Let st 64t - 2t2 represent the height in feet of a paper

Question

1. Find the derivative with respect to x of the functions given below:

a. f(x) = -x² + 3  

b. y = (2x + 3)(5x² - 7x) 

c. y = √x sin(x)

d. s(x) = 2x² + 1/x - 3        

e. y = x⁷ + √7x + (1/ π)     

f. s(x) = 3π tan(x)            

g. f(x) = csc(5x² + 1)        

2. The figure to the right shows two graphs, the graph of a function y = f(x) together with the graph of its derivative f'(x).  Which graph is which?  Explain how you know this.

3. Use implicit differentiation to find dy/dx for the following:

a. sin(y) = x        

b. x² + xy - y³ = 1

4. Let s(t) = 64t - 2t² represent the height in feet of a paper clip that is shot straight up in the air with a rubber band where t is in seconds.

a. Find ds/dt

b. Find d²s/dt²

c. Since the function s(t) represents the height of the paper clip, what does the first derivative represent?

d. Find the paper clip's velocity at t = 5.

e. At what time is the paper clip at its maximum height?

f. What is the maximum height that the paper clip travels?

g. After the paper clip is shot up, in how many seconds does the paper clip hit the ground?