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1. Using the definition of derivative, find f'(x), if f(x) = 2x/(3-5x).
2. A certain student learns a task so that after t weeks the student
has learned
percent of the task.
(a) How many weeks does it take for the student to learn 90% of the task?
(b) Approximate P'(8).
(c) Is your approximation for P'(8) higher than the actual rate or lower? Explain.
(d) What are the units for P'(8)?
(e) It is known that P-1(80) = 15.041. What does this mean? Use units in your
explanation.
(f) Is is also known that (P-1)'(80) = 2? What does this mean, and include units in your explanation.
3. I own a bakery, and am trying to decide how large to make my cookies. I have a graph in
front of me of the revenue R(x), in cents per cookie I can get if make the cookies x grams
per cookie. According to the graph, R(20) = 15 and R'(20) = 1.25.
(a) What do you think the graph of R looks like? Explain your reasoning.
(b) How much do you think I could get per cookie, if I made the cookies 20.5 grams
each? Show all your work, and explain your reasoning.
4. The function y = h(x) is graphed below:
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(a) What is limx->5 f(x)? Why?
(b) What is limx->4 f(x)? Why?
(c) What is limx->6+f(x)? Why?
(d) What is f(5)?
(e) What is f'(7)?
(e) For which values of x is f(x) not continuous?
(f) For which values of x is f(x) differentiable?
(g) For which values a of x does limx->af(x) not exist?
(h) Find a single point for which f(x) < 0, f'(x) > 0, and f''(x) < 0.
(h) Finally, sketch the derivative of f'(x).
