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Applied Calculus Guide Problems

1. Use the definition of the derivative to find f’(x) if 2. Find each of the following limits.

a.

b.

c.

3. Discuss the continuity of the function f defined by

at x=2

4. Differentiate each of the given functions.

a. b.

c. 5.

a. If defines a function implicitly, find b. Find an equation of the tangent line to the curve given in part (a), at the point (1,2)

6. Use logarithmic differentiation to find f’(x) for each of the given functions. a.

b.

7. Find all real numbers x that satisfy each of the given equations. a. b. c.

8. Find each of the following indefinite integrals. a.

b.

c. 9. Evaluate each definite integral.

a.

b.

10. Find the area of the region bounded by the parabolas 11. Suppose that C(x) is the total cost of producing x units of a particular commodity and

p(x) is the unit price at which all x units will be sold. Assume p(x) and C(x) are in dollars. a. Find the marginal cost and the marginal revenue. b. Use marginal analysis to approximate the cost of the 21st unit. c. Find C(21)-C(20). What does this represent? Compare your answer with that to part

(b) d. Use marginal analysis to estimate the revenue derived from the sale of the 21st unit.

What is actual revenue obtained from the sale of the 21st unit?

12. It is estimated that t months from now the population of a certain town will be increasing at the rate of people per month. If the current population is 10,000, what will the population be 8 months from now?

13. Find the first and second partial derivatives of

14. Find the absolute extrema of

15. Given find

a. The intervals on which f is increasing; the intervals on which f is decreasing, b. Where the graph of f is concave upward; where the graph of f is concave downward, c. The relative extrema of f, d. The inflection point of the graph of f, e. Draw a sketch of the graph of f.