1. Review Problems Integration 1. Find the instantaneous rate of change of the function at x = -2 _ 1

2. Review Problems Integration 2. One of these curves is the graph of a function f, another is the graph of f’, and the third is the graph of f”. Which is which? _ A is f B is f’ when C crosses the x-axis B has an extrema, so B is f’ C is f” cannot be f because it has a minimum where no other curve has a zero. Same for f’ 2 A B C

3. Review Problems Integration 3. One of these three curves represents the position of a particle moving in a straight line, another represents the particle’s velocity, and the third represents its acceleration. Which curve is which and why? Curve A is acceleration Curve B is velocity Curve C is the position Neither B nor C crosses t at the points where A has extrema, therefore, A is not acceleration, nor position, so A is s”(t) It crosses the t-axis at the point where B has an extrema and C doesn’t, so B is s’(t), therefore, C is the position function 3 A B C

4. Review Problems Integration 4. The graphs of f and g are shown. If h is defined by h(x) = f(x) g(x), find h’(1) 4 3 2 1 -2 -1 1 2 3 4 5 6 h’(x) = f’(x) g(x) + f(x) g’(x) h’(1) = f’(1) g(1) + f(1) g’(1) = 2 * 1 + 2 * -1 h’(1) = 0 4

5. Review Problems Integration x123 f(x)317 g(x)282 f’(x)457 g’(x)679 h(x) = f(g(x)) h’(x) = f’(g(x)) (g’(x)) h’(1) = f’(g(1)) (g’(1)) = [f’(2)] (6) = 5 * 6 = 30 5 5. The functions f and g are differentiable and defined for all real numbers. The function h is given by h(x) = f(g(x)). Using the values of f, g, f’ and g’ in the table, find h’(1)

6. Review Problems Integration 6. The table shows a few values of the function f and its derivative f’. If h is a function given by What is h’(-1)? 6 x01 f(x)317 f’(x)421 -

7. Review Problems Integration 7. Find the derivative of the function - 7

8. Review Problems Integration 8. Find the derivative of the function f(x)=sin(cos x) f(x) = sin(cos x) = cos(cos x) * -sin x = -sin x * cos (cos x) 8

9. Review Problems Integration 9. Find the derivative of the function _ 9

10. Review Problems Integration  Find the derivative of the function _ 10

11. Review Problems Integration 11. Find the derivative of the function _ 11

12. Review Problems Integration 12. Based on the data in the chart below, estimate by using five subintervals of equal length A. By left-hand Riemann sums Intervals: 8 + 28 + 48 + 44 + 24 = 152 15 14 13 (8,12) 12 (12,11) 11 10 9 8 7 (4,7) 6 (16,6) 5 4 3 (2,0) 2 1 4 8 12 16 20 12 tO2468101214161820 v()24791215119653 48 44 24 8 28

13. Review Problems Integration 12. Based on the data in the chart below, estimate by using five subintervals of equal length B. By Right-hand Riemann Sums Intervals: 28 + 48 + 44 + 24 + 12 = 156 15 14 13 12 (8,12) 11 (12, 11) 10 9 8 7 (4,7) 6 (16,6) 5 4 3 (20,3) 2 1 4 8 12 16 20 13 tO2468101214161820 v()24791215119653 44 24 12 28 48

14. Review Problems Integration 12. Based on the data in the chart below, estimate by using five subintervals of equal length C. By Midpoint Rule Intervals: 16 + 36 + 60 + 36 + 20 = 168 (10, 15) 15 14 13 12 (14, 9) 11 (6, 9) 10 9 8 7 (18, 5) 6 5 4 (2, 4) 3 2 1 4 8 12 16 20 14 tO2468101214161820 v()24791215119653 60 36 20 16 36

15. Review Problems Integration  Based on the midpoint rule, find an estimate of the average velocity over the time interval 0 to 20 inclusive Average Velocity = 15

16. Review Problems Integration 13. A particle moves along a number line such that its position s at any time t, t>0, is given by A. Find the average velocity over the time interval Average velocity 7 moving to the left 16

17. Review Problems Integration 13. A particle moves along a number line such that its position s at any time t, t>0, is given by B. Find the instantaneous velocity at t = 2 Instantaneous velocity Moving to the left 12 17

18. Review Problems Integration 13. A particle moves along a number line such that its position s at any time t, t>0, is given by C. When is the particle at rest? At rest when v(t) = 0 18

19. Review Problems Integration 13. A particle moves along a number line such that its position s at any time t, t>0, is given by D. What is the total distance traveled by the particle over the time interval Use the endpoints 0,5 and when particle stops 1,4 Total distance: t=0 to t=1  12-1=11 t=1 to t=4  12-(-15) = 27 t=4 to t=5  -4 – (-15) = 11 Distance = 49 t=5 s=-4 t=4 s=-15 t=0 t=1 s=1 s=12 -15 -4 0 1 12 19

20. Review Problems Integration 14. Consider the differential equation and let y = f(x) be the solution A. On the axis provided, sketch a slope field on the 14 points indicated - 1 -2 -1 0 1 2 20 x,y-2012 1-4-2024 0Inf 420-2-4

21. Review Problems Integration 14. Consider the differential equation and let y = f(x) be the solution B. For the particular solution with the initial condition f(2)= -1, write the equation of the tangent line to the graph of f at x = 2 21 at point (2,-1) equation: Y + 1 = -4(x – 2)

22. Review Problems Integration 14. Consider the differential equation and let y = f(x) be the solution C. Write the particular solution to the given differential equation with the initial condition f(1) = 1 - 22