Worksheet on Composition of Functions (in preparation for the Chain Rule)

1. Let and . Let 

.

1. Find  and simplify your answer.
 

 
2. Find .
 

 
3. Find .
 

 
4. Find .
 

 
5. Find .
 

 
6. Find  and simplify your answer. Compare with part (c).

2. Suppose the following graphs are given for a car company:

1. Complete the graph below:

 

 
2. If c’(200) = 10 and 200 workers are currently working, approximately how many more cars will be produced by adding another worker?

 
The answer is 10 more cars. The reasoning is as follows: Since c’(200) = 10, the slope of the tangent line to the first graph is 10 when w = 200. This means that you go up by ten on the graph every time you go over by one. So, you go up by 10 cars for each one worker added.

 
3. If p’(4000) = 450 and 4000 cars are currently being produced, approximately how much more profit will be made by producing another car?

 
The answer is $450 more. The reasoning is as follows: Since p’(4000) = 450, the slope of the tangent line to the second graph is 450 when c = 4000. This means that you go up by 450 on the graph every time you go over by one. So, you go up by $450 for each one extra car produced.
 
4. Based on your answers to part (b) and (c), if c’(200) = 10 and p’(4000) = 450 and 200 workers are currently working, approximately how much more profit will be made by adding another worker?

 
If we add another worker, we get ten more cars (from part b). For each of these cars we get $450 (from part c). Altogether we get 450*10 = 4500 more dollars.
 
5. Based on part (d), write a relationship between P’(200), c’(200), and p’(4000).

P’(200) = p’(4000)*c’(200) -- and more importantly, P’(200) = p’(c(200))*c’(200)  

3. For each pair of functions , find a function F such that 

.

1. 

 
2. 

 

 
3. 

 

 
 
4. 

 
 

4. For each function F below, find a pair of functions such that 

.

1. 

 

 
2. 

 

 
3. 

 
4.