







Price before VAT and a discount 
20181115 

From Carol: I bought a computer for £330.
The selling price included VAT at 20%.
Then I was allowed a staff discount of £30.
What was it’s original price before the VAT and staff discount? Answered by Penny Nom. 





Pre tax and post tax percentages 
20180522 

From Tom: Hi,
I work in a clothes shop which can also do mailorders. My Head Office
has sent a report with which we can check the records we keep in store
tally with those at HO.
According to that report we have taken £2578.25 in orders, excluding VAT (20%),
which would therefore be £3093.91 including VAT (2578.25+20%=3093.91).
According to my records, we have taken £3179.32 including VAT, which would be
£2543.45 excluding VAT (3179.3220%=2543.45).
Why is it that the difference between the VAT inclusive figures (3179.323093.91) is +£85.41, but
the difference between the VAT exclusive figures (2543.452578.26) is £34.81?
Now, I can see from the report that 1 order has not been recorded by HO, and I
know that our average order value is around £30, so the £34.81 makes sense.
But surely both figures should be a minus, regardless of whether they include VAT?
I'm sure there is an obvious answer, but I just cannot see it! Any help would be gratefully
received.
Thanks,
Tom Answered by Penny Nom. 





More on the curvature of the Earth 
20180423 

From will: the formula for figuring the earth's curve goes against logic, looking at a fixed point and backup 1mi. the point drops 8" then 16" in the next mi. and 32" in the third mi. why shouldn't it be 24" why is the 8" per mi. squared can you tell me in laymens terms why this is it goes against logic it would seem the correct wat would be to add up 8" per mile as you back up from the fixed point 8" 16" 24" 32" not 8/16/32/64" Answered by Harley Weston. 





The derivative of f(x)=2^x/x 
20171128 

From Chhavi: f(x)=2^x/x.
Find f'(x). Answered by Penny Nom. 





Heat equation 
20171123 

From Max: What does du\dt=a(triangle)^2u mean. Can it be solved for t. Answered by Penny Nom. 





A tangent to a curve 
20171022 

From Jasem:
Suppose that
f(x)=(3x3)^1/2.
(A) Find an equation for the tangent line to the graph of f(x) at x=2
(B) Find all values of xx where the tangent line is horizontal, and enter them as a commaseparated list (e.g., 2,3,6). If there are none, enter none.
Values of x Answered by Penny Nom. 





The derivative of the inverse of a function 
20161028 

From Kate: Hi, I'm in a College level Calculus course and I can't seem to figure out the answer to this question.
Find the derivative of f^1(4) if f(3)=4 and f'(3)=1/7 Answered by Penny Nom. 





Implicit differentiation 
20160606 

From Pranay: Is a circle x^2+y^2=2 a function? If it is not a function,
why is it possible to do implicit differentiation on it?
Thanks. Answered by Penny Nom. 





The derivative of x! 
20160416 

From Sang: How to find the derivative of x! and integral of x! Answered by Penny Nom. 





Prove the earth is round 
20160130 

From Kevin: Working on a project for a science fair.
To prove the earth is round without the ability to get way above the surface.
How could we set up and experiment to see the curve.
My thought is using a telescope at the beach?
Form what we see so far the curve is 8 inches per mile?
So thought is set up a telescope and a target 1 mile away? Answered by Robert Dawson and Chris Fisher. 





A tangent line to a parabola 
20151202 

From pei: Given that the line y=mx5 is a tangent to the curve y=2x^2+3 find the positive value of M. Answered by Penny Nom. 





How much does the Earth curve over a one foot distance? 
20151124 

From Sean: Hi, I am trying to figure out how much the earth curves over a one foot distance. I'd like to be able to draw the exact arc on a piece of paper. I am an artist and am looking to make glass vessels with the exact curvature of the earth. I read on your site that it curves approximately 8 inches per mile. can I just use simple ratios to break it down into inches?? Thank you so much for your help. Answered by Harley Weston. 





The derivative of x^1/3 
20151108 

From omar: hi can help me
am teacher ask me about x^1/3 Derivation definition . Answered by Penny Nom. 





Two cars approaching an intersection 
20150416 

From Engabu: Car A is traveling west at 50km/h & car B is traveling north at 60km/h. both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 3km & car B is 4km from the intersection? Answered by Penny Nom. 





Constructing a box of maximum volume 
20150414 

From Margot: I need to do a PA for maths and I'm a bit stuck.
The PA is about folding a box with a volume that is as big as possible. The first few questions where really easy but then this one came up.
8. Prove by differentiating that the formula at 7 does indeed give you the maximum volume for each value of z. Answered by Penny Nom. 





Angles of elevation and depression 
20150308 

From Timmy: Joel is walking down a street and sees a 115 ft tall building in front of him. He stops 190 feet from the base of the building at the tip of the building's shadow. Round answers to three decimal places.
A. If there was a piece of rope from the top of the building to Joel, how long would it be?
B. What is the angle of elevation from Joel to the top of the building?
C. Margaret says that she could find the angle of depression from the top of the building to Joel by subtracting the angle of elevation from 90°. Is she correct? Explain. Answered by Penny Nom. 





The height of a building 
20150120 

From Emily: A man 2m high observes the angle of elevation to the top of the building to be 70 degrees. And the angle of depression to the bottom of the building to be 19 degrees. How tall is the building? Answered by Robert Dawson. 





Curvature of the Earth 
20141229 

From Jimmy: Both batteries died in my scientific calculator and I have lost my formula for the heigth of the curvature of the earth between two points on the surface. I used degrees and miles. I only had to enter the distance between the two points on the surface and the formula gave me the hieght the earth raised between the two points. Answered by Robert Dawson. 





A ladder against a wall 
20140709 

From thabo: A ladder 6.5m long,leans against a wall so that the top of the ladder is 4.8m from the ground.what is the angle of elevation of the ladder to the top of the wall Answered by Penny Nom. 





Differentiate ln[x(2x4)^1/2] 
20140628 

From Igwe: If y=In[x(2x4)^1/2],find dy/dx at x=3 Answered by Penny Nom. 





The derivative of sin(x) 
20140426 

From Lucky: f(x)=Sin(x), by first principle its f'(x)...show me how to solve such problem. Answered by Penny Nom. 





Curvature of the Earth 
20140328 

From Max:
Recently I read the answer to a question proposed by someone on this site.
The question : What is the rate of curvature per mile on Earth?
The answer given : Use Pythagoras' Theorem to solve for the answer, given a 1 mile side
and a side as the radius. The hypotenuse minus the radius is your answer of drop/mile or curve/mile.
My conjecture : Why go through all of that work if the distance is one? Something like
{1/diameter} would would fine for such a problem. Seems like a lot of work for no reason.
I understand the practical application of Pythagoras' Theorem in this certain situation, as you would need
to use a^2+b^2=c^2 for any distance greater than one [mile]..
It just seems excessive and unnecessary if you're solving for curve / one mile. Answered by Robert Dawson. 





The domain of a derivative 
20131010 

From Renee: I am looking to find the domain of a derivative of a radical function,
one such as: f(x) = the square root of (8 − x).
I am kind of unclear on how domains work for derivative.
I don't understand how you take a function's domain and use that to
find the derivative's domain.
Thanks! Answered by Penny Nom. 





Equal ordinate and abscissa 
20130815 

From sonit: the slope of tangent to the curve y=(4x^2)^1/2 at the point, where the ordinate and abscissa are equal, is Answered by Penny Nom. 





Distance as a function of acceleration 
20130710 

From Tom: If you start at a stoplight and your acceleration is 16t  t^2, how far have you gone after 8 seconds? Answered by Penny Nom. 





The angle of elevation of the sun 
20130703 

From Maurice: A vertical pole with a length of 7m cast a shadow with a length of 5m. Calculate the angle of elevation of the sun and include a diagram. Answered by Penny Nom. 





f(x) + f ''''(x)=0 
20130305 

From Andreea: Hei. I don’t speak lot of english but here is my question,hope u understand: f(x) + f ````(x)=0. so, my question. what is f(x), where f ````(x) is f(x) derivative by four time ? i tried to find the answer and i knew f(x) is something like that f(x)=e^x*sinx but miss something. Answered by Brennan Yaremko. 





The angles of elevation and depression 
20121203 

From Chelsey: a person on a balcony of one building looks towards a second building. if the angle of elevation to the top of the second building is 25 degrees, the angle of depression to the bottom of the second building is 17 degrees, and the balcony of the first building is 22 feet above the ground, what is the height of the second building? Answered by Penny Nom. 





The derivative of y = sin (30º + x) 
20121107 

From Saskia: derivative of y = sin (30º + x) Answered by Harley Weston. 





Differentiation rules 
20121023 

From Morgan: Use the derivative rules to differentiate each of the following:
1. f(x)=1/x1 2. f(x)= sqrt(x) Answered by Penny Nom. 





The height of a building 
20120908 

From Lin: How do surveyors determine a height of a building 150 feet away with an observation angle at 40 degrees?
What is the elevation of that top floor? Answered by Penny Nom. 





5 + 5 + 5  5 + 5 + 5  5 + 5 x 0 = 
20120729 

From Tom: 5 + 5 + 5  5 + 5 + 5  5 + 5 x 0 = Answered by Harley Weston. 





The angular elevation of the sun 
20120714 

From VINEET: WHAT IS ANGULAR ELEVATION OF THE SUM Answered by Penny Nom. 





The derivative of 2sin cubed x  3 sin x 
20120325 

From holly: suppose f(x) = 2sin cubed x  3 sin x
show that f 1(x) = 3 cos x cos 2x Answered by Harley Weston. 





Integral 1/(25x^2)^3/2 
20120222 

From John: Integral 1/(25x^2)^3/2 Answered by Harley Weston. 





Dt[sin t tan (t^2+1)] 
20120221 

From Ayu: Ayu
Dt[sin t tan (t^2+1)]
derivatives Answered by Harley Weston. 





The curvature of the earth 
20120208 

From sean: Question from sean, a student:
Two people 1.8 metres tall walk directly away from each other until they can no longer see each other (due to the curvature of the earth, which has a radius of about 6378 km).
A) Find a function relating the height of two identical objects with the distance between them using the scenario above as an example.
B) Sketch this function (you may use Graphmatica if you wish). Over what domain and range does the function exist?
C) Describe this relation in practical terms. Answered by Harley Weston. 





Notation for the second derivative 
20120206 

From Shafira: In all math textbooks, it is written that d/dx ( d/dx) (y)= d2y/dx2. Why do they write it as d2y/dx2, not as d2y/d2x2? Answered by Robert Dawson. 





The derivative of x^(1/2) 
20120114 

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. 





Implicit differentiation 
20111020 

From Monica: Find dy/dx in terms of x and y, if sin(xy)=(x^2)y. Answered by Penny Nom. 





The derivative of f(x) = (x+1)^1/2 
20110905 

From Carla: Find the derivative using the limit process of
f(x) = (x+1)^1/2 Answered by Harley Weston. 





An antiderivative of the square root of (8t + 3) 
20110419 

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. 





A stone is dropped into a lake 
20110324 

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. 





What is the height of the building? 
20110205 

From Myra: The angle of elevation is 45 degrees. The distance from George's foot to the top of the building is 50m. What is the height of the building? Answered by Penny Nom. 





Angle of elevation 
20101210 

From PANKAJ: angle of elevation of the sun perpendicular72 and base 88 find angle Answered by Penny Nom. 





A building and a flag pole 
20100909 

From paul: A flag pole and a building stand on the same horizontal level. From the point p at the bottom of the building,the angle of elevation of the top t of the flag pole is 65 degrees. From the top q of the building the angle of elevation of the point t is 25 degrees.If the building is20 meters high. Calculate the distance pt Answered by Penny Nom. 





80,000 yards of dirt 
20100812 

From Jeremy: If I have an area 50' x 1200', what would be the elevation with 80,000 yards of dirt on it. Answered by Robert Dawson. 





The integral of (x^2*exp(x)/(exp(x)1)^2 
20100809 

From sujoy: please find this integral for me
int(x^2*exp(x)/(exp(x)1)^2 Answered by Robert Dawson. 





Using the limit definition 
20100606 

From Meagan: Using the limit definition find the derivative of 3/(2x^2) Answered by Harley Weston. 





Curvature of the Earth 
20100529 

From grant: if you have 2 plumb structures 10ft. tall, how far apart will they be when they are 1 inch out of parallel at the top Answered by Tyler Wood. 





Line of sight 
20100506 

From David: I live in St. Joseph, Michigan and there is an ongoing argument regarding lineofsight over the horizon.
Standing on a 200 foot high bluff here, people swear they can see the top of the Willis (nee Sears) Tower in Chicago, which is about 1653 feet high.
It is my contention that this is actually a "refracted reflection" and not directlineofsight.
So, to settle the argument, I'd sure like some simple explanation for this, even ifand I hope notI am incorrect.
David Answered by Harley Weston. 





The height of a flag shaft 
20100425 

From Sarah: A man standing 20metres away from a tower observes the angles of elevation to the top and bottom of a flag shaft standing on the tower as 62degrees and 60degrees respectively. Calculate the height of the flag shaft.' Answered by Penny Nom. 





The derivative of y=x^x 
20100409 

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+h1)h+...x^x)/h
This is also equal to lim a>0 lim h>0 (x^(x+a)+(x+h)x^(x+h1)h...x^x)/h.
Evaluating for a=0 you get lim h>0 (x^x+(x+h)x^(x+h1)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+h1)... which when evaluated for h=0 gives us: x(x^(x1)). 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. 





The derivative of cos^3x 
20100406 

From Erson: Find y' of the given function: y = cos^3x. Answered by Harley Weston. 





How far apart are the two girls? 
20100118 

From benny: Debby and john are looking up at their house from the backyard. From Debby's
point of view, the top of the house is at an angle of elevation of 40 degrees
From Johns point of view, directly closer to the house, it is 60 degrees. The
house is 15m high. How far apart are the two girls? Answered by Robert Dawson. 





The second derivative of y = x³ (x² + 5) 
20091214 

From Kyrie: Find d²y/dx² for this function:
y = x³ (x² + 5) Answered by Penny Nom. 





Graphing y=(4x^2)^5 
20091025 

From natalie: I want to graph the curve of y=(4x^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. 





Differentiating y= square root(x1) 
20090929 

From edith: describe the xvalues at which f is differentiable.
y= square root(x1) Answered by Penny Nom. 





An antiderivative problem 
20090813 

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. 





Application of Derivatives of Trig Functions 
20090521 

From Alannah: I have a word problem from my Calculus textbook that I can't figure out.
Triangle ABC is inscribed in a semicircle with diameter BC=10cm. Find the value of angle B that produces the triangle of maximum area.
I am supposed to set up an equation for the area of the triangle A=b x h/2 using Trig functions based on angle B to represent the base and height but I'm not sure how to do this when the side length given is not the hypotenuse. Answered by Janice Cotcher. 





A vertical radio tower is located on the top of a hill 
20090421 

From Rafael: A vertical radio tower is located on the top of a hill that has an angle of elevation of 10 degrees. A 70foot guy wire is attached to the tower 45 feet above the hill.
a. Make a drawing to illustrate the situation
b. What angle does the guy wire make with the side of the hill?
c. How far from the base of the tower is the guy wire anchored to the hill?
What confuses me about this problem is the visual situation. Isn't the angle of the guy wire with the side of the hill the same as the angle of elevation? And if not, then how is one supposed to find the other angles without any more information? Answered by Harley Weston. 





The derivative using limits 
20090421 

From Kirstin: I am trying to take the limit of f(x) = [f(x+h)f(x)] / h
If you try taking the limit by substituting the limiting value h=0, you get 0/0,
which of course is not the right answer. You rewrite f(x+h)f(x) so it has a factor
of h in it, which you cancel with the h in the denominator before you substitute h=0.
But I am not sure how to do this. Thanks. Answered by Robert Dawson. 





The second derivative of h(x)=f(g(x)) 
20090216 

From Kristina: If h(x)=f(g(x)), and is differentiable, then find h"(x). Answered by Robert Dawson. 





A definite integral 
20090209 

From Mathata: Evaluate: integral from 0 to 1, x^2 e^x^3dx Answered by Harley Weston. 





What's the derivative of y = e^2xe^2x / e^2x+e^2x? 
20090202 

From Heather: What's the derivative of y = e^2xe^2x / e^2x+e^2x ? Answered by Harley Weston. 





Partial derivatives 
20090117 

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 
20090112 

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. 





Determine y'' by implicitly differentiating twice 
20090104 

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. 





Confusion in a multiple choice question 
20081114 

From BJ: My son got this math problem which he could mostly solve. Here it is:
The highest location in a certain country is 4525 m above sea level. The lowest point in the same country is 192 m below sea level.
a) Find the difference of the two elevations. His answer: a(b)= a+b or 4525(192)= 4525+192=4717 m. No problem.
b) A city is 2221 m above sea level. Is this elevation closer to the highest point or the lowest point?
His answer: highest point because: 45252221= 2304 m (closest to the highest point) and 2221(192)=2221+192= 2413 m. (farthest to the lowest point). OK so far.
But then he was given 4 choices for this question with no other information:
a) 4717 m; lowest b) 4333 m; lowest c) 4333 m; highest d) 4717 m; highest
What does it mean? What's the connection with the rest of the problem? Answered by Harley Weston. 





Antiderivative of 1/(x(1  x)) 
20081022 

From Matt: derivative of dx/(x(1x))
From what I've seen I should break apart the equation as such
derivative of dx/x  dx/(1x)
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. 





Concavity and the second derivative 
20081015 

From Christina: I'm having trouble solving for a second derivative for the following graphing question.
f(x) = (X^2+2x+4)/2x
using the quotient rule, I found:
f'(x) = (x^24)/(2x^2)
however, using the quotient rule again I can't seem to solve it (concavity):
f'''(x)=[(2x)(2x^2)(x^24)(4x)]/[(2x^2)^2]
f''(x)=[(4x^3(4x^3 16x)]/4x^4
f''(x)=16x/4x^4
f''(x)=4/x^3
and making the equation equal to zero result in 0=4 which doesn't seem to make sense... Answered by Penny Nom. 





How tall is the wall? 
20080929 

From ash: you and bob are separated by a tall wall you stand 10 feet further from the wall
than bob your angle of elevation is 37 degrees and his 44 degrees
how tall is the wall? Answered by Penny Nom. 





What is the derivative of (2^sinx)/(logbase4(2x+1))? 
20080916 

From Jesse: What is the derivative of (2^sinx)/(logbase4(2x+1)) Answered by Harley Weston. 





Angle of elevation 
20080909 

From kristy: A man on the tenth floor of a building shouts down to a person on the street. If the angle of elevation from the street to the man in the building is 35° and the man in the building is 40 feet up, about how far away from the building is the person on the street? Answered by Penny Nom. 





The height of a tree 
20080909 

From danice: At a certain time of day, the angle of elevation of the sun is 30°. A tree has a shadow that is 25 feet long. Find the height of the tree to the nearest foot. Answered by Penny Nom. 





f(x)=sin^3(3x^2) find f ' (x) 
20080421 

From Michael: f(x)=sin^3(3x^2) find f ' (x) Answered by Harley Weston. 





f(x) =ax^blnx 
20080413 

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. 





The chain rule 
20080410 

From joey: pls help me to Differentiate
y=(3x^24x)^8 Answered by Harley Weston. 





Distance seen 
20080318 

From Nev: formule: Distance Seen
S = 1.225 x square root of H
S = Distance seen in Miles
H = Height in Feet
What dose 1.225 Relate To ? Answered by Stephen La Rocque. 





Angle of Elevation 
20080129 

From Rita: Uluru or Ayers Rock is a sacred place for Aborigines of the western desert of Australia.
ChunWei uses a surveying device to measure the angle of elevation to the top of the rock to be 11.5 degrees.
He walks half a mile closer and measures the angle of elevation to be 23.9 degrees.
How high is Ayers Rock in feet? Answered by Stephen La Rocque. 





Differentiate 
20071228 

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. 





A curve sketch 
20071122 

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. 





The derivative 
20071119 

From ralf: Find the derivative of the function
1. y=1+2x^{8}
2. y=(1+2x )^{8} Answered by Harley Weston. 





f(x+y) = f(x) + f(y) + 2xy 
20071101 

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 
20071027 

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. 





The derivative of f(x)=1/(x1) 
20070921 

From Michelle: im having trouble finding the derivative of f(x)=1/(x1) using the f(x+h)f(x)/h method. Answered by Stephen La Rocque. 





Differentiate x^(1/3) using first principles 
20070914 

From Sheila: our teacher gave us this question as a challenge and even he couldnt figure it out:
Differentiate x^(1/3) [aka the cube root of x] using first principles. i know the answer is 1/(3.x^2/3), but how is it possible using first principles? Answered by Harley Weston. 





The elevation of the sun 
20070910 

From Elena: The "angle of elevation" of an object about you is the angle between a horizontal line of sight between you and the object. (See figure) After the sun rises, its angle of elevation increases rapidly at first, then more slowly, reaching a maximum near noontime. Then the angle decreases until sunset. The next day the phenomenon repeats itself. Assume that when the sun is up, its angle of elevation (E) varies sinusoidally with the time of day. Let t be the number of hours that has elapsed since midnight last night. Assume that the amplitude of this sinsoid is 60 degrees, and the maximum angle of elevation occurs at 12:45 p.m.. Assume that at this time of year the sinusoidal axis is at E=5 degrees. The period is, of course, 24 hours.
a. Sketch a graph of this function
b. What is the realworld significance of the t  intercepts?
c. What is the real world significance of the portion of the sinusoid, which is below the taxis?
d. Predict the angle of elevation at 9:27 a.m., and at 2:30 p.m.
e. Predict the time of sunrise
f. As you know, the maximum angle of elevation increases and decreases with the change of the seasons. Also, the times of sunrise and sunset change with the seasons. What one change could you make to your mathematical model that would allow you to use it for predicting the angle of elevation of the sun at time on any day of the year. Answered by Harley Weston. 





The height of a pole 
20070802 

From lalaine: Hi, this is my problem..
From a point 50.2 m to the pole, a student measured the angle of elevation to the top of the pole to be 32°. Find the height of the pole if the student's height from his feet to his eyes is about 4 ft. Answered by Penny Nom. 





Using the chain rule to solve a derivative 
20070729 

From Charles: I need to find the derivative fo the following function.
_______________________
\/ ______________
\/ (x  1) / (x + 2) + 1 Answered by Stephen La Rocque. 





Angle of inclination from the horizontal 
20070718 

From Joyce: In flying upward for 1260 yards along a straight inclined path airplane rises
156 yards. Find the climbing angle
( the angle of inclination from the horizontal)
Thank you in advance Answered by Stephen La Rocque. 





Height of a tower from two observations 
20070716 

From joyce: An observer wishes to determine the height of a tower. He takes sight @
the top of the tower from A & B w/c are 5oft. apart @
the elevation on a direct line w/ the tower.
The vertical angle @ point A is 30 degrees & the point B is 40 degrees.
What is the height of the tower?
Find the value of x in angle tangent 40 degrees and 30 degrees?
Show the solution of the value of x? Answered by Stephen La Rocque. 





Height of an antenna (angle of elevation) 
20070716 

From Fhay: An antenna stands on the edge of the top of a 52 story building from a point 320 ft. from the base of the building, the angle of elevation to the top of the antenna is 64 degrees in each story is 12 ft. high. Find the height of the antenna Answered by Stephen La Rocque. 





The two towers (angles of elevation trigonometry) 
20070714 

From joyce: The angle of elevation of tower B from the top of tower A is 28 degrees
and the angle of elevation of the top of tower A from the base is 46 degrees
Find the height of tower A if tower B is 120 m high? Answered by Stephen La Rocque. 





Implicit Derivatives 
20070713 

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 
20070709 

From Bob: What is the derivative of the function a sub n = [n/(n+1)]^n ? Answered by Stephen La Rocque. 





Log base 2 of log base 2 of x 
20070627 

From alex: y = log base 2 of lag base 2 of x
The slope of the tangent to the given curve at its xintercept is..? Answered by Harley Weston. 





Angles of depression 
20070613 

From Phonda: The pilot of a small private plane can look forward and see the control tower for a small airstrip. Beyond that is a large factory that is 3 milies from the airstrip. The angles of depression are 12.5 degrees and 4.8 degrees respectively.
Find the airplane's altitude, to the nearest ten feet. Answered by Stephen La Rocque. 





Maximizing the volume of a cone given the slant length 
20070514 

From Christina: A coffee filter for a new coffee maker is to be designed using a conical filter. The filter is to be made from a circle of radius 10cm with a sector cut from it such that the volume of coffee held in the filter is maximised. Determine the dimensions of the filter such that the volume is maximised. Answered by Stephen La Rocque and Kerstin Voigt. 





The second derivative 
20070414 

From Gerry: In mathematical context,what do you understand by the term "Second Derivative" Answered by Penny Nom. 





y = sin(2x) 
20070322 

From bader: sin(2x)
find dx/dy Answered by Penny Nom. 





Angle of elevation 
20070313 

From Joslyn: A ship at sea sights a 12m high lighthouse on a cliff which is 80m above sea level.
If the angle of elevation to the top of the lighthouse is 27 degrees, calculate the distance from the ship to the shore. Answered by Haley Ess. 





An excavation 
20070307 

From Brandon: you have and excavation area 140' long by 20' wide by 3' deep. What is the excavated volume? Answered by Penny Nom and Brennan Yaremko. 





Angle of elevation 
20070205 

From Zee: A 55 ft. flagpole casts a 25 ft. shadow. Calculate the angle of elevation to the sun to the nearest degree. Answered by Stephen La Rocque. 





Derivatives 
20070201 

From Jacob: Find the derivative of: y= pi^2+x^2+3xy+sin(y^2) Answered by Penny Nom. 





Percent elevation gain 
20070124 

From Susan: okay... so if I hike Gregory Canyon and it's a 1.1 mile hike with a 900 foot elevation gain, what's the percent elevation gain? Answered by Stephen La Rocque. 





Differentiate Y= sin3x + cos7x 
20060822 

From james: Differentiate the function of x using the basic rules.
Y= sin3x + cos7x Answered by Stephen La Rocque. 





livinia the housefly finds herself caught in the oven 
20060627 

From Danielle:
livinia the housefly finds herself caught in the oven at the point (0,0,1). the temperature at the points in the oven is given by the function
T(x,y,z) = 10 (xe^{(y2)} + ze^{(x2)})
where the units are in degrees celsius.
(i.)so if livinia begins to move towards the point (2,3,1) at what rate in deg/cm does she find the temp. changing?
(ii.)in what direction should she move in order to cool off as rapidly as possible?
(iii.)suppose that livinia can fly at a speed of the square root of 2 (cm/sec.) if she moves in the direction of part (ii) at what rate will she find the temp. to be changing?
Answered by Penny Nom. 





differentiate the volume of a cylinder with V respect to h 
20060524 

From A student: differentiate the volume of a cylinder with V respect to h Answered by Stephen La Rocque. 





Rate of ladder falling 
20060430 

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. 





The waterway between Lake Huron and Lake Superior 
20060321 

From Trenae: the waterway between lake huron and lake superior separates the u.s and canada.it is usually 13 feet above the water when its closed and each section is 210 feet long if the angle of elevation is 70 degrees then what is the distance from the top of the drawbridge to the water and the width of the gap created by the 2 sections of the bridge Answered by Penny Nom. 





Differentiation, powers and logs 
20060106 

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 word problems 
20051114 

From Jennifer:
1. Every other person on a school's parent advisory committee is surveyed to determine how many people support passage of a school bond to build a new elementary school. Is this a good sample? Why or why not?
2.What is the difference in elevation between the highest point in California,Mount Whitney,which towers 4421 meters above sea level, the lowest point in California,Death Valley, which lies 86 meters below sea level?
Answered by Harley Weston. 





The elevation of the top of the house 
20051113 

From Chloe: Karen is standing 23 metres away from the base of a 23 metre high house. Assume that Karen's eyes are 1.5 metres above ground. Find the elevation of the top of the house from Karen's eye line. Answered by Penny Nom. 





Notation for the second derivative 
20051108 

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 
20051027 

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 
20051018 

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. 





How do you differentiate y=(x)^(x^x)? 
20050914 

From Calebius: How do you differentiate y=(x)^{(xx)}? Answered by Penny Nom. 





Differentiating F(x,y) = 0 
20050123 

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. 





ln(x)/x 
20041211 

From Tina: What is the derivative of (ln x)/x? The double derivative? Answered by Penny Nom. 





The third derivative 
20041015 

From Holly: Why would you ever take the third derivative? Answered by Harley Weston. 





differentiate Y=X^X^X 
20040913 

From Kunle: differentiate Y=X^X^X Answered by Penny Nom. 





Differentiation 
20040804 

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. 





Angles of elevation and depression 
20040518 

From Anjum: what is the difference between an angle of elevation and angle of depression? Answered by Penny Nom. 





Some calculus problems 
20040401 

From Weisu:
I have questions about three word problems and one
regular problem, all dealing with derivatives.
 Find all points on xy=e^{xy} where the tangent line
is horizontal.
 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?
 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?
 The production P of a company satisfies the
equation P=x^{2} + 0.1xy + y^{2}, 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 derivative 
20040331 

From A student: What is the nth derivative of f(x) =(2x)/(1(x^{2}))? Answered by Harley Weston. 





A partial derivative 
20040319 

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. 





Acres and square miles 
20040217 

From Richard:
How many acres are in .17 square miles? and How many acres are in .6 square miles?
The two areas I am requesting information about are Vatican City and San Marino, the two smallest countries in the world. If I can transfer the sq. miles into acres , I can relate the size of these countries to our school grounds and the students will better understand their sizes.
Answered by Penny Nom. 





Areas, perimeters and derivatives 
20040215 

From Geoff: A recent lecture in my calculus class led me to realize that the derivative of the area of a circle, with respect to the radius is equal to the circumfrence. This also holds true for the relationship between the volume of a sphere and the surface area of that sphere:
why do these hold true? why is it only for circular objects? Answered by Harley Weston. 





The derivative of x to the x 
20040214 

From Cher: what about the derivative of x to the power x? Answered by Penny Nom. 





How far can you see? 
20031215 

From Judy:
How far apart, assuming no obstacles, can two people stand and still see each other?
i know this deals with the curvature of earth, but i can't figure out the formulas involved.
Answered by Chris Fisher. 





Functions, graphs and derivatives 
20031005 

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 
20031001 

From A student:
find the slope of the tangent to each curve at the given point f(x)=square root 16x, where y=5 Answered by Penny Nom. 





Hidden by the curvature of the Earth 
20030423 

From Shirley: There are 2 six foot men. What would the distance be between them before one could not be seen because of the curvature of the earth? Answered by Penny Nom. 





A two stage rocket 
20021126 

From Hoda: a two stage rocket accelerates in free space by ejecting fuel at a constant relative speed , v(ex). the full fuel load makes up 80% of the initial mass of the entire two stage rocket . the rocket accelerates from rest until at the end of the first stage when 75% of its fuel has been burnt. find an expression for the speed of the rocket at the end of the first stage in terms of v(ex). Answered by Claude Tardif. 





The slope of a tangent line 
20020304 

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. 





A tangent line 
20011121 

From A student: write an equation of the line tangent to the graph of
e^{y} + ln(xy) = 1 + e at (e,1) Answered by Harley Weston. 





4 sinx cosy = 1 
20011010 

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. 





When will the ship disappear? 
20011010 

From Stacy: If the sail of a ship were a 100 ft. tall and you were a mouse at the edge of the shore looking out at it, how far out would the ship be when it disappears? ( your eye level is level with the water.) Answered by Harley Weston. 





(x^25x6)/(x6) 
20011002 

From Bill: given f(x) = (x^{2}5x6)/(x6) find f'(6). Answered by Harley Weston. 





Conservation, consumption and population growth 
20010604 

From Steve: I'm trying to quantify the relation between conservation/consumption and population growth. For instance let's consider California: The 2000 census states that California's population grew from 29,760* in 41990 to 33,871 in 42000. I want to find r or rate of growth per year. Based on the exponential growth formula for population growth: . . . Answered by Penny Nom. 





Differentiation 
20010417 

From Esther: Could you please tell me what the first derivative is of the following: y = 2/(2x+e^{2x}) Is it (1+xe^{2x})/(2x+e^{2x})^{2} or perhaps 4(1+e^{2x})/(2x+e^{2x})^{2} ? I am a little confused between the two! Answered by Harley Weston. 





The angle of elevation 
20010308 

From Jeffrey: At a Certain time, a vertical pole 3m tall cast a 4m shadow. What is the angle of elevation of the sun? Answered by Harley Weston. 





The domain of the derivative 
20010222 

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} 
20010217 

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=x^{n } 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. 





Motivators 
20010116 

From Michelle Stapley: Do you know of any (or where I can find) student motivators for math at the secondary level? Basically any way I can motivate my students to WANT to learn math. Answered by Penny Nom. 





Net, gross and vat 
20001120 

From Chris: I have my gross but want to find out the net so I know how much the VAT is? Answered by Penny Nom. 





Concavity 
20001022 

From Alex: the question is: on what interval is f(x)=(x^{2})(e^{x})? ive found the 2nd derivative which is e^{x}(x^{2}+4x+2) and i did the quadratic to get 22^{0.5} and 2+2^{0.5}, but i dont know what the interval is. Answered by Harley Weston. 





Derivatives, there must be an easier way 
20000906 

From Brad Goorman: The direction read: Take the derivative of each expression. y = {1+[x+(x^{2} +x^{3})^{4}]^{5}}^{6}
Answered by Harley Weston. 





A derivative problem 
20000604 

From Jeff Ellis: If F(x)=(4+x)(3+2x^{2})^{2}(2+3x^{3})^{3}, find F'(0) Answered by Harley Weston. 





Thearcius Functionius 
20000503 

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 
20000503 

From Bonnie Null: I am to find the indefinite integral of: (e^{x}  e^{x})^{2} dx Answered by Claude Tardif. 





y = x^x^x^x... 
20000405 

From Michael Hackman: Find the derivative of: y = x^x^x^x... on to infinity. Answered by Claude Tardif. 





Filling a hole 
20000401 

From John McNeill: I have an area 20 feet long and 12 feet wide It starts out at a depth of 4 inches and ends up at 0 inches. How much sand do I need to fill the excavation. Answered by Penny Nom. 





A calculus problem 
19991208 

From JT Wilkins: These are the questions:  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  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. 





The chain rule 
19991203 

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





Two calculus problems 
19991201 

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 yaxis 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 
19991116 

From Gina Renicker: The derivative of: y=e^{(xlnx)} and y=x^{2arctan(x1/2)} Answered by Harley Weston. 





Ben's observation 
19991028 

From Emily Nghiem and Ben Rose: As a teacher at a school called Educere in Houston, I have a ninthgrade student who discovered the following shortcut last year as an eightgrader. What he noticed is that given any two consecutive integers (or n and n+1 for any rational number greater than or equal to 2), the difference between their squares was equal to the sum of the two numbers. . .
Answered by Chris Fisher and Penny Nom. 





Clockwise or Counterclockwise? 
19991027 

From Tim: A particle moves around the circle x^{2} + y^{2} = 1 with an xvelocity component dx/dt = y  Find dy/dt
 Does the particle travel clockwise or counterclockwise around the circle? Why?
Answered by Harley Weston. 





Derivatives with logs 
19991026 

From Kate: What is the derivative of 5 to the 5x2 at x equals 0.8? Answered by Harley Weston. 





Parametric Equations 
19990806 

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 : yc/p=xp^2cp^3 Answered by Harley Weston. 





A calculus problem 
19990722 

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 xaxis. the point (0,9) is on the curve. Find the values of p,q and r. Answered by Harley Weston. 





Graphing the Derivative 
19990118 

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 
19990116 

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. 
19980827 

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. 





Curvature of the Earth 
19980316 

From Robert Dyck: How can I find the curvature per mile of the earths surface? What is it? Answered by Harley Weston. 





A Tightrope Walker. 
19980219 

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. 





Mathematical Induction and the Derivative 
19970318 

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. 





Derivées partielle 
19991019 

From Arnaud Flandin: Quel est la definition des derivées partielle Answered by Claude Tardif. 

