







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. 





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. 





Odds and evens in an n by n+1 table 
20100121 

From Shankar: The boxes of an n * (n+1) table ( n rows and n+1 columns) are filled with integers.
Prove that one can cross out several columns ( not all of them !) so that after this operation
all the sums of the numbers in each row will be even. Answered by Robert Dawson. 





Friends in a class 
20090914 

From pran: it is known that among any group of three students in a class two of them are friends. the total number of students is 25. prove that there is a student who has at least 12 friends Answered by Claude Tardif. 





Choices at a restaurant 
20090413 

From Rob: There is a restaurant you get:
Rice/Noodles (1) 
Main Ingredient (any) 
Sauce (1) 
1 
1 
1 
2 
2 
2 
3 
3 
3 
4 
4 
4 
5 
5 
5 
6 
6 
6 
7 
7 


8 


9 


10 


11 


12 

So the question is how many different combinations are there. You can only have 1 rice/noodles in a selection and only 1 sauce in a selection but you can have between 1 and all twelve mains in a selection. there are 7 rice/noodles , 12 mains and 6 sauces. How many possibilies. I did it mentally in the restuarant, no pen, paper or calculater and i got 3276..i think thats wrong. please help
Rob Answered by Robert Dawson, Stephen La Rocque and Claude Tardif. 





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. 





How many two digit numbers contain at least one 7? 
20070906 

From Janet: How many two digit numbers contain at least one number seven? Answered by Penny Nom. 





Choosing a bicycle 
20070328 

From Jackie: A specific brand of bike comes in two frames, for males or females. Each frame comes in a choice of two colors, red and blue, and with a choice of three seats, soft, medium, and hard.
a) Use the counting principle to determine the number of different arrangements of bicycles that are possible.
b) Construct a tree diagram illustrating all the different arrangements of bicycles that are possible. c) List the sample space. Answered by Penny Nom. 





Three dice 
20040510 

From A student: If one has 3, 6 sided dice what is the probability of the numbers that are rolled to total 4 through 10 inclusively?
Subsequent to this, what is the probability to do this consecutively...say 3 times? Answered by Peny Nom. 





Divisibility by 2 or 5 or both 
20031030 

From Abdu: How many positive integers less than 1,001 are divisible by either 2 or 5 or both? Answered by Penny Nom. 





Friends and enemies 
20030324 

From Becky:
Consider a room that contains six people. Any two people are either friends of each other, or they are enemies. A. Argue why there are three people, all who are friends, or there are at least three people, all who are enemies
B. Rephrase the situation using graph terminology, using all of these terms correctly: vertex, edge, graph, complement, clique, independent set, and bipartite. Answered by Penny Nom. 





Investing $5,000 
20010709 

From A student: A principal amount of $5,000 was invested in a savings account for 5 years.The interest earned was $500.Use the simple interest formula to find the annual rate of interest. Answered by Penny Nom. 





Danging couples 
20010606 

From Danielle: How many boygirl dancing couples could be formed if 85 boys and 102 girls attend a school dance? Answered by Penny Nom. 





Your DJ Business 
20010530 

From Linda: Your DJ Business has 6 rap, 10 rock, 6 alternative, 8 oldies, and 5 country CD singles. How many different 10song sets can the DJ play, if she plays 2 singles from each category? and How many different 10song sets if she plays exactly 3 rap singles and 4 rock singles in each set? Answered by Leeanne Boehm. 





Choosing a car 
20010428 

From Ashley: THE CAR DEALERSHIP IN TOWN OFFERS 32 DIFFERENT MODELS OF VIHICLES.EACH MODEL HAS A CHOICE OF EIGHT INTERIOR COLORS,EIGHT EXTERIOR COLORS,AND ALSO THE OPTION OF AUTOMATIC OR MANUAL TRANSMISSION. HOW MANY COMBINATIONS ARE POSSIBLE? Answered by Penny Nom and Andrei Volodin. 





Permutations and Multiplication Principle 
20000922 

From Candice: A forester selects 4 pink and 4 white dogwoods. The trees are to be planted in row. If a tree is distinguished by color only, in how many ways can the eight dogwoods be planted? How many of these arrangements have at least two trees of the same color side by side? Answered by Denis Hanson and Claude Tardif. 





towers of cubes 
19991005 

From Sanker: I need help to solve this Rules for bulding towers of cubes rule 1  The number of cubes on the bottom layer is always one less than the number of squares on the grid  rule 2  Each new layer is made with one cube less than the layer underneath it.   Investigate how many different arrangements there are of 4 cubes on top of 5 cubes on a two by three grid
 investigate the number of different arrangements of six cubes on top of seven cubes on a two by four grid
 investigate the relation between the number of arrangements of cubes and the size of the grid
 when there are two layers of cubes
 when there are more than two layers of cubes
Answered by Walter Whiteley. 





Square Roots and Functions. 
19970423 

From Ed: 1. In most texts the solution to a question such as square root x = 6 is x is undefined. Yet when teaching to solve xsquared = 36 x = +6 or 6 There appears to be a contradiction here. My question is when, where and why do we use the principle square root, not both + and ? This often occurs as the extraneous root in the solution of radical equations and in stating the domain and range of functions involving square roots. 2. Are there any simple rules for determining whether equations are functions without graphing them and doing a vertical line test? Answered by Harley Weston. 

