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Review Problems Integration

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

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Review Problems Integration

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

  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’ 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 A B C

  4. Review Problems Integration • 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

  5. Review Problems Integration 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. 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)? -

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

  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)

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

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

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

  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 48 44 24 28 8

  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 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 60 36 20 36 16

  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 =

  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 • Find the average velocity over the time interval Average velocity 7 moving to the left

  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

  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

  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

  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 -1

  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 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 -

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