1. Limit & Derivative Problems Problem…Answer and Work… 1. 1. 1

2. Limit & Derivative Problems Problem…Answer and Work… 2. 2. 2

3. Limit & Derivative Problems Problem…Answer and Work… 3. 3. 3

4. Limit & Derivative Problems Problem…Answer and Work… 4.Consider the function given by Is f(x) continuous at x=1? Justify. 4. 4

5. Limit & Derivative Problems Problem…Answer and Work… 5.Is the function given by continuous for all x? If not, where are the discontinuities? Are they removable? 5. 5

6. Limit & Derivative Problems Problem…Answer and Work… 6.Let the piecewise function f be defined as follows: Which of the following is true about the function f? I. f(2) = 2 II. III. f(x) is continuous at x = 2 A. I only B. III only C. I and II only D. I and III only E. I, II, and III 6. Test: f(2) = 2? Yes, so I is true Test: Test: f(x) is continuous at x = 2? Does the lim f(x) = f(2)? 4 is not equal to 2 No, so III is false Answer is A) I only 6

7. Limit & Derivative Problems Problem…Answer and Work… 7. What is the value of a for which f(x) is continuous for all values of x? A. -2 B. -1 C. 0 D. ½ E. 1 7. To be continuous at x = 1 7

8. Limit & Derivative Problems Problem…Answer and Work… 8. Find the cartesian coordinates of the point on the graph of where the instantaneous rate of change of f is equal to 5 8. to find y substitute x = ½ in the original function f(x) Ans: (1/2, 11/4) 8

9. Limit & Derivative Problems Problem…Answer and Work… 9. Which of the following directly describes the discontinuities associated with a. A hole at x = 3, a vertical asymptote at x = 3 b. Holes at x = -3 and x = 3 c. A hole at x = 3, a vertical asymptote at x = -3 d. Vertical asymptotes at x = 3 and x = -3 e. No discontinuities 9. Hole at x = 3 because we factored out (x – 3) There is a vertical asymptote at x = -3 9

10. Limit & Derivative Problems Problem…Answer and Work… 10. Given the piecewise function For what values of a and b is f(x) differentiable at x = 1? A. a = 2 b = -3 B. a = 2 b = -2 C. a = -2 b = 1 D. a = 3 b = -1 E. a = 5 b = 8 10. Differentiability implies continuity To be differentiable x = 1 Solve for a when b = 1 a – 1 = -3 a = -2 Ans: C 10

11. Limit & Derivative Problems Problem…Answer and Work… 11. Which of the following is (are) true about the function I. It is continuous at x = 0 II. It is differentiable at x = 0 III. A. I only B. II only C. I and III only D. II and III only E. I, II, III 11. Test 1: Continuous at x = 0 yes Test 2: Differentiable at x = 0? No Test 3: Yes Ans: C 11

12. Limit & Derivative Problems Problem…Answer and Work… 12. To apply either the Mean Value Theorem or Rolle’s Theorem to a function f, certain requirements regarding the continuity and differentiability of the function must be met. Which of the following states the requirements correctly? A. f is continuous on (a, b) and differentiable on (a, b) B. f is continuous on (a, b) and differentiable on [a, b] C. f is continuous on (a, b) and differentiable on [a, b) D. f is continuous on [a, b] and differentiable on (a, b) E. f is continuous on [a, b] and differentiable on [a, b] 12. Look at the definition of Rolle’s Theorem and the Mean Value Theorem f is continuous on [a, b] and differentiable on (a, b) Ans: D 12

13. Limit & Derivative Problems Problem…Answer and Work… 13. Let f be the function defined by A. Determine the x and y intercepts, if any. Justify your answer. 13. A 13

14. Limit & Derivative Problems Problem…Answer and Work… 13. Let f be the function defined by B. Write an equation for each vertical and each horizontal asymptote. Justify your answer. 13. B Vertical asymptote Horizontal asymptote 14

15. Limit & Derivative Problems Problem…Answer and Work… 13. Let f be the function defined by C. Determine the intervals on which f is increasing or decreasing. Justify your answer. 13. C 15

16. Limit & Derivative Problems Problem…Answer and Work… 13. Let f be the function defined by D. Determine the relative minimum and maximum points, if any. Justify your answer. 13. D Relative minimum occurs at x = -2 when x = -2 16

17. Limit & Derivative Problems Problem…Answer and Work… 13. Let f be the function defined by E. Determine the intervals on which f is concave up or concave down. Justify your answer. 13. E 17

18. Limit & Derivative Problems Problem…Answer and Work… 13. Let f be the function defined by F. Determine any points of inflection 13. F Point of inflection when x = 3 18

19. Limit & Derivative Problems Problem…Answer and Work… 14. On the interval [1, 3], what is the average rate of change for the functions, if 14. 19

20. Limit & Derivative Problems Problem…Answer and Work… 15. Is the function defined by continuous at x = 4? Justify your answer. 15. 20