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

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

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

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) h’(x) = f’(x) g(x) + f(x) g’(x) h’(1) = f’(1) g(1) + f(1) g’(1) = 2 * * -1 h’(1) = 0 4

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 = 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)

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 -

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

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

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

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

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

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: = (8,12) 12 (12,11) (4,7) 6 (16,6) (2,0) tO v()

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: = (8,12) 11 (12, 11) (4,7) 6 (16,6) (20,3) tO v()

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: = 168 (10, 15) (14, 9) 11 (6, 9) (18, 5) (2, 4) tO v()

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

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

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

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

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=

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 x,y Inf

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)

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) =