MA113:025–027 Russell Brown Review 2 September 16, 1998

1. Estimate the following limits. Your answers should be correct to 2 decimal places.

lim_{x→0} sin(e^{x} − 1)/x,  lim_{x→0} \frac{2^{x} − 3^{x}}{x} .

2. Find the following limits using the limit laws. Explain each step of your solution:

lim_{x→2} \frac{x − 2}{x^{2} − 4},  lim_{x→2} \frac{x + 2}{x^{2} + 4},  lim_{x→2} \frac{x + 3}{x − 2}.

3. Use the squeeze theorem to evaluate the following limits. lim_{x→∞} \frac{sinx}{x}, lim_{x→0} x sin(1/x).

4. Let f(x) = 2^{x}. Using the definition, estimate the derivative at x = 2.

5. Let f(x) = \sqrt{2x}. Find the derivative, f′(x), using the definition. Give the domain of the derivative.

6. State the definition of the derivative of f at a.

7. If f is differentiable at a, is f continuous at a? If your answer is yes, provide a proof. If your answer is no, provide an example.

8. On which intervals are the following functions continuous? On which intervals are the functions differentiable.

f(x) = |x|, g(x) = \frac{1}{x^{2} − 4}.

9. Show that the equation cos x = x has a solution.

10. Let f(x) = e^{2x}. a) Estimate f′(0). Explain how you arrived at your answer. b) Use the tangent line at 0 to approximate e^{1.02} and e^{0.99}.

11. If the prevailing price for milk is p dollars per gallon, then a store will sell Q(p) gallons of milk. a) What are the units for Q′(p)? b) Do you expect Q′(p) > 0 or Q′(p) < 0? Why?

12. Below are the graphs of f, f′ and f′′. Which is which? Explain your answer.

13. Below is the graph of a function f. Sketch the graph of f′.

14. Below is the graph of a function f′. Sketch two possibilities for the graph of f.