution and one that does not involve multiples of the first occurence, as 16, 10 cutout 2 is a multiple of 8,5 cutout 1.

Answered by Walter Whiteley.
Water flowing out of a tank 2013-11-03
From Carolyn:
The flow of water out of a hole in a tank is known to be proportional to the square root of the height of water above the hole. That is,

dV/dt (proportional to) sq root (h)

The tank has a constant cross-sectional area A, show that the height of water in the tank is given by

h = ((-kt+C)/2)^2

If the tank is 9 metres high, and it takes 5 hours for it to drain from full to half full, how much longer will we have to wait until it is completely empty?

Answered by Penny Nom.
Maximize the volume of a cone 2013-10-09
From Conlan:
Hi I am dong calculus at school and I'm stumped by this question:

A cone has a slant length of 30cm. Calculate the height, h, of the cone if the volume is to be a maximum.

If anyone can help me it would be greatly appreciated.

thanks.

Answered by Penny Nom.
Equal ordinate and abscissa 2013-08-15
From sonit:
the slope of tangent to the curve y=(4-x^2)^1/2 at the point, where the ordinate and abscissa are equal, is
Answered by Penny Nom.
Differentiate x^x - 2^sinx 2013-08-09
From tarun:
derivative of x^x - 2^sinx
Answered by Penny Nom.
Tangents to the curve y = x^3 2013-03-24
From Ethan:
How many tangent lines to the curve y = x^33 pass through the point (2, 4)? For each such line, and the exact coordinates of the point of tangency on the curve.
Answered by Penny Nom.
Related rates 2013-02-17
From Ishaak:
A hemispherical bowl is filled with water at a uniform rate. When the height of water is h cm the volume is π(rh^2-1/3 h^3 )cm^3, where r s the radius. Find the rate at which the water level is rising when it is half way to the top, given that r = 6 and the bowl fills in 1 minute.
Answered by Penny Nom.
Integration from 0 to 2pi of 1/(3cos x + 2) dx 2013-02-04
From ankit:
Integration from 0 to 2pi of 1/(3cos x + 2) dx
Answered by Harley Weston.
Maximize profit 2013-01-19
From Chris:
A firm has the following total revenue and total cost function.
TR=100x-2x^2
TC=1/3x^3-5x^2+30x
Where x=output
Find the output level to minimize profit and the level of profit achieved at this output.

Answered by Penny Nom.
An integral 2012-12-16
From Slavena:
integration of (lnx)^2 / x dx
Answered by Penny Nom.
An area bounded by lines 2012-12-16
From sidra:
find area bounded by functions:
y=x
y=2x
and y=5-x

Answered by Penny Nom.
A max/min problem 2012-12-14
From bailey:
A right angled triangle OPQ is drawn as shown where O is at (0,0). P is a point on the parabola y = ax – x^2 and Q is on the x-axis.

Show that the maximum possible area for the triangle OPQ is (2a^3)/(27)

Answered by Penny Nom.
The derivative of y = sin (30º + x) 2012-11-07
From Saskia:
derivative of y = sin (30º + x)
Answered by Harley Weston.
An implicit differentiation problem 2012-10-26
From Katie:
find y' of x^2y-2y^3=3x+2y
Answered by Harley Weston.
How fast is the distance between the aircraft and the car increasing? 2012-10-24
From Steven:
At a certain instant an aircraft flying due east at 240 miles per hour passes directly over a car traveling due southeast at 60 miles per hour on a straight, level road. If the aircraft is flying at an altitude of .5mile, how fast is the distance between the aircraft and the car increasing 36 seconds after the aircraft passes directly over the car?
Answered by Penny Nom.
Differentiation rules 2012-10-23
From Morgan:
Use the derivative rules to differentiate each of the following:
1. f(x)=1/x-1
2. f(x)= sqrt(x)

Answered by Penny Nom.
A tangent to f(x) = 1/x 2012-09-04
From Steven:
Consider the graph of the function f(x) = 1/x in the first quadrant, and a line tangent to f at a point P where x = k. Find the slop of the line tangent to f at x = k in terms of k and write an equation for the tangent line l in terms of k.
Answered by Penny Nom.
A volume of revolution 2012-07-15
From Tewodros:
Let f(x) = e^x and g(x) = x^1/2 both be defined on [0,1]. Consider the region bounded by f(x), g(x), x = 0, x = 1. Rotate this region about the y-axis and determine the volume using the shell method.
Answered by Harley Weston.
Two cars approach a right-angled intersection 2012-04-10
From Michael:
Two cars approach a right-angled intersection, one traveling south a 40km/h and the other west at 70km/h. When the faster car is 4km from the intersection and the other case if 3km from the intersection, how fast is the distance between the car cars changing?
Answered by Penny Nom.
A maximization problem 2012-04-09
From Nancy:
After an injection, the concentration of drug in a muscle varies according to a function of time, f(t). Suppose that t is measured in hours and f(t)=e^-0.02t - e^-0.42t. Determine the time when the maximum concentration of drug occurs.
Answered by Penny Nom.
The spread of a rumor 2012-04-09
From Roohi:
The function f(t) = a/(1+3e^(-bt)) has also been used to model the spread of a rumor. Suppose that a= 70 and b=3 0.2. Compute f(2), the percentage of the population that has heard the rumor after 2 hours. Compute f'(2) and describe what it represents. Compute lim t approaches infinity and describe what it represents.
Answered by Penny Nom.
The period T of a pendulum 2012-03-27
From Ashley:
The period T of a pendulum is given in terms of its length, l, by T=2pi sqrt(l/g) where g is the acceleration due to gravity(a constant)
a. find dT/dl
b. what is the sign of dT/dl
c. what does the sign of dT/dl tell you about the period of the pendulums?

Answered by Penny Nom.
The derivative of x^-(1/2) 2012-01-14
From Eric:
I have an problem figuring out the derivative of the negative square root of x i.e. x^-(1/2) using the first principle.
Could someone please show me?
Thanks in advance!

Answered by Harley Weston.
Lost in the woods 2012-01-12
From Liz:
I am lost in the woods. I believe that I am in the woods 3 miles from a straight road. My car is located 6 miles down the road. I can walk 2miles/hour in the woods and 4 miles/hour along the road. To minimize the time needed to walk to my car, what point on the road should i walk to?
Answered by Harley Weston.
A volume of revolution 2012-01-11
From john:
find volume of solid generated by revolving the region in the first quadrant bounded by the curve y squared=x cubed, the line x=4 and the x-axis about the line y=8. The answer in the back of the book is 704 pi divided by5
Answered by Penny Nom.
Water is flowing into a cup 2011-12-19
From Tim:
A cup has a radius of 2" at the bottom and 6" on the top. It is 10" high. 4 Minutes ago, water started pouring at 10 cubic " per minute. How fast was the water level rising 4 minutes ago? How fast is the water level rising now? What will the rate be when the glass is full?
Answered by Penny Nom.
A cube of ice is melting 2011-12-05
From Emily:
a cube of ice (i.e.) each side is of the same length) is melting at a rate such that the length of each side is decreasing at a rate of 5cm per hour. how fast is the volume of the cube decreasing (in cubic cm per hour) at the instant the length of each side is 25cm?
Answered by Penny Nom.
Water pouring into a conical tank 2011-11-21
From Patience:
Hi my name is patience and I'm having a problem with this question.
Water pours into a conical tank of semi vertical angle 30 degrees at the rate of 4 cm^3/s, where h is the depth of the water at time t. At what rate is the water rising in the tank when h = 10 cm?
Thank you

Answered by Penny Nom.
Lines tangent to y^2=4x 2011-11-11
From Reuchen:
Find equations of the lines tangent to y^2=4x and containing (-2,1).
Answered by Penny Nom.
A spherical ball in a conical wine glass 2011-10-26
From Jules:
A heavy spherical ball is lowered carefully into a full conical wine glass whose depth is h and whose generating angle (between the axis and a generator) is w. Show that the greatest overflow occurs when the radius of the ball is (h*sin(w))/(sin(w)+cos(2w)).
Answered by Claude Tardif.
Implicit differentiation 2011-10-20
From Monica:
Find dy/dx in terms of x and y, if sin(xy)=(x^2)-y.
Answered by Penny Nom.
A hemispherical bowl with a lead ball inside 2011-09-27
From Jean:
"(a) Water is being poured into a hemispherical bowl of radius 3 inch at the rate of 1 inch^3/s. How fast is the water level rising when the water is 1 inch deep ?

(b) In (a), suppose that the bowl contains a lead ball 2 inch in diameter, and find how fast the water level is rising when the ball is half submerged."

Answered by Penny Nom.
The derivative of f(x) = (x+1)^1/2 2011-09-05
From Carla:
Find the derivative using the limit process of
f(x) = (x+1)^1/2

Answered by Harley Weston.
The height of a fluid in a horizontal tank 2011-07-24
From jason:
Same set up as many others, cylindrical tank on its side, but I am interested in defining the change in volume and/or fluid level as a function of time at a constant volumetric outflow. I plan on hooking a pump to the tank so "gpms' will be constant. I have a couple different sized tanks and pumps so I want a general equation. Thanks for your help.
Answered by Harley Weston.
A line tangent to f(x)=1/x 2011-06-05
From Michael:
A line tangent to f(x)=1/x in the first quadrant creates a right triangle with legs the x-axis and the y-axis. Prove that this triangle is always 2 square units regardless of where the point of tangency is.
Answered by Penny Nom.
An antiderivative of the square root of (8t + 3) 2011-04-19
From Caitlyn:
I know how to take an antiderivative. But this one's stumping me. I need it to finish a problem. What's the antiderivative of the square root of (8t + 3)

~Caitlyn=

Answered by Penny Nom.
Designing a tin can 2011-03-31
From Tina:
A tin can is to have a given capacity. Find the ratio of the height to diameter if the amount of tin ( total surface area) is a minimum.
Answered by Penny Nom.
A stone is dropped into a lake 2011-03-24
From AnneMarie:
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 25 cm/s. Find the rate at which the area within the circle is increasing after 4s.
Answered by Penny Nom.
At what rate is the grain pouring from the chute? 2011-02-26
From MJ:
Suppose that grain pouring from a chute forms a conical heap in such a way that the height is always 2/3 the radius of the base. At the moment when the conical heap is 3 m high, its height is rising at the rate of 1/2 m/min. At what rate (in m^3/min) is the grain pouring from the chute?
Answered by Penny Nom.
Integrating ln^3x/x 2011-01-14
From ken:
y=ln^3x/x from x=1 to x=11
Answered by Penny Nom.
What is the maximum weekly profit? 2010-10-10
From Joe:
A local artist sells her portraits at the Eaton Mall. Each portrait sells for $20 and she sells an average of 30 per week. In order to increase her revenue, she wants to raise her price. But she will lose one sale for every dollar increase in price. If expenses are $10 per portrait, what price should be set to maximize the weekly profits? What is the maximum weekly profit?
Answered by Stephen La Rocque and Penny Nom.
A Taylor polynomial for (lnx)/x 2010-09-29
From Dave:
I have a series problem that I cannot solve. The problem asks for you to compute a Taylor polynomial Tn(x) for f(x) = (lnx)/x. I calculated this poly out to T5(x) and attempted to use this to identify a pattern and create a series in order to calculate Tn(x). However, the coefficients on the numerator out to F5prime(x) are as follows: 1, -3, 11, -50, 274... Ok, so the negative is an easy fix -> (-1)^n-1. But the other coefficients are stumping me. I can't see any sort of pattern there and I've tried every trick I know. Is there another way to go about this? Thanks!
Answered by Chris Fisher.
limit as x approaches a of ((x^(1/2))-(a^(1/2)))/(x-a)? 2010-09-29
From emily:
limit as x approaches a of ((x^(1/2))-(a^(1/2)))/(x-a)?
Answered by Penny Nom.
Continuity 2010-09-18
From Carina:
Hi. My name's Carina and I'm currently a sophomore in high school. I'm having a lot of difficulties in AP Calculus with continuity, one-sided limits, and removable discontinuities. Basically, I have no idea how to do them or even what they are. I read the lesson but I still don't get it. Can someone put it in simpler terms so I can understand how to complete my questions? Thank you!
Answered by Robert Dawson.
Maximizing the volume of a cylinder 2010-08-31
From Haris:
question: the cylinder below is to be made with 3000cm^2 of sheet metal. the aim of this assignment is to determine the dimensions (r and h) that would give the maximum volume. how do i do this? i have no idea. can you please send me a step-to-step guide on how t do this? thank you very much.
Answered by Penny Nom.
A max min problem 2010-08-19
From Mark:
a rectangular field is to be enclosed and divided into four equal lots by fences parallel to one of the side. A total of 10000 meters of fence are available .Find the area of the largest field that can be enclosed.
Answered by Penny Nom.
Maximize the floor area 2010-07-07
From shirlyn:
A rectangular building will be constructed on a lot in the form of a right triangle with legs of 60 ft. and 80 ft. If the building has one side along the hypotenuse, find its dimensions for maximum floor area.
Answered by Penny Nom.
A max/min problem 2010-06-12
From valentin:
What is the maximum area of an isosceles triangle with two side lengths equal to 5 and one side length equal to 2x, where 0 ≤ x ≤ 5?
Answered by Harley Weston.
Integration of sin^3 (2x) 2010-05-29
From ascher:
how do you integrate this equation ∫ sin^3 (2x) dx
Answered by Robert Dawson and Penny Nom.
An optimization problem 2010-05-23
From Marina:
Hello, I have an optimization homework assignment and this question has me stumped..I don't even know

A hiker finds herself in a forest 2 km from a long straight road. She wants to walk to her cabin 10 km away and also 2 km from the road. She can walk 8km/hr on the road but only 3km/hr in the forest. She decides to walk thru the forest to the road, along the road, and again thru the forest to her cabin. What angle theta would minimize the total time required for her to reach her cabin?
I'll do my best to copy the diagram here:

                             10km
Hiker_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _Cabin
      \                           |                              /
       \                          |                             /
     f  \                      2km                          /
         \                        |                           /
theta   \___________________________ /
                            Road


Answered by Penny Nom.
The rate of change of y with respect to x 2010-04-29
From Tom:
I just had a quick calc question about wording that wasn't ever addressed in class. When the book says "the rate of change of y with respect to x", should it be considered how fast y is changing in comparison to x?

I ask because the textbook says that "y is changing 3 times faster than x, so the rate of change of y with respect to x is 3." I'm use to rate being like velocity, as in units of distance per units of time. All we're told in class is that it's the slope of the tangent line, I was hoping you could clarify for me what exactly is meant by the wording of a "rate of change of something with respect to something else". More specifically, what "rate" and "with respect to" mean within this context?

Thanks for your time

Answered by Harley Weston.
A rectangular garden 2010-04-25
From Billy:
Tanisha wants to make a rectangular garden with a perimeter of 38 feet. What is the greatest area possible that tanisha can make the garden?
Answered by Penny Nom.
Integrate the ((4th root of x^3)+1) dx 2010-04-12
From Bridget:
integrate the ((4th root of x^3)+1) dx
Answered by Tyler Wood.
The derivative of y=x^x 2010-04-09
From David:
So, its David, and I was wondering about the derivative of y=x^x. I have often seen it be shown as x^x(ln(x)+1), but when I did it through limits it turned out differently. Here's what I did:
It is commonly know that df(x)/dx of a function is also the limit as h->0 of f(x+h)-f(x)/h. To do this for x^x you have to start with lim h->0 ((x+h)^(x+h)-x^x)/h. The binomial theorem then shows us that this is equal to lim h->0 (x^(x+h)+(x+h)x^(x+h-1)h+...-x^x)/h
This is also equal to lim a->0 lim h->0 (x^(x+a)+(x+h)x^(x+h-1)h...-x^x)/h.
Evaluating for a=0 you get lim h->0 (x^x+(x+h)x^(x+h-1)h...x^x)/h
Seeing as the last 2 terms on the numerator cancel out you can simplify to a numerator with h's is each of the terms, which you can then divide by h to get:
lim h->0 (x+h)x^(x+h-1)... which when evaluated for h=0 gives us: x(x^(x-1)). This statement is also equal to x^x.
This contradicts the definition of the derivative of x^x that is commonly shown. So, my question is: can you find any flaws in the logic of that procedure? I do not want to be shown how to differentiate x^x implicitly because I already know how to do that.

Answered by Robert Dawson.
A max min problem 2010-04-06
From Terry:
The vertex of a right circular cone and the circular edge of its base lie on the surface of a sphere with a radius of 2m. Find the dimensions of the cone of maximum volume that can be inscribed in the sphere.
Answered by Harley Weston.
The derivative of cos^3x 2010-04-06
From Erson:
Find y' of the given function: y = cos^3x.
Answered by Harley Weston.
Sand falling off a conveyer 2010-04-02
From Katherine:
sand is falling off a conveyer onto a pile at the rate of 1.5 cubic feet per minute. The diameter of the base is approximately twice the altitude. At what rate is the height of the pile changing when it is 10 feet high?
Answered by Penny Nom.
The integral of X^3/the square root of 1-x^2 dx 2010-03-07
From William:
The integral of X^3/the square root of 1-x^2 dx.
Answered by Harley Weston.
A cone circumscribed about a given hemisphere 2010-01-19
From Neven:
The cone of smallest possible volume is circumscribed about a given hemisphere. What is the ratio of its height to the diameter of its base?
(G.F.Simmons, Calculus with Analytic Geometry, CH4 Applications of Derivatives)

Answered by Chris Fisher.
f(x)=x+2sinx 2009-12-12
From amroziz:
for which values of x does the graph of f(x)=x+2sinx have horizontal tangent
Answered by Harley Weston.
How fast is the distance between the two cars decreasing? 2009-12-08
From Jenny:
Two cares are on a collision course toward point P. The paths of the two cars make a 30 degree angle with each other. The first car is 40 km from P, and traveling toward P at 16 km/hour. The second car is 50 km from P, traveling at 20 km/hour. How fast is the (straight line) distance between the two cars decreasing. (Hint: Law of Cosines)
Answered by Harley Weston.
The triangle formed by the tangent and the coordinates axes 2009-11-23
From Nirmala:
Given that y=1/x, x is not equal to zero. Prove that the area of the triangle formed by the tangent and the coordinates axes is 2.
Answered by Harley Weston.
f(x)= (e^x) / [(e^x)+(ex^2)] 2009-11-10
From natalie:
I'm trying to graph the function, f(x)= (e^x) / (e^x)+(ex^2) [e to the x divided by e to the x plus e times x squared] I know that there aren't any vertical asymptotes, but is there a horizontal asymptote? and also, I'm stuck on finding the concavity for this graph. I tried to find f "(x), but it came out to be really long and I am not sure how to find the x values for f "(x) without using a graphic calculator. thanks, natalie
Answered by Chris Fisher and Harley Weston.
At what rate are the people moving apart? 2009-11-01
From saira:
A man starts walking north at 4 ft/s from a point P. 5 minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 minute after the woman starts walking ?
Answered by Harley Weston.
Painting a dome 2009-10-30
From Jessica:
A hemispherical dome with a radius of 50 ft will be given a coat of paint .01 inch thick. The Contractor for the job wants to estimate the number of gallons of paint needed. Use a differential to obtain an estimate (231 cubic inches/gallon) HINT: Approximate the change in volume of hemisphere corresponding to increase of .01 inch in the radius.
Answered by Robert Dawson.
Graphing y=(4-x^2)^5 2009-10-25
From natalie:
I want to graph the curve of y=(4-x^2)^5 without using a graphing calculator. To do this, I'm suppose to find: domain, y and x intercepts, asymptotes, intervals of increase/decrease, local max/min, concavity and points of inflection. I got all the way to the step where I'm solving the concavity and I'm stuck. I found the f"(x) and it came out to be really large polynomial. I want to know how I can solve for the x of f"(x) without the use of a graphing calculator, when the polynomial has x^6 and x^8. Thank you so much, natalie
Answered by Harley Weston.
A max/min problem 2009-10-12
From avien:
a rectangle has a line of fixed length Lreaching from the vertex to the midpoint of one of the far sides. what is the maximum possible area of such a rectangle? SHOW SOLUTION USING CALCULUS
Answered by Penny Nom.
A line tangent to a parabola 2009-10-01
From kanchan:
for what value of c a line y=mx+c touches a parabola y^2=4a(x-a)
Answered by Penny Nom.
solve integral of ( x^2+x+1)^5 2009-09-18
From jaka:
solve integral of ( x^2+x+1)^5
Answered by Robert Dawson.
An antiderivative problem 2009-08-13
From Indrajit:
∫4e^x + 6e^-x/(9e^x + 4e^-x)dx = Ax + Bloge(9e2x - 4) + C

then A=?......B=?.....C=?

plz solve it...."^" stands for "to the power of"....

Answered by Harley Weston.
A rectangular pen 2009-08-13
From Kari:
A rectangular pen is to be built using a total of 800 ft of fencing. Part of this fencing will be used to build a fence across the middle of the rectangle (the rectangle is 2 squares fused together so if you can please picture it). Find the length and width that will give a rectangle with maximum total area.
Answered by Stephen La Rocque.
Torricelli's trumpet 2009-07-29
From Gary:
I was reading about torricelli's trumpet which is described by the equation1/x which is then rotated around the x axis which results in a figure which looks like a trumpet. Now in order to find the volume the integral 1/x^2 dx is used which diverges when integrated so the volume is finite.However if you integrate 1/x dx which is the formula on the plane the answer diverges. Now if you took an infinite area then rotated it around the x axis shouldn't you get an infinite volume? Notice the area I am talking about is under the line 1/x not the surface area of the trumpet which is what the painters paradox is about What am I missing? Thanks
Answered by Robert Dawson.
The integral of x^x 2009-06-18
From ANGIKAR:
what would be the integration of (X^Xdx)?

give answer in details.

Answered by Robert Dawson and Harley Weston.
differentiate y sin[x^2]=x sin[y^2] 2009-05-11
From mamiriri:
derivate y sin[x^2]=x sin[y^2]
Answered by Harley Weston.
The integral of a to power x squared 2009-04-28
From JIM:
WHEN I ATTENDED U.OF T. (TORONTO ) MANY YEARS AGO WE WERE TOLD THE FOLLOWING INTEGRAL COULD NOT BE SOLVED : a to power x squared . is this still true ?

CURIOUS , JIM

Answered by Robert Dawson.
A max-min problem 2009-04-20
From Charlene:
A fixed circle lies in the plane. A triangle is drawn inside the circle with all three vertices on the circle and two of the vertices at the ends of a diameter. Where should the third vertex lie to maximize the perimeter of the triangle?
Answered by Penny Nom.
Sand falls from a conveyor belt 2009-04-01
From Tracy:
Sand falls from a conveyor belt at the rate of 10 cubic feet per minute onto a conical pile. The radius of the base is always equal to half the pile's height. How fast is the height growing when the pile is 5ft high?
Answered by Stephen La Rocque.
A spherical Tootsie Roll Pop 2009-04-01
From Tracy:
A spherical Tootsie Roll Pop you are sucking on is giving up volume at a steady rate of .8 ml/min. How fast will the radius be decreasing when the Tootsie Roll Pop is 20 mm across?
Answered by Harley Weston.
An isosceles triangle 2009-03-26
From sela:
An isosceles triangle has two equal sides of length 10 cm. Theta is the angle between two equal sides.
a) Express area of a triangle as a function of theta
b) If theta is increasing at a rate of 10 degrees/minute, how fast is area changing at the instant theta=pi/3?
c) at what value of theta will the triangle have the maximum area?

Answered by Penny Nom.
A max-min problem 2009-03-24
From Jay:
Determine the area of the largest rectangle that can be inscribed between the x-axis and the curve defined by y = 26 - x^2.
Answered by Harley Weston.
A common tangent to two curves 2009-03-02
From Jay:
For what values of a and b will the parabola y = x^2 + ax + b be tangent to the curve y = x^3 at (1,1)?
Answered by Penny Nom.
Implicit differentiation 2009-03-01
From Emily:
determine the derivative y' at the point (1,0)
y= ln(x^2+y^2)

y'(1)= ??

Answered by Stephen La Rocque.
Implicit differentiation 2009-02-18
From Sunny:
Find slope of the tangent line to the curve 2(x^2+y^2)2=25(x^2–y^2) at (3,-1)
Answered by Robert Dawson and Harley Weston.
The area between the x-axis and a curve 2009-02-18
From Lauren:
This is from a homework question I can't figure out.

Let R be the region in the fourth quadrant enclosed by the x-axis and the curve y= x^2 - 2kx, where k > 0. If the area of the region R is 36 then what is the value of k?

Answered by Robert Dawson.
The second derivative of h(x)=f(g(x)) 2009-02-16
From Kristina:
If h(x)=f(g(x)), and is differentiable, then find h"(x).
Answered by Robert Dawson.
A definite integral 2009-02-09
From Mathata:
Evaluate: integral from 0 to 1, x^2 e^x^3dx
Answered by Harley Weston.
A trig limit 2009-02-05
From Samantha:
lim x-> 0 ( ( r*cos(wt +h) + r*cos(wt) )/ h )

Where r & w are constants.

Answered by Harley Weston.
A point on -8x^2+5xy+y^3=-149 2009-02-04
From Vivian:
Consider the curve defined by -8x2+5xy+y3=-149
a) find dy/dx
b) Write an equation for the line tangent to the curve at the point (4,-1)
c) There is a number k so that the point (4.2,k) is on the curve. Using the tangent line found in part b), approximate the value of k.
d) write an equation that can be solved to find the actual value of k so that the point (4.2,k) is on the curve
e) Solve the equation found in part d) for the value of k

Answered by Harley Weston.
limit sinx/x 2009-01-30
From Jackie:
how to evaluate limit sinx/x as x tends to zero if x is in degrees
Answered by Stephen La Rocque and Harley Weston.
An integral from 1 to infinity 2009-01-24
From Ray:
Determine the area bounded by the x-axis and the curve y=1/(x^2) from x=1 to x=infinity.
A. 1.00
B. infinity
C. indeterminate
D. 2.00

Answered by Harley Weston.
Archimedes' formula for parabolic arches 2009-01-23
From La:
Use calculus to verify Archimedes' formula for y=9-x^2. Prove Archimedes' formula for a general parabolic arch.
Answered by Harley Weston.
In the shadow of a flagpole 2009-01-22
From La:
How fast is the length of the shadow of an 18 foot flagpole growing when the angle of elevation of the sun is 45 degrees and is decreasing at a rate of 10 degrees per hour?
Answered by Harley Weston.
Partial derivatives 2009-01-17
From Meghan:
I have a question I've been working at for a while with maxima/minima of partial derivatives.

"Postal rules require that the length + girth of a package (dimensions x, y, l) cannot exceed 84 inches in order to be mailed. Find the dimensions of the rectangular package of greatest volume that can be mailed. (84 = length + girth = l + 2x + 2y)"

Answered by Harley Weston.
Negative rate of change 2009-01-12
From hemanshu:
when i have to find rate of change of decrease in any value my ans comes in negative why??????????
Answered by Penny Nom.
What is the maximum revenue? 2009-01-09
From Kristy:
A skating rink manager finds that revenue R based on an hourly fee x for skating is represented by the function R(x) = -200x^2 + 1500x

What is the maximum revenue and what hourly fee will produce maximum revenues?

Answered by Harley Weston.
A max/min problem 2009-01-09
From Angelica:
have 400 feet of fence. Want to make a rectangular play area. What dimensions should I use to enclose the maximum possible area?
Answered by Robert Dawson.
The area of a region bounded by two curves 2009-01-07
From Rogerson:
Find the area, S, enclosed by the given curve(s) and the given line.
y = x^2 - x - 1, y = x+2

Answered by Harley Weston.
A kennel with 3 individual pens 2009-01-06
From Jean:
An animal clinic wants to construct a kennel with 3 individual pens, each with a gate 4 feet wide and an area of 90 square feet. The fencing does not include the gates. Write a function to express the fencing as a function of x. Find the dimensions for each pen, to the nearest tenth of a foot that would produce the required area of 90 square feet but would use the least fencing. What is the minimum fencing to the nearest tenth?
Answered by Harley Weston.
The area enclosed by a curve and the x-axis 2009-01-04
From Rogerson:
Find the area, S, enclosed by the curve y = -x^2 + 6x - 5 and the x-axis in the interval 0≤x≤4.
Answered by Harley Weston.
Determine y'' by implicitly differentiating twice 2009-01-04
From Walter:
Given x^3 - 3xy + y^3 = 1 , determine y'' by implicitly differentiating twice. I cannot solve this. Would you be kind enough to perform the mathematics and show the steps involved in obtaining the solution?
Answered by Harley Weston.
The area of a region in the plane 2009-01-03
From Rogerson:
Find the area, S, of the shaded region enclosed by the given cureve, the given line and the x-axis.

y = -x^2 + 1
line x = 2

Answered by Harley Weston.
Pouring angles for a crucible 2008-12-20
From Richard:
I am trying to work at pouring angles and volume left in during pouring a crucible, The crucible is cylindrical and flat bottomed.

I know the diameter, radius and volume of the crucibles. and the volume of liquid going into it.

So lets say the crucible is only half full firstly I need to work out the angle just before its going to pour. ( I can work this out as long as there is a certain volume of liquid if its not enough I cant do it)

Now the problem I also need to work out how much I should tilt the crucible to allow a certain amount out and be able to do this untill the volume reaches 0 at 90' turn. This is where I am stuck.

The reason for needing to be able to work this out is so i can develop a constant flow for example 10Kg of metal per second. Thank you very much for you time

Answered by Harley Weston.
Integral of cos^2 X between pi/2 and 0 2008-12-18
From Wanda:
Integral or Area of cos^2 X between pi/2 and 0.

The answer that I got is -pi/4. Is this correct? If not, how did you come up with your answer?

Answered by Robert Dawson.
A sphere in a can of water 2008-12-12
From Meghan:
A cylindrical can open at the top has (inside) base radius equal to 1. The height of the can is greater than 2. Imagine placing a steel sphere of radius less than 1 into the can, then pouring water into the can until the top of the sphere is just covered.

What should be the radius of the sphere so the volume of water used is as large as possible?

Answered by Harley Weston.
How fast is the distance between the airplanes decreasing? 2008-11-10
From Crystal:
At a certain instant, airplane A is flying a level course at 500 mph. At the same time, airplane B is straight above airplane A and flying at the rate of 700 mph. On a course that intercepts A's course at a point C that is 4 miles from B and 2 miles from A. At the instant in question, how fast is the distance between the airplanes decreasing?
Answered by Harley Weston.
A trig limit 2008-11-04
From Teri:
Although I have this problem completely worked out in front of me I still cannot understand how it was done. The problem is:
Find the limit.
lim x->0 sin2x/tan7x.

Answered by Harley Weston.
Separating variables 2008-11-04
From Terry:
by separating variables solve the initial value problem

(x+1)y' + y = 0 y(0) = 1

Answered by Harley Weston.
Taxes in Taxylvania 2008-10-22
From April:
Taxylvania has a tax code that rewards charitable giving. If a person gives p% of his income to charity, that person pays (35-1.8p)% tax on the remaining money. For example, if a person gives 10% of his income to charity, he pays 17 % tax on the remaining money. If a person gives 19.44% of his income to charity, he pays no tax on the remaining money. A person does not receive a tax refund if he gives more than 19.44% of his income to charity. Count Taxula earns $27,000. What percentage of his income should he give to charity to maximize the money he has after taxes and charitable giving?
Answered by Harley Weston.
Antiderivative of 1/(x(1 - x)) 2008-10-22
From Matt:
derivative of dx/(x(1-x))

From what I've seen I should break apart the equation as such derivative of dx/x - dx/(1-x) and then get the 2 corresponding log functions.

If that is correct why does this factoring work, if that is incorrect what is the proper way to find the derivative.

Answered by Harley Weston.
The slope of a tangent line 2008-10-18
From Amanda:
If f(x)=square root of (x+4), and the slope of the tangent line at x=21 was 1/n for some integer n, then what would you expect n to be?
Answered by Stephen La Rocque.
The biggest right circular cone that can be inscribed in a sphere 2008-09-08
From astrogirl:
find the volume of the biggest right circular cone that can be inscribed in a sphere of radius a=3
Answered by Harley Weston.
[f(x)-f(1)]/(x-1) 2008-08-14
From katie:
Evaluate (if possible) the function of the given value of the independent variable:

f(x)=(x^3)-x:

[f(x)-f(1)]/(x-1)

Answered by Penny Nom.
Integral of X^2 2008-07-28
From Hemanshu:
Integral of X^2
Answered by Janice Cotcher.
The maximum range of a projectile 2008-07-22
From kwame:
the range R of projectile fired with an initial velocity Vo ,at an angle of elevation (@ )theta from the horizontal is given by the equation R = (Vo(squared) sin2theta)/g. where g is the accelation due to gravity . Find the angle theta such that the projectile has maximum range .
Answered by Harley Weston.
A square and a circle 2008-07-20
From kobina:
4 ft of a wire is to be used to form a square and a circle. how much of the wire is to be used for the square and how much should be used for the square in order to enclose the maximum total area
Answered by Harley Weston.
A difference quotient 2008-07-10
From Rita:
Find the difference quotient of f, that is, find [f (x + h) - f (x)]/h, where h does not = 0 for the given function. Be sure to simplify.

f(x) = 1/(x + 3)

Answered by Janice Cotcher.
A dog tied to a round building 2008-07-08
From maitham:
i have this question which i don't know how to solve it :

One dog was linked to the outer wall of a building round of 20 meters in diameter. If the length of chain linking the dog sufficient turnover of half the distance around the building, What area can guard dog?

they said that we can solve it by integral .. can you solve it for me?

Answered by Harley Weston.
The rate of change in the depth of the water 2008-06-12
From Liz:
A rectangular pool 50ft long and 30ft. wide has a depth of 8 ft. for the first 20 ft. for its length and a depth of 3 ft. on the last 20ft. of its length and tapers linearly for the 10 ft in the middle of its length. the pool is being filled with water at the rate of 3ftcubed/ min at what rate is the depth of the water in the pool increasing after 15 hours?
Answered by Harley Weston.
The length of a shadow 2008-05-27
From Simon:
A figure skater is directly beneath a spotlight 10 m above the ice. IF she skates away from the light at a rate of 6m/s and the spot follows her, how fast is her shadow's head moving when she is 8m from her starting point? The skater is (almost) 1.6m tall with her skates on.
Answered by Stephen La Rocque and Harley Weston.
The weight of a concrete column 2008-05-11
From russell:
a cylindrical form is filled with a slow curing concrete. The base of the form is 10 ft in radius, and height is 25 ft. while the concrete hardens, gravity causes the density to vary from a density of 90 lbs/ft^3 at the bottom to a density of 50 lb/ft^3 at the top. Assume that the density varies linearly from the top to the bottom, and compute the total weight of the resulting concrete column
Answered by Harley Weston.
A lidless box with square ends 2008-04-28
From Chris:
A lidless box with square ends is to be made from a thin sheet of metal. Determine the least area of the metal for which the volume of the box is 3.5m^3. I did this question and my answer is 11.08m^2 is this correct? If no can you show how you got the correct answer.
Answered by Stephen La Rocque and Harley Weston.
At what value of t is the maximum acceleration? 2008-04-25
From Mary:
Velocity of a function (which is the first derivative of its position) is defined over the interval 0 to 12 using the following piecewise function: v(t)=-1 from 0 to 4, v(t)=x-5 from (4 to 8 and v(t)=-x+11 from (8 to 12. At what value of t is the maximum acceleration?
Answered by Stephen La Rocque.
A volume of revolution 2008-04-24
From Sabahat:
Hi, i have a region enclosed by both axes, the line x=2 and the curve y=1/8 x2 + 2 is rotated about the y-axis to form a solid . How can i find the volume of this solid?. (Please note that y equation is read as y =1 over 8 times x square plus 2.) I will be really grateful if you answer this question. :)
Answered by Harley Weston.
An open box 2008-04-23
From Le:
Metal Fabrication; If an open box is made from a tin sheet 8 in square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made.
Answered by Harley Weston.
What is the integral of 13sin^3(x)*cos^7(x)dx? 2008-04-22
From Cathrine:
I am having trouble integrating this problem. It says to evaluate the integral but I don't know what to do or how to do it.

It is the integral of
13sin^3(x)*cos^7(x)d

Answered by Harley Weston.
f(x)=sin^3(3x^2) find f ' (x) 2008-04-21
From Michael:
f(x)=sin^3(3x^2) find f ' (x)
Answered by Harley Weston.
The area bounded by 3 curves 2008-04-13
From Sabahat:
Hi, I have enclosed a diagram.
The diagram shows the curve y=(2x-5)4. The point P has co-ordinates (4,81) and the tangent to the curve at P meets the x-axis at Q.

Find the area of the region (shaded in the diagram) enclosed between the curve, PQ and the x-axis . (Please note that the equation y is read as y=2x -5 whole raise to power 4.)

Answered by Stephen La Rocque.
f(x) =ax^blnx 2008-04-13
From charles:
supposef(x) =ax^blnx is a real- valued function. Determine exact values(not decimal approximations) fro nonzero constants a and b so that the function f has a critical point at x=e^3 and a maximum value of 1/2e
Answered by Harley Weston.
A volume of revolution 2008-04-04
From ted:
Consider the region bounded by y=x^2 + 1, y=5-3x and y=5. Sketch and shade the given region; then set up but dont evaluate teh integrals to find the following:

a) The volume of the solid generated by rotating the region about the line y=5

b) the volume of the solid generated by rotating the region about the y-axis

Answered by Penny Nom.
lim as x approaches infinite of 5x + 2/x-1 2008-04-04
From Jordan:
how to solve this.

lim as x approaches infinite of 5x + 2/x-1

Answered by Stephen La Rocque and Harley Weston.
The integral of dx / (4x^2 - 25)^3/2 2008-04-01
From Meghan:
I have a question from the trigonometric substitution of my calculus course.

integral of dx / (4x^2 - 25)^3/2

Answered by Harley Weston.
A max-min problem 2008-03-27
From LSL:
show that of all rectangle with a given area, the square has the smallest perimeter.
Answered by Penny Nom.
A train and a boat 2008-03-15
From Sabrina:
A railroad bridge is 20m above, and at right angles to, a river. A person in a train travelling at 60 km/h passes over the centre of the bridge at the same instant that a person in a motorboat travelling at 20km/h passes under the centre of the bridge. How fast are the two people separating 10s later?
Answered by Harley Weston.
What point on the graph y = e^x is closest to the origin? 2008-03-03
From elvina:
What point on the graph y = e^x is closest to the origin? Justify your answer.
Answered by Stephen La Rocque.
A Norman window 2008-02-25
From Jason:
If the perimeter of a Norman window is 20 feet, what is the maximum area of the window?
Answered by Stephen La Rocque.
A ball bearing is placed on an inclined plane 2008-02-15
From Leah:
A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is x. The distance (in meters) that the ball bearing rolls in t seconds is s(t) = 4.9(sin x)t^2. What is the speed of the ball bearing, and what value of x will produce the maximum speed at a particular time?
Answered by Penny Nom.
Two regions with equal area 2008-02-13
From James:
There is a line through the origin that divides the region bounded by the parabola y=3x-5x^2 and the x-axis into two regions with equal area. What is the slope of that line?
Answered by Harley Weston.
Integration by parts 2008-01-30
From seth:
hi i really dont understand integr ation by parts. for example, the integral(t^2sintdt. i have u=t^2 and v'=sint also u'=t^/3 v=-cost for the formula i have uv-integralvu' dx this is all well and good but i cant get it right.
Answered by Harley Weston.
Inflection points 2008-01-25
From Armando:
Hi, Im trying to write a program that takes an equation ( f(x) = 0 ) and returns a list of the inflexion points in a given interval. there must be (I think) a mathematical method or algorithm to do this, probably involving the (second) derivate of the function. However I have not found such a method yet. Any help on this will be much appreciated.
Answered by Stephen La Rocque and Harley Weston.
Maximize income 2008-01-18
From Chris:
Lemon Motors have been selling an average of 60 new cars per month at $800 over the factory price. They are considering an increase in this markup. A marketing survey indicates that for every $20 increase, they will sell 1 less car per month. What should their new markup be in order to maximize income?
Answered by Stephen La Rocque and Harley Weston.
The integral of 1/ (x(x+1)^0.5) 2007-12-29
From Nooruddin:
Integral of
dx / x(x+1)^0.5
(boundaries are 5 and 3)

Answered by Harley Weston.
Differentiate 2007-12-28
From taiwo:
i am finding it difficult to use first principle to differentiate this question: y=xcos2x. can u help me.
Answered by Penny Nom.
lim sinx/(x +tanx) 2007-12-16
From shimelis:
i have problem how do you solve this equation lim sinx/(x +tanx)
Answered by Harley Weston.
Maximize the product 2007-11-25
From David:
Hi i have this site call calcchat.com, but i dont understand how they explained this can you take a look? The question is: Direction: Find two positive numbers that satisfy the given requirements. The sum is S and the product is a maximum this is what they did 1) Let x and y be two positive numbers such that x + y = S 2)P = xy 3) = x (S - x) 4) =Sx - x^2 5)...etc. the thing i dont get is how did they go from step 2 to step 3 and also i know this sound dumb but how did they get step 2? =)
Answered by Harley Weston.
A rectangular plot of farmland 2007-11-25
From Christy:
A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?
Answered by Harley Weston.
A curve sketch 2007-11-22
From Ahson:
Find critical points, determine the monotonicity and concavity and sketch a graph of f(x) with any local maximum, local minimum and inflection points labeled:

1. f(x) = x^4 - x^3 - 3x^2 + 1

Answered by Harley Weston.
A rectangle in an ellipse 2007-11-18
From David:
I need to find the max area of a rectangle inscribed in an ellipse with the equation x^2+4y^2=4.. What I have so far is f(x,y)=4xy g(x,y)=x^2+4y^2-4=0,

y=sqrtx^2-4/4 f'(x)=2x^2/sqrt-4x^2+2(sqrt-4+x^2).

What I need to know is how to finish the problem and find the actual mas area of the rectangle. David

Answered by Penny Nom.
lim [x + squareroot(x^2 + 3)] as x->-inf 2007-11-16
From David:
Find the limit. (Hint: treat the expression as a fraction whose denominator is 1, and rationalize the numerator.)

lim [x + squareroot(x^2 + 3)] as x->-inf

i got to

lim -3/(x - squareroot(x^2 + 3)) as x->-inf

but i'm having trouble understanding why the answer is 0 plz explain thx

Answered by Harley Weston.
Local maxima, minima and inflection points 2007-11-13
From Russell:
let f(x) = x^3 - 3a^2^ x +2a^4 with a parameter a > 1.

Find the coordinates of local minimum and local maximum

Find the coordinates of the inflection points

Answered by Harley Weston.
Maximize his profit 2007-11-12
From apoorva:
During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that he lost two sales per day.

a. Find the demand function, assuming it is linear.
b. If the material for each necklace costs Terry $6, what should the selling price be to maximize his profit?

Answered by Penny Nom.
Family of functions 2007-11-12
From Russell:
Consider the family of functions
f(t)= Asin3t + Acos3t +Bsin8t + Bcos8t

find exact values of parameters A and B so that f(0) = 2 and f ' (0) = 1

Answered by Stephen La Rocque.
Two integrals 2007-11-09
From Akilan:
how to integrate these (tan x)^6(sec x)^4 and sinh(x)(cosh(x))^2.

Please send me how to do this question. Having exams on Monday. Please help.

Answered by Harley Weston.
Increasing and decreasing for functions 2007-11-09
From David:
Direction: Identify the open intervals on which the function is increasing or decreasing.

f(x)=1/(x^2)
f'(x)= -2/(x^3)

i understand how to get up until there, and the undf. is x=0, but now i'm having problem setting up the number table chart. i cant remember how, and where to place the increase and decrease + - the chart, for example <---------0----------> where would the increase and the decrease be place?

Answered by Harley Weston.
f(x+y) = f(x) + f(y) + 2xy 2007-11-01
From Marcia:
For all real numbers x and y, let f be a function such that f(x+y) = f(x) + f(y) + 2xy and such that the limit as h -> 0 of f(h) / h = 7, find: f(0), use the definition of the derivative to find f'(x), and find f(x).
Answered by Penny Nom.
How to solve related rates problems 2007-10-27
From David:
Can you plz explain how and where you come up with an equation to solve this?
Find the rate of change of the distance between the origin and a moving point on the graph of y = sin x if dx/dt = 2 centimeters per second.

Answered by Stephen La Rocque.
lim x->1 (root x - x^2)/{1 - root x) 2007-10-16
From Meghan:
Hi! I have a question from my Calculus textbook that I've been picking at for a while and I'm stuck.

lim x->1 (root x - x^2)/{1 - root x).

Answered by Stephen La Rocque and Penny Nom.
The average rate of change of a function 2007-10-11
From vern:
Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. f(X)=sinX for the inverval [0,pi/6]?
Answered by Harley Weston.
The tangent to y = x^3 at x = 0 2007-09-04
From Amit:
consider the equation = x^3. The equation of tangent to this curve (which is smmetrical in Ist and IVth quadrant) at (0,0) is y=0, which is x-axis. but graphically one can visulize that x-axis intersects the curve, so how can it be the tangent to the curve. Please help.
Answered by Harley Weston.
f(x) = (x^4) - 4x^3 2007-07-22
From Michael:
I'm a student who needs your help. I hope you'll be able to answer my question. Here it is: Given the function f(x)=(x^4)-4x^3, determine the intervals over which the function is increasing, decreasing or constant. Find all zeros of f(x) and indicate any relative minimum and maximum values of the function.
Any help would be appreciated. Thank you for your time.

Answered by Harley Weston.
A normal to a curve 2007-07-16
From Samantha:
The function f is defined by f:x --> -0.5x^2 + 2x + 2.5
Let N be the normal to the curve at the point where the graph intercepts the y-axis. Show that the equation of N may be written as y = -0.5x + 2.5.
Let g:x--> -0.5x + 2.5
(i) find the solutions of f(x) = g(x)
(ii) hence find the coordinates of the other point of intersection of the normal and the curve

Answered by Penny Nom.
The isosceles triangle of largest area with perimeter 12cm 2007-07-16
From sharul:
find the dimension of isosceles triangle of largest area with perimeter 12cm
Answered by Harley Weston.
Implicit Derivatives 2007-07-13
From Charles:
I need help computing y' by implicit differentiation the question is: y^2 + x/y + 4x^2 - 3
Answered by Stephen La Rocque.
Derivative of a Function 2007-07-09
From Bob:
What is the derivative of the function a sub n = [n/(n+1)]^n ?
Answered by Stephen La Rocque.
Using calculus to prove the formula for the area of a triangle 2007-07-04
From Apratim:
Using calculus how can one show that the area of any triangle is 1/2 times its base times its height?
Answered by Stephen La Rocque.
Log base 2 of log base 2 of x 2007-06-27
From alex:
y = log base 2 of lag base 2 of x The slope of the tangent to the given curve at its x-intercept is..?
Answered by Harley Weston.
sin|x| and cos|x| 2007-06-25
From Mac:
Can anyone tell me whether sin|x| and cos|x| is differentiable at x=0 ? As far as i know, cos(x) and sin(x) is differentiable at all x.
Answered by Penny Nom and Stephen La Rocque.
Limits as x approaches a constant 2007-06-25
From Mac:
can you please tell me what is the reason they say "denominator is a negative quantity" in the solution 11 and "denominator is a positive quantity" solution 10 ??
If i guess correctly, for solution 10, its because of x^2 in the denominator.

Answered by Penny Nom.
Two tangent lines to y=x^3 2007-06-07
From stephanie:
find the equations of two tangent lines to the y=x^3 function through the point (2,8)
Answered by Penny Nom.
The limit of a rational function 2007-05-28
From Imad:
     3 _______ 3 _______
lim \/ 1 + x -    \/ 1 - x
x->0 ---- ----------------------------
              x

Answered by Penny Nom.
A circular blob of molasses 2007-05-28
From Julie:
A circular blob of molasses of uniform thickness has a volume of 1 m^3. The thickness of the molasses is decreasing at a rate of 0.1 cm/hour. At what rate is the radius of the molasses increasing when the radius is 8 m?

Thanks,
Julia

Answered by Penny Nom.
Optimization - carrying a pipe 2007-05-05
From A student:
A steel pipe is taken to a 9ft wide corridor. At the end of the corridor there is a 90° turn, to a 6ft wide corridor. How long is the longest pipe than can be turned in this corner?
Answered by Stephen La Rocque.
Maximize the volume of a cone 2007-04-27
From ashley:
hello, I've been stumped for hours on this problem and can't quite figure it out. The question is: A tepee is a cone-shaped shelter with no bottom. Suppose you have 200 square feet of canvas (shaped however you like) to make a tepee. Use calculus to find the height and radius of such a tepee that encloses the biggest volume. Can you help??
Answered by Stephen La Rocque and Penny Nom.
A cylinder inside a sphere 2007-04-25
From Louise:
i need to find the maximum volume of a cylinder that can fit inside a sphere of diamter 16cm
Answered by Penny Nom.
Liquid is being poured into the top of a funnel 2007-04-19
From neroshan:
Liquid is being poured into the top of a funnel at a steady rate of 200cm^3/s. The funnel is in the shape of an inverted right circular cone with a radius equal to its height. It has a small hole at the bottom where the liquid is flowing out at a rate of 20 cm^3/s. How fast is the height of the liquid changing when the liquid in the funnel is 15 cm deep? At the instant when the height of the liquid is 25cm, the funnel becomes clogged at the bottom and no more liquid flows out. How fast does the height of the liquid change just after this occurs?
Answered by Penny Nom.
Minimum cost for a fixed volume 2007-04-18
From James:
My question goes: A silo is to be constructed and surmounted by a hemisphere. The material of the hemisphere cost twice as much as the walls of the silo. Determine the dimensions to be used of cost is to be kept to a minimum and the volume is fixed.
Answered by Penny Nom.
The second derivative 2007-04-14
From Gerry:
In mathematical context,what do you understand by the term "Second Derivative"
Answered by Penny Nom.
What is the limit of 3.x^(3/x) as x approaches +infinity? 2007-04-11
From Teodora:
What is the limit of 3.x^3/x as x approaches +infinity ?
Answered by Haley Ess.
Find the volume of the solid 2007-04-07
From tricia:
a solid is constructed so that it has a circular base of radius r centimeters and every plane section perpendicular to a certain diameter of the base is a square, with a side of the square being a chord of the circle. find the volume of the solid at first i thought the length of a side of the square would be r, but that isn't awlays be true- only when the chord is in the center. so how can i solve this without any values? i dont understand the relationship between the chord and radius, except that the radius intercepts the chord at the midpoint. i know i hav to take the integral to get the volume, but how do i even find the area of one of the squares? please help, thanks, tricia
Answered by Penny Nom.
y = sin(2x) 2007-03-22
From bader:
sin(2x) find dx/dy
Answered by Penny Nom.
At what rate is the area of the triangle changing? 2007-02-24
From mac:
two sticks 3.5 feet long are hinged together and are stood up to form an isosceles triangle with the floor. The sticks slide apart, and at the moment when the triangle is equilateral, the angle is increasing at the rate of 1/3 radian/sec. At what rate is the area of the triangle increasing or decreasing at that moment?
Mac

Answered by Penny Nom.
Find the area of the triangle 2007-02-20
From Christina:
Graph the function f(X)= x+1/x-1 and graph the tangent line to the function at the points A:(2,3) and B:(-1,0). The point of intersection of the two tangent lines is C. Find the area of the triangle ABC.
Answered by Stephen La Rocque.
The volume of a frustum of a pyramid 2007-01-17
From Sam:
Find the volume of a frustum of a pyramid with square base of side b, square top of side a, and height h.
Answered by Penny Nom.
Integrate x^8 (x^8 + 2)^2 ((x^8 + 2)^3 + 1)^4 2007-01-09
From James:
How do you integrate x^8 (x^8 + 2)^2 ((x^8 + 2)^3 + 1)^4
Answered by Penny Nom.
What are the dimensions of the most economical container? 2007-01-04
From Ashely:
A cylindrical container costs $2.00 per square foot for the sides and $3.00 a square foot for the top and bottom. The container must hold 100 cubic feet of material. What are the dimensions of the most economical container.
Answered by Stephen La Rocque.
Rolle's Theorem 2006-12-07
From Erika:
If f(x) = (x^2)(square root of [3-x]) on the interval [0,3] is given, Does Rolle's Theorem apply? If yes, find any values of c such that f '(c)=0
Answered by Penny Nom.
A Norman window 2006-11-30
From Joe:
a norman window is a rectangle with a semicircle on top. If a norman window has a perimeter of 28, what must the dimensions be to find the maximum possible area the window can have?
Answered by Stephen La Rocque.
Tangent lines 2006-11-09
From Melissa:
let f be a function with f(1)=4 such that for all points (x,y) on the graph of f the slope is given by (3x^(2)+1)/(2y)

a.)Find the slope of the graph of f at the point where x=1.
b.)Write an equation for the line tangent to the graph of f at x=1 and use it to approximate f(1.2)
c.) Find whether f is concave up or concave down when x=1. Is your answer in part b an overestimate or an underestimate?

Answered by Stephen La Rocque.
Water is being pumped into the pool 2006-10-24
From Jon:
A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deeps at the deep end. Water is being pumped into the pool at 1/4 cubic meters per minute, an there is 1 meter of water at the deep end.

a) what percent of the pool is filled?

b) at what rate is the water level rising?

Answered by Stephen La Rocque.
An approximation 2006-10-22
From Ellen:
consider the curve -8x^2 +5xy+y^3 +149 =0 Write an equation for the line tangent to the curve at (4, -1) use this equation to approximate the value of K at the point (4.2, K)
Answered by Penny Nom.
How fast is the water level rising when the water is 1 meter deep? 2006-10-19
From Don:
The cross section of a 5-meter trough is an isosceles trapezoid with a 2-meter lower base, a 3-meter upper base and an altitude of 2 meters. Water is running into the trough at a rate of 1 cubic meter per minute. How fast is the water level rising when the water is 1 meter deep?
Answered by Stephen La Rocque.
The velocity of a pendulum, part II 2006-09-07
From Erin:
We saw the question in your database about the velocity of a pendulum swinging.....It is the same exact question....but there is another question......it says....

"estimate the instantaneous rate of change of d with respect to t when t = 1.5. At this time, is the pendulum moving toward or away from the wall? Explain."

Answered by Harley Weston.
Differentiate Y= sin3x + cos7x 2006-08-22
From james:
Differentiate the function of x using the basic rules.

Y= sin3x + cos7x

Answered by Stephen La Rocque.
How fast is the water level rising 2006-08-12
From Erin:
Water runs into a conical tank at the rate of 9ft3/min. The tank stands point down and has a height of 10 ft. and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft. deep? (V=1/3 pi r2 h).
Answered by Penny Nom.
An Integral 2006-07-30
From Aniket:

I am Aniket studing in 12 th standard At Mumbai
I have following integration problem please give me a solution

integral of 1/under root of (5x2 - 2x) dx


Answered by Penny Nom.
Minimizing a cost 2006-07-25
From Edward:
The cost of running a car at an average speed of V km/h is given by c= 100 + (V2 / 75) cents per hour. Find the average speed (to the nearest km/h) at which the cost of a 1000 km trip is a minimum.
Answered by Stephen La Rocque.
differentiate the volume of a cylinder with V respect to h 2006-05-24
From A student:
differentiate the volume of a cylinder with V respect to h
Answered by Stephen La Rocque.
integral of tan^4 x 2006-05-14
From Aqil:
integral of tan4 x
Answered by Penny Nom.
Rate of ladder falling 2006-04-30
From Harsh:
A ladder 4 m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 30 cm/s, how quickly is the top of the ladder sliding down the wall when the bottom of the ladder is 2 m from the wall?
Answered by Stephen La Rocque.
Find the point of inflexion for the curve y = e^x/(x^2-1) 2006-03-31
From Sam:
Hi, i am trying to find the point of inflexion for the curve y = ex/(x2-1) and i got a really complex expression for y". I can't seem to solve x4-4x3+4x2+4x+3=0 so does that mean there is no point of inflexion?
Answered by Penny Nom.
A fence around a pen 2006-03-30
From Daryl:
I hope you can help me out with the attached problem, It has been driving me crazy.
Answered by Stephen La Rocque and Penny Nom.
The box of maximum volume 2006-02-01
From Elizabeth:
A box factory has a large stack of unused rectangular cardboard sheets with the dimensions of 26 cm length and 20 cm width.
The question was to figure what size squares to remove from each corner to create the box with the largest volume.
I began by using a piece of graph paper and taking squares out. I knew that the formula L X W X H would give me volume. After trial and error of trying different sizes I found that a 4cm X 4cm square was the largest amount you can take out to get the largest volume. My question for you is two parts

First: Why does L X H X W work? And second, is their a formula that one could use, knowing the length and width of a piece of any material to find out what the largest possible volume it can hold is without just trying a bunch of different numbers until you get it. If there is, can you explain how and why it works.

Answered by Penny Nom.
Differentiation, powers and logs 2006-01-06
From Claudia:

Question: how do I find the derivative of

x* ln(x+(e^2))^2

x^lnx

x^(e^(-x^2))


Answered by Penny Nom.
Two related rates problems 2005-12-29
From Shimaera:

#1. A manufacturer determines that the cost of producing x of an item is C(x)=0.015x2+12x+1000 and the price function is p(x)=250+2x/10. Find the actual and marginal profits when 500 items are produced.

#2. At 9 a.m a car is 10km directly east of Marytown and is traveling north at 100 km/h. At the same time, a truck leaves Marytown traveling east at 70 km/h. At 10 a.m, how is the distance between the car and the truck changing?


Answered by Penny Nom.
The Mean Value Theorem 2005-12-22
From Candace:
Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Find all numbers "c" that satisfy the Mean Value Theorem. 11. f(x)=3x2 + 2x +5 [-1, 1]
Answered by Penny Nom.
A max-min problem 2005-12-16
From Julie:
A car travels west at 24 km/h. at the instant it passes a tree, a horse and buggy heading north at 7 km/h is 25 km south of the tree. Calculate the positions of the vessels when there is a minimum distance between them.
Answered by Penny Nom.
Mrs. Faria lives on an island 2005-12-15
From Julie:
Mrs. Faria lives on an island 1 km from the mainland. She paddles her canoe at 3 km/h and jogs at 5 km/h. the nearest drug store is 3 km along the shore from the point on the shore closest to the island. Where should she land to reach the drug store in minimum time?
Answered by Penny Nom.
Notation for the second derivative 2005-11-08
From Mussawar:
my question is d/dx( dy/dx) = d2y/dx2. why it is not equal to d2y/d2x.
Answered by Penny Nom.
Velocity and acceleration 2005-10-27
From Candace:
When taking the integral of the position function, you get the velocity function, and the same for velocity to acceleration. So when you do each of these, you get a function. But when you integrate on a graph, you get an area under a curve. The area is un units squared- where do the units go when you make it an equation? How can a function be an area?
Answered by Harley Weston.
Can we take the derivative of independent variable 2005-10-18
From Mussawar:
why we take derivative of dependent variable with respect to independent variable .can we take the derivative of independent with respect to dependent.if not why.
Answered by Walter Whiteley.
U'(X) - U(X) = 0; U(0) = 2 2005-09-23
From David:
Out of interest could you please answer the following questions?

U'(X) - U(X) = 0; U(0) = 2

and

U''(X) - U'(X) = 0; U'(0) = U(0) = 2


Answered by Harley Weston.
A point is moving on the graph of x^3 + y^2 = 1 in such a way that 2005-09-17
From Gina:
A point is moving on the graph of x3 + y2 = 1 in such a way that its y coordinate is always increasing at a rate of 2 units per second. At which point(s) is the x coordinate increasing at a rate of 1 unit per second.
Answered by Penny Nom.
How do you differentiate y=(x)^(x^x)? 2005-09-14
From Calebius:
How do you differentiate y=(x)(xx)?
Answered by Penny Nom.
At what rate is the circumference of the circle increasing? 2005-08-08
From John:

A mathematics professor is knitting a sweater. The main part of the sweater is knit in a large spiral, ending up with a diameter of 30 inches. She knits at a constant rate of 6/7 square inches per minute.

1. At what rate is the circumference of the circle increasing when the diameter is 2 inches?

2. How long will it take her to finish this piece of the sweater?


Answered by Penny Nom.
A lighthouse is located on a small island,... 2005-07-14
From Brittnee:
A lighthouse is located on a small island, 3 km away from the nearest point P on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?
Answered by Penny Nom.
The volume of a hopper 2005-05-28
From Brian:
I would like to know the volume of this rectangular hopper. can you help
Answered by Penny Nom.
Logarithmic differentiation 2005-05-23
From Richard:
I need to convince myself that I understand the process of differentiating y=xx.
The specific question is that if I have to take the logarithm of both sides of the equation how can differentiate the following?
y= {(x+2)(x+2)}/{(x+1)(x+1)} - {(x+1)(x+1)}/(xx),
I have an idea that the differential of this fairly complex function is itself ... am I right or wrong.

Answered by Penny Nom.
L'hopital's rule 2005-05-15
From Abraham:
Find the limit of [(1/(x+4))-(1/4)]/x as x approaches zero. How do you use l"hopital's rule to find this limit. I know how to do it with multiplying everything by 4(x+4), and getting the answer, -1/16.But how do you apply derivatives with l'hopitals rule to this type of problem?
Answered by Penny.
A Taylor series for ln(x) 2005-04-16
From Anood:
i have to represent ln(x) as a power series about 2

i`m not getting the final answer which is ln 2+ sigma (((-1)(n+1)/ (n*2n))*(x-2)n). i don`t get the ln 2 part

i show you my trial

f(x)= ln x.

f-(x)=(1/x) .

f--(x)= (-1/x2)*1/2!

f---(x)= (2/x3)*1/3!

f----(x)= (-6/x4)* 1/4!

so the pattern shows me that f(n)= ((-1)(n+1))/xn *n)

so f(2)= sigma ((-1)(n+1))/2n *n) *(x-2)n

so as you see i don`t get ln 2


Answered by Penny Nom.
Differentiating F(x,y) = 0 2005-01-23
From Jacob:
In calculus, we often mention to the students that if F(x,y) = 0, then we can differentiate both sides and still get an equality. The problem is that we can't perform the same operation on F(x) = 0, say x = 0, otherwise 1 = 0, which is absurd. What is the reason?
Answered by Walyer Whileley and Harley Weston.
Three calculus problems 2004-12-09
From Ashley:
Hi, I am having a lot of trouble with three calculus questions and was wondering if you could help
Answered by Penny Nom.
Implicit differentiation 2004-10-24
From Emily:
If x^3+3xy+2y^3=17, then in terms of x and y, dy/dx =
Answered by Penny Nom.
The integrating factor method 2004-08-05
From A student:
Whilst using the integrating factor method, I am required to integrate a function multipled by another function.

say f(t) = exp(kt) and some other function g(t); where exp = exponential and k is some constant.

Integral f(t)*g(t) dt or Integral exp(kt)*g(t) dt

What would the result of this integral be? I have never met an integral like this before. Would it simply be exp(kt)*g(t)/k? More specifically, the problem and my attempted answer is in PDF format:

In my attempted solution, I am unsure about the last two lines I have written out, as it relates to integrating a function multipled by another function.

Answered by Harley Weston.
Integrating e^sin(x) 2004-08-04
From A student:
I need to know that how to solve the integral " e^sin x",
Answered by Harley Weston.
Differentiation 2004-08-04
From A parent:
I am a parent trying to understand higher level of maths and would be very grateful if you could help with differentiating the following functions, identifying general rules of calculus:

a) f(x)=e^2^xIn(cos(8x))

b) f(x)=secx/SQRTx^4+1

Answered by Penny Nom.
Maximizing the angle to the goal mouth 2004-05-15
From Yogendra:
You are running down the boundary line dribbling the ball in soccer or hockey. Investigate where in your run the angle the goal mouth makes with your position is at a maximum.
Answered by Penny Nom.
Related rates and baseball 2004-04-26
From Bethany:
A baseball diamond is the shape of a square with sides 90 feet long. A player running from second to third base at a speed of 28 feet/ second is 30 feet from second base. At what rate is the player's distance from home plate changing?
Answered by Penny Nom.
A changing rectangle 2004-04-03
From A student:
The width x of a rectangle is decreasing at 3 cm/s, and its length y is increasing at 5 cm/s. At what rate is its area A changing when x=10 and y=15?
Answered by Penny Nom.
Some calculus problems 2004-04-01
From Weisu:

I have questions about three word problems and one
regular problem, all dealing with derivatives.

  1. Find all points on xy=exy where the tangent line
    is horizontal.
  2. The width x of a rectangle is decreasing at 3 cm/s,
    and its length y is increasing at 5 cm/s. At what rate
    is its area A changing when x=10 and y=15?
  3. A car and a truck leave the same intersection, the
    truck heading north at 60 mph and the car heading west
    at 55 mph. At what rate is the distance between the
    car and the truck changing when the car and the truck
    are 30 miles and 40 miles from the intersection,
    respectively?
  4. The production P of a company satisfies the
    equation P=x2 + 0.1xy + y2, where x and y are
    the inputs. At a certain period x=10 units and y=8
    units. Estimate the change in y that should be made to
    set up a decrease of 0.5 in the input x so that the
    production remains the same.

If you could just give me some hints on these
questions, I'd really appreciate it. Thanks!


Answered by Penny Nom.
A partial derivative 2004-03-19
From Penny Nom:
Is it possible to differentiate the following equation, if so could you please explain.

S=SQRT(T-(5/X^2))

I would like the derivative of S with respect to X.

Answered by Harley Weston.
The derivative of x to the x 2004-02-14
From Cher:
what about the derivative of x to the power x?
Answered by Penny Nom.
A pyramid-shaped tank 2004-02-13
From Annette:
The base of a pyramid-shaped tank is a square with sides of length 9 feet, and the vertex of the pyramid is 12 feet above the base. The tank is filled to a depth of 4 feet, and water is flowing into the tank at a rate of 3 cubic feet per second. Find the rate of change of the depth of water in the tank. (Hint: the volume of a pyramid is V = 1/3 B h , where B is the base area and h is the height of the pyramid.)
Answered by Harley Weston.
Some trig problems 2004-01-18
From Weisu:

I have some questions about pre-calculus.

(1) (2(cos(x))^2)+3sin(x)-1=0
(2) sin(x)cos(x)=(1/2)
(3) 3sin(x)=1+cos(2x)
(4) tan(x)*csc(x)=csc(x)+1
(5) sin(arccsc(8/5))
(6) tan(arcsin(24/25))
(7) arccos(cos(11pi/6))
the last problem uses radian measure.


Answered by Penny Nom.
A riddle 2003-11-19
From Sarah:
Ok, our teacher gave us this riddle, and I cannot for the life of me figure it out. He said that there are three problems with the following proof:
Answered by Penny Nom.
The sketch of a graph 2003-10-07
From A student:
I was wondering how do you figure out if a graph has a horizontal tangent line. One of my homework problem was to sketch the graph of the following function; (4/3)x3-2x2+x. I set f''(x) ( the second derivative) of the function equal to zero and got the inflection point:(1/2,1/6). Also i am having trouble finding the concavity for x>1/2 and x<1/2, i am getting a different answer from the back of the book, the graph i draw looks completely different from the correct answer.
Answered by Penny Nom.
Indeterminate forms 2003-10-06
From A teacher:
Is it possible for me to find any geometrical interpretation without using calculus to explain indeterminate forms?
Answered by Chris Fisher.
Functions, graphs and derivatives 2003-10-05
From Jathiyah:
I wanted to know how would you tell (on a graph diplaying two funtions), which funtion is the derivative of the other?
Answered by Walter Whiteley.
The slope of a tangent 2003-10-01
From A student:

find the slope of the tangent to each curve at the given point

f(x)=square root 16-x, where y=5


Answered by Penny Nom.
A helicopter rises vertically 2003-09-02
From Kate:
A helicopter rises vertically and t seconds after leaving hte ground its velocity is given in feet per second by v(t) = 8t + 40 / (t+2)2 How far above the ground will the helicopter be after 3 seconds?
Answered by Penny Nom.
Two precalculus problems 2003-08-04
From Kate:

Please help me verify the identity:
cos2x(sec2x-1)=sin2x

Also I am having trouble withdetermining whether f(x) is odd, even, or neither
f(x)=x3-x


Answered by Penny Nom.
Natural logarithms 2003-07-22
From Amanda:
I'm going into my senior year of high school. I will be taking AP calculus, and my teacher gave us some homework over the summer. However, there are two things that I do not understand how to do. The first is, she wants us to be able to generate a unit circle by hand using 30, 60 and 90 degree triangles. I have used the unit circle in trigonometry, however I was never taught how to draw it. Secondly, I need to know how to do natural logarithms without a calculator. I was not taught how to do this, and the worksheet I was given only showed me how to complete them using a calculator.
Answered by Claude Tardif.
Odd powers of sine and cosine 2003-06-25
From Antonio:
Can you please tell me how to integrate a trig function involving sine and cosine? I know if the powers of both the sine and cosine are even and nonnegative, then I can make repeated use of the power-reducing formulas. But for the question I have on my hand, the powers of both sine and cosine are odd: ( sin3x + cos7x ) dx.
Answered by Harley Weston.
Integrating e^x sin(x) 2003-05-03
From Lech:
I am having trouble integrating the following expression by parts:

ex sin(x)

I used the integrator at http://www.integrals.com/ to find the solution,

? 1/2 ex cos(x) + 1/2 ex sin(x).

This is easy to confirm by differentiation, however I am confounded as how to arrive at the answer.


Answered by Penny Nom and Claude Tardif.
The volume of air flowing in windpipes 2003-05-02
From James:
The volume of air flowing in windpipes is given by V=kpR4, where k is a constant, p is the pressure difference at each end, R is the radius. The radius will decrease with increased pressure, according to the formula: Ro - R = cp, where Ro is the windpipe radius when p=0 & c is a positive constant. R is restricted such that:
0 < 0.5*Ro < R < Ro,
find the factor by which the radius of the windpipe contracts to give maximum flow?

Answered by Penny Nom.
Integration of 1/(2+cos(x)) 2003-01-07
From A student:
integral from pi to 0 of

dx/(2+cos x)

i used the substitution t=tan(x/2) and i ended up with

integral from +infinity to 0 of

2dt/(t2+3)

which looks like an inverse tan function , and i ended up with sqr(27)/2 pi , which is not the same as my calculator's answer , so i suspct i am doing some thing wrong. can some one tell me where i am going wrong please.
Answered by Penny Nom.

Differentiating inverses 2002-11-20
From Amy:
f(x)= x3+x+1, a=1 find g'(a) (g = f -1). I am having trouble finding g(a).
Answered by Penny Nom.
A max/min problem 2002-09-21
From Evelina:
A window is the shape of a rectangle with an equilateral triangle on top. The perimeter of the window is 300 cm. Find the width that will let the maximum light to enter.
Answered by Penny Nom.
Integrating x^x 2002-06-18
From Jeremy:
I am a student at the University of Kansas and I am wondering if there is a general anti-derivative for x x (i.e. the integral of x x dx)? I've looked in a bunch of Table of Integrals and have found nothing (can you guys find it?), so I'm sort of wondering if this may be a research type question.
Answered by Claude Tardif.
A good rule of thumb when driving 2002-06-13
From Lisa:
A good rule of thumb when driving is that you should be about one car length away from the car in front of you for every 10 miles per hour that you are travelling. Suppose you follow this rule perfectly (so you are exactly the correct distance away). You are waiting at a stop light with your front bumper just touching the car in front of you. The light turns green and the car in front accelerates at a constant rate "r". Calculate how you should accelerate in order to follow the rule.
Answered by Penny Nom.
A spotlight shines on a wall 2002-05-25
From Barb:
A spotlight on the ground shines on a wall 12m away. If a man 2m tall walks from the spotlight toward the bldg at a speed of 1.6 m/s, how fast is his shadow on the bldg decreasing when he is 4m from the bldg?
Answered by Penny Nom.
What is Calculus About? 2002-05-13
From A student:
I am a high school senior and have to write an essay answering the question "What is Calculus?" I need some ideas. Thanks
Answered by Walter Whiteley.
A rectangular marquee 2002-05-07
From Alyaa:
a marquee with rectangular sides on a square base with a flat roof is to be constructed from 250 meters square of canvas. find the maximum volume of the marquee. i find this topic so hard
Answered by Harley Weston.
How will I use calculus in my career? 2002-05-06
From Meridith:
How will I, hopefully a future secondary mathematics teacher, use calculus in my career if I'm not teaching calculus?
Answered by Walter Whiteley.
Arc length 2002-04-17
From Vix:
Find the point on the curve r(t)=(12sint)i-(12cost)j+5tk at a distance 13pi units along the curve from the point (0,-12,0) when t=0 in the direction opposite to the direction of increasing arc length.
Answered by Harley Weston.
Related rates 2002-04-17
From Molly:
A tanker spilled 30 ft cubed of chemicals into a river, causing a circular slick whose area is expanding while its thickness is decreasing. If the radius of the slick expands at the rate of 1 foot per hour, how fast is them thickness of the slick decreasing when the area is 100 feet squared?
Answered by Penny Nom.
The slope of a tangent line 2002-03-04
From Ridley:
Suppose a function f(x) has the line 3x+4y=2 as its tangent line at x=5. Find f'(5).
Answered by Harley Weston.
Getting to B in the shortest time 2001-12-19
From Nancy:
A motorist in a desert 5 mi. from point A, which is the nearest point on a long, straight road, wishes to get to point B on the road. If the car can travel 15 mi/hr on the desert and 39 mi/hr on the road to get to B, in the shortest possible time if......

A.) B is 5 mi. from A

B.) B is 10 mi. from A

C.) B is 1 mi. from A


Answered by Penny Nom.
A lighthouse and related rates 2001-11-29
From Melissa:
A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline, and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?
Answered by Penny Nom.
A tangent line 2001-11-21
From A student:
write an equation of the line tangent to the graph of

ey + ln(xy) = 1 + e at (e,1)

Answered by Harley Weston.
Asymptotes 2001-11-09
From Frank:

given the function:

f(x) = (x2) / (x-1)

the correct answer to the limit of f(x) as x approaches infinity is:

y = x+1

all math references point to this answer and the method they all use is long division of x-1 into x2

however if one were to multiply both the numerator and denominator by 1/x and then take the limit, one gets:

y=x

how can the descrepency between the two answers be explained?


Answered by Chris Fisher and Penny Nom.
A lighthouse problem 2001-11-02
From A student:
A lighthouse at apoint P is 3 miles offshore from the nearest point O of a straight beach. A store is located 5 miles down the beach from O. The lighthouse keeper can row at 4 mph and walk at 3.25 mph.

a)How far doen the beach from O should the lighthouse keeper land in order to minimize the time from the lighthouse to the store?

b)What is the minimum rowing speed the makes it faster to row all the way?

Answered by Harley Weston.
Concavity of f(g) 2001-10-25
From Troy:
Suppose f & g are both concave upward on (-infinity,infinity). Under what condition on f will the composite function h(x)= f(g(x)) be concave upward?
Answered by Walter Whiteley.
Maximize the area 2001-10-13
From Mike:

I have no clue how to do this problem. Here is what the professor gave to us:

A=LW

C=E(2L+2W) + I(PL)

Where P = # of partitions

E= cost of exterior of fence

I = cost of interior of fence

C = total cost of fence

.
.
.

Answered by Harley Weston.
4 sinx cosy = 1 2001-10-10
From A student:
How would i differentiate the following example in terms of t (x and y are functions of t)

4 sinx cosy = 1


Answered by Claude Tardif.
The height of the lamppost 2001-10-02
From Werner:
I am working on question 51,section 3.7 ,page 191 of Stewart's Single Variable Calculus. The question involves a lamp post which is casting a shadow around the eliipse whose formula is x2 + 4*y2 = 5. I have found the derivative of the elllipse both explicitly: x/4(((5-x2)/4)0.5) and implicitly : y' = - x/(4*y).
Answered by Harley Weston.
(x^2-5x-6)/(x-6) 2001-10-02
From Bill:
given f(x) = (x2-5x-6)/(x-6) find f'(6).
Answered by Harley Weston.
The Mean Value Theorem 2001-07-23
From Corrie:
I need to find if the mean value theorem exists. and if so, find all values c guaranteed by the theorem.

f(x) = |x2-25| on the interval [-10,0]


Answered by Harley Weston.
Area between curves 2001-06-13
From Phil:

question 1

find the area bound by the curves y = x2 + 2x + 3 and y = 2x + 4

question 2

Find the volume generated by rotating the curve x2 + y2 = 9 about the x-axis

Answered by Harley Weston.
National consumption function 2001-05-09
From Brian:
If consumption is $11 billion when disposable income is 0 and the marginal propensity to consume is dC/dy = 1/(2y+4)1/2+0.3(in billions of dollars), find the national consumption function.
Answered by Harley Weston.
Maximize profit 2001-05-09
From Brian:
The marginal cost for a certain product is given by MC = 6x+60 and the fixed costs are $100. The marginal revenue is given by MR = 180-2x. Find the level of production that will maximize profit and find the profit or loss at that level.
Answered by Harley Weston.
The average value of a continuous function 2001-05-08
From Esther:
The average value of a continuous function y = f(x) on the interval [a,b] is given by ________________?
Answered by Harley Weston.
A Taylor series 2001-04-27
From Karan:
Given the following information of the function
  1. f''(x) = 2f(x) for every value of x

  2. f(0) = 1

  3. f(0) = 0
what is the complete Taylor series for f(x) at a = 0

Answered by Harley Weston.
Oil revenue 2001-04-21
From Brian:
Suppose that t months from now an oil well will be producing crude oil at the rate of r(t), not a constant, barrels per month and that the price of crude oil will be p(t), not a constant, dollars per barrel. Assume that the oil is sold as soon as it is extracted from the ground.
  1. Find an expression for the total revenue from the oil well, R(t).

  2. A certain oil well that currently yields 400 barrels of crude oil a month will run dry in 2 years. The price of crude oil is currently $18 per barrel and is expected to rise at a constant rate of 3 cents per barrel per month. What will be the total revenue from this well? {Hint: Model the degraded production rate with the equation:
    r(t) = (A-Bt)e0.04t}

Answered by Harley Weston.
Differentiation 2001-04-17
From Esther:
Could you please tell me what the first derivative is of the following:

y = 2/(2x+e2x)

Is it (1+xe2x)/(2x+e2x)2 or perhaps -4(1+e2x)/(2x+e2x)2 ? I am a little confused between the two!


Answered by Harley Weston.
Integration by parts 2001-04-09
From A student:
how do you integrate x tan-1x dx, i know it can be done by integration by parts maybe, but i'm not sure....
Answered by Claude Tardif and Harley Weston.
The domain of a function 2001-04-08
From Mina:
Let f(x) = (2x2+3x-17)/(x+5)
What is the domain of f? What are the values of x for which f'(x) does not = 0?

Answered by Harley Weston.
The normal to a curve 2001-04-08
From Varenne:
I am having SO much trouble tackling this question and don't know what the right answer is... can you help me out? The question is

Find the equation of the normal to the curve y=(x-2)2/(1-x)2 that is parallel to the line x+4y+7=0

Answered by Harley Weston.
Common tangents 2001-04-08
From Anne:
I have been working on this problem for a while but I'm not sure I'm getting the right answer: Find the common tangents of 2y=x2 and 2y=-x2-16 Thanks for the help. :)
Answered by Harley Weston`.
Airflow in windpipes 2001-03-25
From Ena:
The volume of air flowing in windpipes is given by V=kpR4, where k is a constant, p is the pressure difference at each end, R is the radius. The radius will decrease with increased pressure, according to the formula: Ro - R = cp, where Ro is the windpipe radius when p=0 & c is a positive constant. R is restricted such that:
0 < 0.5*Ro < R < Ro,
find the factor by which the radius of the windpipe contracts to give maximum flow?

Answered by Harley Weston.
A jogger 2001-03-12
From Bill:
At time t=0 a jogger is running at a velocity of 300 meters per minute. The jogger is slowing down with a negative acceleration that is directly propotional to time t. This brings the jogger to a stop in 10 minutes.

a) write an expression for the velocity of the jogger at time t.

b) what is the total distance traveled by the jogger in that 10-minute interval.


Answered by Harley Weston.
The domain of the derivative 2001-02-22
From Wayne:
I know that the domain of f'(x) is a subset of the domain of f(x). Is it necessarily true that the subset will have at most one less element than the domain of the original function?
Answered by Harley Wesston.
Differentiation of y = x n 2001-02-17
From Jashan:
i am studying differentation at the moment i have drawn some graphs such as y=x 2. i have found the formula for the gradient of this curve, this being 2x obtained by using differentation, but i need to know the general case for the formula where y=xn in order for me to understand this topic more throughly, i would also like to know how u derived this general formula
Answered by Harley Weston.
A quartic equation 2001-02-15
From George:
Let P(x) = x4 + ax3 + bx2 + cx + d. The graph of y = P(x) is symmetric with respect to the y-axis, has a relative max. at (0,1) and has an absolute min. at (q, -3)

a) determine the values for a, b c, and d using these values, write an equation for P(x)
b) find all possible values for q.


Answered by Harley Weston.
Find an exprression for f(x) 2001-02-07
From A 12th grade AP Calc student:
Let f be the function defined for all x > -5 and having the following properties. Find an expression for f(x).

i) f ''(x) = 1/ (x+5)1/3 for all x in the domain of f

ii) the line tangent to the graph of f at (4,2) has an angle of inclination of 45 degress.


Answered by Harley Weston.
Height of the lamp 2000-12-31
From Joey:
The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2 + 4y2 < 5. If the point (-5,0) is on the edge of the shadow, how far above the x-axis is the lamp located?
Answered by Harley Weston.
How do you integrate secant(theta)? 2000-12-22
From Robert Williamson:
How do you integrate secant(theta)?

I know the answer is ln [sec(theta) + tan(theta)] but how do you get there?


Answered by Claude tardif.
A limit using l'hopital's rule 2000-12-13
From Wassim:
I need to know how to solve the:
limits of (x ( to the power lamda) -1 )/LAMDA when lamda tends to zero ( the answer is that the functional form is ln x) and I still don't know how using hopital rule leads to this answer.

Answered by Harley Weston.
A non-integrable function 2000-12-03
From Mark Spilker:
I have a proof that I cannot do here it goes.
Let F(x)= 1 if x is a rational number 0 if x is an irrational number Prove the function is not intregrable on the interval (0,1). Hint: Show that no matter how small the norm of the partition, the Riemann Sum for the SigmaNotation F(wi) DeltaiX is not unique.

Answered by Harley Weston.
Comparing an integral and a sum 2000-11-21
From Douglas Norberg:
A fellow teacher asked me about a problem she wanted to give to her students. It involved whether to take a million dollars or a penny doubled a number of times. I was able to determine the number must have been .01 * 230 which is about $10 million and a lot more than $1 million. To check that I was right I used a spreadsheet and did a Riemann sum.

When I finished I reasoned that I had done the task in several steps and I could have done it in 1 step. Thus I integrated .01 * 2x from 0 through 30 but the number I got was $15,490,820.0324. Why the difference?


Answered by Harley Weston.
Concavity 2000-10-22
From Alex:
the question is: on what interval is f(x)=(x2)(ex)?

ive found the 2nd derivative which is ex(x2+4x+2) and i did the quadratic to get -2-20.5 and -2+20.5, but i dont know what the interval is.
Answered by Harley Weston.

Dividing a region in half 2000-09-21
From Kerry:
There is a line through the origin that divides the region bounded by the parabola y=x-x2 and the x-axis into two regions with equal area. What is the slope of the line?
Answered by Penny Nom.
A cycloid in Cartesian form 2000-09-20
From Billy:
The parametric equation of cycloid is given:
x=r(t-sint)
y=r(1-cost)

How to eliminate t?


Answered by Harley Weston.
A proof that 1=2 2000-09-19
From sporky:
Why does the proof for 1=2 not work?

x = 1
x2 = 1
x = x2
1 = 2x (derivitive)
1 = 2(1)
1 = 2 ???

please tell me where the false logic is.


Answered by Walter Whiteley.
Derivatives, there must be an easier way 2000-09-06
From Brad Goorman:
The direction read: Take the derivative of each expression.

y = {1+[x+(x2 +x3)4]5}6


Answered by Harley Weston.
Velocity of a pendulum 2000-08-28
From Mekca:
A pendulum hangs from the ceiling. as the pendulum swings, its distance,d cm, form one wall of the room depends on the number of seconds,t, since it was set in motion. assume that the equation for d as a function of t is: d=80+30cos3.14/3t, t>0.

estimate the instantaneous rate of change of d at t=5 by finding the average rates for t=5 to 5.1, t=5 to 5.01, and t=5 to 5.001.


Answered by Harley Weston.
Some trigonometry 2000-08-11
From Angela:
I have some PreCal questions. I am a student at the secondary level. I would be very grateful for your help.

Solve the equation for theta (0 <= theta < 2pi).

tan2(theta) = 3 I know sec2(theta) -1 = tan2(theta)

.
.
.
Answered by Harley Weston.

L'Hospital's Rule 2000-07-19
From Dan Krymkowski:
The limit of the following as x goes to infinity is 2*y. Y is a constant.

lim 2*x*log(x/(x-y)) = 2*y


Answered by Harley Weston.
Divisors of 2000 2000-06-06
From Amanda Semi :
  1. find the product of all the divisors of 2000
  2. dog trainer time has 100m of fencing to enclose a rectangular exercise yard. One side of the yard can include all or part of one side of his building. iff the side of his building is 30 m, determine the maximum area he can enclose

Answered by Claude Tardif.
A derivative problem 2000-06-04
From Jeff Ellis:
If F(x)=(4+x)(3+2x2)2(2+3x3)3, find F'(0)
Answered by Harley Weston.
Calculus Research Questions 2000-05-22
From William Wright:
I am a Calculus Teacher, and me and my class ran into these two problems without solutions in my manual, we got answers, but are unable to check them. If anyone gets this email and can respond to this with the solutions it be greatly appreciated.
.
.
.

Answered by Harley Weston.
Radioactive decay 2000-05-18
From Catherine Sullivan:
Please help me with the following: The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to carbon-12 at a rate proportional to the amount of C-14 present, with a half life of 5730 years. Suppose C(t) is the amount of C-14 at time t.
  1. Find the value of the constant k in the differential equation: C'=-kC
  2. In 1988 3 teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of C-14 contained in freshly made cloth of the same material. How old is the Shroud according to the data?

Answered by Harley Weston.
Related Rates 2000-05-07
From Derek:
How can you show that if the volume of a balloon is decreasing at a rate proportional to its surface area, the radius of the balloon is shrinking at a constant rate.
Answered by Harley Weston.
An improper integral 2000-05-04
From A high school senior:
Hi, I am a high school senior and I need help stugying for a final. I am stuck on one of the questions on my review sheet. Does the improper integral from 5 to infinity of (38/97)x converge or diverge? If it converges I also need to know how to find the approximate value accurate to .01 of its actual value.
Answered by Harley Weston.
Thearcius Functionius 2000-05-03
From Kevin Palmer:
With the Olympics fast approaching the networks are focusing in ona new and exciting runner from Greece. Thearcius Functionius has astounded the world with his speed. He has already established new world records in the 100 meter dash and looks to improve on those times at the 2000 Summer Olympics.

Thearcius Functionius stands a full 2 meters tall and the networks plan on placing a camera on the ground at some location after the finish line(in his lane) to film the history making run. The camera is set to film him from his knees(0.5 meters up from the ground) to 0.5 meters above his head at the instant he finishes the race. This is a total distance of two meters(the distance shown by the camera's lens).
Answered by Harley Weston.

An indefinite integral 2000-05-03
From Bonnie Null:
I am to find the indefinite integral of: (ex - e-x)2 dx
Answered by Claude Tardif.
Minimizing the metal in a can 2000-05-02
From May Thin Zar Han:
A can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can.
Answered by Harley Weston.
Two calculus problems 2000-05-01
From Kaushal Shah:
How Do WE Integrate the following Functions,
  1. Integral xtanx dx
  2. How was natural base "e" discovered and why e=2.7.......

Answered by Claude Tardif.
The area of a triangle using calculus 2000-04-15
From Todd Bowie :
Hi, I am not a student but am reviewing calculus for an upcoming interview. I would like to know how to derive the area of a triangle using calculus. Thanks!
Answered by Patrick Maidorn.
y = x^x^x^x... 2000-04-05
From Michael Hackman:
Find the derivative of: y = x^x^x^x... on to infinity.
Answered by Claude Tardif.
Riemann sums 2000-03-30
From Joshua D. Parham:
If n is a positive integer, then
 
     lim   (1/n)[1/(1+1/n) + 1/(1+(2/n) + ... + 1/(1+n/n)] 
n->infinity 

can be expressed as the integral from 1 to 2 of 1/x dx
Answered by Penny Nom.

Functions that satisfy f' = f 2000-03-16
From Kevin Palmer :
Recently my calculus teacher asked his students to try and find any functions whose derivatives where the exact same as the original function.

The only function then I have determined that statement to be accurate in is all the natural exponential functions. Ex. f(x) = ex, f'(x) = ex

If possible could you please email me all the functions that you can find in which the original function and its derivative is identical.


Answered by Claude Tardif.
Maximize 2000-03-12
From Tara Doucet:
My question is Maximize Q=xy^2 (y is to the exponent 2) where x and y are positive integers such that x + y^2 ( y is to the exponent 2)=4
Answered by Harley Weston.
A mixture problem 2000-03-06
From Rebecca Edwards:
A tank in which cholocate milk is being mixed contains a mixture of 460 liters of milk and 40 liters of chocolate syrup initially. Syrup and milk are then added to the tank at the rate of 2 liters per minute of syrup and 8 liters of milk per minute. Simultaneously the mixture is withdrawn at the rate of 10 liters per minute. Find the function giving the amount of syrup in the tank at time t.
Answered by Harley Weston.
Two calculus problems 2000-03-03
From Tara Doucet:
  1. The height of a cylinder with a radius of 4 cm is increasing at rate of 2 cm per minute. Find the rate of change of the volume of the cylinder with respect to time when the height is 10 cm.

  2. A 24 cm piece of string is cut in two pieces. One piece is used to form a circle and the other to form a square. How should the string be cut so the sum of the areas is a maximum?


Answered by Harley Weston.
Slant height of a cone 2000-02-24
From Jocelyn Wozney :
I need help with this problem for my high school calculus class. Any help you can give me will be greatly appreciated-I am pretty stumped. "Express the volume of a cone in terms of the slant height 'e' and the semi-vertical angle 'x' and find the value of 'x' for which the volume is a maximum if 'e' is constant.
Answered by Harley Weston.
Some integration problems 2000-02-23
From Tim Valentine:
I am having a great deal of difficulty with the following integrals, can you help? I think they need the use of trig substitution or integration by parts but I cannot figure out how to begin. Thanks!

The integral of 1/(2+3x2) dx.

and

The integral of x * square root of (4x+5) dx.


Answered by Harley Weston.
A moving point on the graph of y=sinx 2000-02-22
From Veronica Patterson:
Find the rate of change of the distance between the origin and a moving point on the graph of y=sinx if dx/dt=2 centimeters per second.
Answered by Harley Weston.
The quotient rule 2000-02-21
From Charlene Anderson:
Question: I came across a question in our book that states: Let Q(x) = N(x) / D(x) Then re-write Q(x) in a form that can utilize the Power and Product Rules. Use this rearranged form to derive the Quotient Rule.

The Quotient Rule can be derived from the Power Rule and the Product Rule.

One must also use the chain rule too, right?


Answered by Harley Weston.
Play ball 2000-02-03
From Jessie:
Here's a calc question that is probably a lot easier than I am making it. If you have a legendary "baseball problem" for the related rates section of Calc I, and you are given that the runner is running from 2nd to 3rd base at a given rate, and the umpire is standing at home plate, and you are given the distance between the bases on the field, how do you find the rate of change of the angle between the third base line (from the point of the umpire) and the runner? Here is a sample prob: Runner is moving from 2nd to 3rd base at a rate of 24 feet per second. Distance between the bases is 90 feet. What is the rate of change for the angle (theta, as described previously) when the runner is 30 feet from 3rd base?
Answered by Harley Weston.
Functions 2000-01-23
From Tara:
Hi my name is Tara, I have two math problems that I need help with in my calculus math class.

  1. If f(x)= x - 2 show that (x+3)f(x)-(x+2)f(x+1)+4=0

  2. Graph this function and use the graph to determine the range y=2x2 - 8x - 3

Answered by Harley Weston.
The limit of f(x)/x 2000-01-22
From Laurent Jullien:
I would appreciate help to prove that a twice continuously differentiable convex function from R+ to R has the property that f(x)/x has a limit when x tends to infinity.
Answered by Claude Tardif.
Why study calculus? 2000-01-05
From Trlpal:
I am a high school senior enrolled in a pre-calculus class. Could you tell me what the benefits of taking calculus are and why it would be important to take the class.
Answered by Walter Whiteley and Harley Weston.
A decreasing ellipsoid 1999-12-15
From A student instructor:
The volume of an ellipsoid whose semiaxes are of the lengths a,b,and c is 4/3 *pi*abc. Suppose semiaxes a is changing at a rate of A cm/s , the semiaxes b is changing at B cm/s and the semiaxes c is changing at C cm/s . If the volume of the ellipsoid is decreasing when a=b=c what can you say about A,B,C? Justify.
Answered by Harley Weston.
Two calculus problems 1999-12-13
From Alan:
I have 2 questions that are very new to me, they were included on a quiz and the material was never covered. Our teacher never explained the purpose and detailed explanation of how to solve the problem. Could you help? Thanks.

Question 1:
A ball is falling 30 feet from a light that is 50 feet high. After 1 sec. How fast is the shadow of the ball moving towards the light post. Note that a ball moves according to the formula S=16t^2

Question 2:
How many trapezoids must one use in order for the error to be less than 10^-8 if we want to find the area under the curve Y=1/X from 1 to 2. Find the exact area, Graph the function and use the trap rule for the "N" that you found.


Answered by Harley Weston.

A calculus problem 1999-12-08
From JT Wilkins:
These are the questions:

  1. Show that there exists a unique function that meets the following requirements:

    a) f is differentiable everywhere
    b) f(0)= f'(0)= 0
    c) f(x+y)= f(x)+ f(y), for all real values of x,y

  2. Consider the function F: R-->R (All Reals)

    F(x) = 0, for x irrational & 1/q, x=p/q gcd(p,q)=1 q > 0

    a)determine the values x where f is continuous, respectively discontinuous.
    b)determine the values x when f is differentiable and for each of these values compute f'(x).


Answered by Penny Nom.
Advanced Calculus 1999-12-07
From Kay:
Hi, my name is Kay. Please help---these problems are driving me crazzzzy!!!! Your help would be greatly appreciated!
  1. Let a,b be contained in R, a
  2. .
    .
    .

Answered by Claude Tardif.
The chain rule 1999-12-03
From Jennifer Stanley:
This problem is making me dizzy. I would greatly appreciate a little help!

Express the derivative dy/dx in terms of x.

y=u^2(u-u^4)^3 and u=1/x^2
Answered by Harley Weston.

Two calculus problems 1999-12-01
From O'Sullivan:
Question #1 Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely? It's under the chain rule section of differentiation if that any help.

I've set up a ratio and tried to find the constant but am stuck.

Question #2 The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x^2 + 4y^2 < or= 5. If the point (-5,0) is on the edge of the shadow, how far above the x axis is the lamp located?

The picture shows an x and y axis with only the points -5 and 3 written on the x axis. the lamp is on the upper right quadrant shining down diagonally to the left. There's an ellipse around the origin creating the shadow. It's formula is given as x^2 + 4y^2=5.


Answered by Harley Weston.
Two derivatives 1999-11-16
From Gina Renicker:
The derivative of:

y=e(xlnx) and y=x2arctan(x1/2)
Answered by Harley Weston.

Area of a circle and an inequality 1999-10-30
From Adam Anderson:
I have two problems.

The first: prove that the area of a cirlce is pi times radius squared without using calculus.

The second: show that ln(x) < x - 1 for all x > 0.


Answered by Harley Weston.

Clockwise or Counterclockwise? 1999-10-27
From Tim:
A particle moves around the circle x2 + y2 = 1 with an x-velocity component dx/dt = y
  1. Find dy/dt

  2. Does the particle travel clockwise or counterclockwise around the circle? Why?

Answered by Harley Weston.
Derivatives with logs 1999-10-26
From Kate:
What is the derivative of 5 to the 5x-2 at x equals 0.8?
Answered by Harley Weston.
l'Hospital's Rule 1999-10-18
From Yannick Gigandet:
How can I solve these two limits :
  1. lim when n approches 1 of n[a1/n -1]

  2. lim when x approches 0 of (eax - ebx) / x
Thanks for the answer!

Answered by Harley Weston.
Length of a line 1999-10-10
From Dagmara Sarudi:
My question has to do with the length of a diagonal. This problem came up when I thought about the shortest distance between two points, for example walking from one point to another in my neighborhood. I can choose a zig zag route and assuming the blocks I walk are exactly the same length, it shouldn't matter what route I took, the distance I travel should still be the same when I reached my goal. If, on the other hand I could travel in a diagonal line, the distance would be shorter. But what if, in my zig zag motion, the sections get so small the route approaches a diagonal. Shouldn't it be that each separate section added together equals the value of the two original sides? Or would it suddenly equal the value of the diagonal (which, of course was shorter than the two sides added together)?
What gives?


Answered by Chris Fisher and Harley Weston.

A trig limit 1999-10-06
From Yannick Gigandet:
What is the limit, as x approaches pi/3, of (1-2cosx) / sin(x-(pi/3)) ?
Answered by Penny Nom.
Two limits 1999-10-02
From Jennifer:
How do I find

lim (1-cosx)/(x^2) as x-> 0

and

lim (tan3x)/x as x->0
Answered by Harley Weston.

Distance between the windows 1999-09-19
From Lawrence:
An observer on level ground is at distance d from a building. The angles of elevation to the bottom of the windows on the second and third floors are a and b respectively. Find the distance h between the bottoms of the windows in terms of a b and d
Answered by Harley Weston.
2 to the x and x squared 1999-09-17
From John:
For what values of x is 2 to the exponent x greater than x squared?
Answered by Harely Weston.
Parametric Equations 1999-08-06
From Nicholas Lawton:
Show that an equation of the normal to the curve with parametric equations x=ct y=c/t t not equal to 0, at the point (cp, c/p) is :

y-c/p=xp^2-cp^3
Answered by Harley Weston.

A calculus problem 1999-07-22
From Nicholas Lawton:
The curve y= e^x(px^2+qx+r) is such that the tangents at x=1 and x=3 are parallel to the x-axis. the point (0,9) is on the curve. Find the values of p,q and r.
Answered by Harley Weston.
The shortest ladder 1999-06-26
From Nicholas:
A vertical wall, 2.7m high, runs parallel to the wall of a house and is at a horizontal distance of 6.4m from the house. An extending ladder is placed to rest on the top B of the wall with one end C against the house and the other end, A, resting on horizontal ground. The points A, B, and C are in a vertical plane at right angles to the wall and the ladder makes an angle@, where 0<@
Answered by Harley Weston.
Even and Odd Function 1999-06-17
From Kent:
There is one function with the domain of all real numbers that is both even and odd. Please give me the answer to this question before I go insane.
Answered by Penny Nom.
A circle in a square 1999-05-26
From Jose V Peris:
A circle is inscribed in a square. The circumference of the circle is increasing at a constant rate of 6 inches per second. As the circle expands, the square expands to maintain the condition of tangency.

find the rate at which the perimeter of the square is increasing.

find the rate of increase in the area enclosed between the circle and the square at the instant when the area of the circle is 25(pi) square inches.
Answered by Harley Weston.

Related rates 1999-05-13
From Tammy:
The sides of a rectangle increase in such a way that dz/dt=1 and dx/dt=3*dy/dt. At the instant when x=4 and y=3, what is the value of dx/dt? (there is a picture of a rectangle with sides x and y, and they are connected by z, which cuts the rectangle in half)
Answered by Harley Weston.
A Polar Plot 1999-05-06
From Irene:
Consider the polar equation r=2-3Cos(theta/2) In the interval [o, 4Pi], how would you find the area of one of the leaves and also the length of one of the edges of a leaf?
Answered by Harley Weston.
Radius of convergence 1999-04-21
From Nowl Stave:
Why is the radius of convergence of the first 6 terms of the power series expansion of x^(1/2) centered at 4 less than 6?
Answered by Harley Weston.
The average rate of change of a function 1999-04-20
From Tammy:
Suppose that the average rate of change of a function f over the interval from x=3 to x=3+h is given by 5e^h-4cos(2h). what is f'(3)? I would appreciate any help with this question.
Answered by Harley Weston.
Graphing the Derivative 1999-01-18
From Milena Ghebre:
This question has been nagging me for sometime now.

Is there a way of finding out the derivative of a function, just by looking at the graph of it?
Answered by Walter Whiteley.

Calculus 1999-01-16
From Kaylea Rankin:
Differentiate the following.

y = 1 /(2+3/x)
Answered by Jack LeSage and Penny Nom.

The area and the circumference of a circle. 1998-08-27
From Jason Wright:
I was looking at the relationship of the area of a circle and the circumference when I realized that 2*pi*r is the derivative of pi*r^2. I was wondering if there is any connective deep dark meaning as to why this appears to be related. Thanks for any help you can give me!
Answered by Walter Whiteley.
Volumes of Revolution 1998-07-24
From Lorraine Wall:
I'm on the section fpr The Computation of Volumes of Solids of Revolution and the following question is giving me problems:

-Consider the region in the first quadrant bounded by the x-and y-axes, the vertical line x=3, and the curve y=1/(xsquared + 3) I can determine the volume of the solid by rotating the region about the y-axis using the shell method but I can't seem to be able to get started with the volume when rotated about the x-axis.
Answered by Harley Weston.

Calculus problems 1998-07-13
From Lorraine:
I'm stuck again. Can you help?

This involves integration using the method of partial fractions

the integral of:
7x(to the 5th) - 2x(cubed) + 3 dx
--------------------------------------
x(to the fourth) - 81

Do I have to do long division to reduce the numerator to the fourth power?

the integral of:
4- 16x +21x(squared) + 6x(cubed) - 3x(fourth) dx
----------------------------------------------------
x(cubed)(x - 2)(squared)

Lorraine
Answered by Harley Weston.

A Calculus Problem 1998-06-28
From Lorraine:
I'm a post-secondary student taking calculus by correspondence. I'm stuck on the following question (and similar ones) Can you help?

Evaluate the following indefinite integral:

d(theta)
----------
1 + sin (theta)

(It says to multiply both numerator and denominator by: 1 - sin(theta)

Thanks
Lorraine
Answered by Harley Weston.

A trig limit 1998-05-28
From Ann:
This problem is a calculus 1 limit problem-high school level. I'm teaching myself calc over the summer and I'm already stumped.

find the limit

 lim sec^(2)[(sqrt2)(p)]-1 p-->0 --------------------- 1-sec^(2)[(sqrt3)(p)] 
I'm Ann.

Answered by Harley Weston.
A Tightrope Walker. 1998-02-19
From Amy Zitron:
A tightrope is stretched 30 feet above the ground between the Jay and the Tee buildings, which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A....
Answered by Harley Weston.
Some Calculus Problems. 1997-10-30
From Roger Hung:
  1. What real number exceeds its square by the greatest possible amount?

  2. The sum of two numbers is k. show that the sum of their squares is at least 1/2 k^2.

  3. .
    .
    .

Answered by Penny Nom.
A Trigonometric Limit 1997-09-18
From Brian Ray:
What is the limit, as x approaches 0, or tan^23x/x^2? (read, tan squared 3x over...)?
Answered by Harley Weston.
A Limit Problem 1997-09-16
From Robert Reny:
what is the limit, as x approaches 0, of 3x/2x-[x]? [] means absolute value.
Answered by Harley Weston.
Mathematical Induction and the Derivative 1997-03-18
From Shuling Chong:
"Obtain a formula for the nth derivative of the product of two functions, and prove the formula by induction on n."

Any educated tries are appreciated.
Answered by Penny Nom.

isomorphisme 2000-08-10
From Romain Kroes:
Pour les beoins d'un ouvrage d'économie que je suis en train de terminer, pouvez-vous me dire qui est (sont) l'inventeur de l' "isomorphisme" en mathématiques (calcul tensoriel)?
Answered by Claude Tardif.
Derivées partielle 1999-10-19
From Arnaud Flandin:
Quel est la definition des derivées partielle
Answered by Claude Tardif.
 
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