Chapter 4 Test Review

1. Chapter 4 Test Review Page 256: 3, 17, 23, 27, 35, 37, 39, 49, 65

2. Find the intervals on which the function is (a) increasing, (b) decreasing, (c) concave up, (d) concave down. Then find any (e) local extreme values, and (f) inflection points. Critical values: Where does the derivative equal zero, or where is the derivative undefined?

3. Find the intervals on which the function is (a) increasing, (b) decreasing, (c) concave up, (d) concave down. Then find any (e) local extreme values, and (f) inflection points.

4. Find the intervals on which the function is (a) increasing, (b) decreasing, (c) concave up, (d) concave down. Then find any (e) local extreme values, and (f) inflection points.

5. Use the derivative of the function y = f (x) to find the point at which f has a (a) local maximum, (b) local minimum, or (c) point of inflection. Critical values: Where does the derivative equal zero, or where is the derivative undefined? (a) No local maximum (b) Local maximum (and absolute) minimum at x = –1.

6. Use the derivative of the function y = f (x) to find the point at which f has a (a) local maximum, (b) local minimum, or (c) point of inflection. (c) Points of Inflection at x = 0 and x = 2

7. Find the function with the given derivative whose graph passes through the point P.

8. Find the linearization L(x) of f (x) at x = a.

9. 35. Connecting f and f ’ The graph of f ‘ is shown in Exercise 33. Sketch a possible graph of f given that it is continuous with domain [–3, 2] and f(–3) = 0.

10. 37. Mean Value Theorem Let f (x) = xlnx. • The function is continuous on the interval [0.5, 3] and differentiable over (0.5, 3).

11. 37. Mean Value Theorem

12. 37. Mean Value Theorem (c) Secant line AB

13. 37. Mean Value Theorem (d) Tangent line parallel to AB

14. 39. Approximating Functions Let f be a function with f’ (x) = sin x2 and f (0) = –1. • Find the linearization of f at x = 0. • Approximate the value of f at x = 0.1.

15. 39. Approximating Functions Let f be a function with f’ (x) = sin x2 and f (0) = –1. (c) Is the actual value of f at x = 0.1 greater than or less than the approximation in (b)? Greater than the approximation since f ’(x) is actually positive over the interval (0, 0.1) and the estimate is based on the derivative being 0.

16. 49. Inscribing a Cylinder Find the height and radius of the largest right circular cylinder that can be put into a sphere of radius as shown.

17. 65. Estimating Change Write a formula that estimates the change that occurs in the volume of a right circular cone when the radius changes from a to a + dr and the height does not change. When the radius changes from a to a + dr, the volume change is approximately