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Mathematics LibreTexts

1.3.3E: Rates of Change and Behavior of Graphs

  • Page ID
    30240
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    SECTION 1.3 EXERCISE

    1. The table below gives the annual sales (in millions of dollars) of a product. What was the average rate of change of annual sales

      a) Between 2001 and 2002?
      b) Between 2001 and 2004?
      year 1998 1999 2000 2001 2002 2003 2004 2005 2006
      sales 201 219 233 243 249 251 249 243 233
    2. The table below gives the population of a town, in thousands. What was the average rate of change of population

      a) Between 2002 and 2004?
      b) Between 2002 and 2006?
      year 2000 2001 2002 2003 2004 2005 2006 2007 2008
      population 87 84 83 80 77 76 75 78 81
    3. Based on the graph shown, estimate the average rate of change from \(x = 1\) to \(x = 4\).屏幕快照 2019-06-09 下午8.02.50.png
    4. Based on the graph shown, estimate the average rate of change from \(x = 2\) to \(x = 5\).







      Find the average rate of change of each function on the interval specified.
    5. \(f(x)=x^{2}\) on [1, 5]
    6. \(q(x)=x^{3}\) on [-4, 2]
    7. \(g(x)=3x^{3} -1\) on [-3, 3]
    8. \(h(x)=5 - 2x^{2}\) on [-2, 4]
    9. \(k(t)=6t^{2} +\dfrac{4}{t^{3} }\) on [-1, 3]
    10. \(p(t)=\dfrac{t^{2} - 4t + 1}{t^{2} + 3}\) on [-3, 1]

      Find the average rate of change of each function on the interval specified. Your answers will be expressions involving a parameter (\(b\) or \(h\)).
    11. \(f(x)= 4x^{2} -7\) on [1, \(b\)]
    12. \(g(x)= 2x^{2} -9\) on [4, \(b\)]
    13. \(h(x)= 3x + 4\) on [2, 2 + \(h\)]
    14. \(k(x)= 4x - 2\) on [3, 3 + \(h\)]
    15. \(a(t)=\dfrac{1}{t + 4}\) on [9, 9 + \(h\)]
    16. \(b(x)=\dfrac{1}{x + 3}\) on [1, 1 + \(h\)]
    17. \(j(x)=3x^{3}\) on [1, 1 + \(h\)]
    18. \(r(t)=4t^{3}\) on [2, 2 + \(h\)]
    19. \(f(x)=2x^{2} + 1\) on [\(x\), \(x + h\)]
    20. \(g(x)=3x^{2} - 2\) on [\(x\), \(x + h\)]

      For each function graphed, estimate the intervals on which the function is increasing and decreasing.
    21. 屏幕快照 2019-06-09 下午8.13.25.png
    22. 屏幕快照 2019-06-09 下午8.13.46.png
    23. 屏幕快照 2019-06-09 下午8.15.37.png
    24. 屏幕快照 2019-06-09 下午8.16.18.png

    For each table below, select whether the table represents a function that is increasing or decreasing, and whether the function is concave up or concave down.

    屏幕快照 2019-06-09 下午8.18.21.png

    For each function graphed, estimate the intervals on which the function is concave up and concave down, and the location of any inflection points.

    33. 屏幕快照 2019-06-09 下午8.19.37.png

    34. 屏幕快照 2019-06-09 下午8.20.03.png

    35. 屏幕快照 2019-06-09 下午8.20.28.png

    36. 屏幕快照 2019-06-09 下午8.20.52.png

    Use a graph to estimate the local extrema and inflection points of each function, and to estimate the intervals on which the function is increasing, decreasing, concave up, and concave down.

    37. \(f(x) = x^4 - 4x^3 + 5\)

    38. \(h(x) = x^5 + 5x^4 + 10x^3 + 10x^2 - 1\)

    39. \(g(t) = t \sqrt{t + 3}\)

    40. \(k(t) = 3t^{\dfrac{2}{3}} - t\)

    41. \(m(x) = x^4 + 2x^3 - 12x^2 - 10x + 4\)

    42. \(n(x) = x^4 - 8x^3 + 18x^2 - 6x + 2\)

    Answer

    1. a) 6 million dollars per year
    b) 2 million dollars per year

    3. \(\dfrac{4 - 5}{4 - 1} = -\dfrac{1}{3}\)

    5. 6

    7. 27

    9. \(\dfrac{352}{27}\)

    11. \(4b + 4\)

    13. 3

    15. \(-\dfrac{1}{13h + 169}\)

    17. \(9 + 9h + 3h^2\)

    19. \(4x + 2h\)

    21. Increasing: (-1.5, 2). Decreasing: \((-\infty, -1.4) \cup (2, \infty)\)

    23. Increasing: \((-\infty, 1) \cup (3,4)\). Decreasing: \((1, 3) \cup (4, \infty)\)

    25. Increasing, concave up

    27. Decreasing, concave down

    29. Decreasing, concave up

    31. Increasing, concave down

    33. Concave up \((-\infty, 1)\). Concave down \((1, \infty)\). Inflection point at (1, 2)

    Screen Shot 2019-08-20 at 11.41.13 AM.png35. Concave down \(-\infty, 3) \cup (3, \infty)\)

    37. Local minimum at (3, -22).
    Inflection points at (0, 5) and (2, -11).
    Increasing on \(3, \infty)\). Decreasing \((-\infty, 3)\)
    Concave up \((-\infty, 0) \cup (2, \infty)\). Concave down (0, 2)

    39. Local minimum at \((-2, -2)\)Screen Shot 2019-10-01 at 8.53.49 AM.png
    Decreasing (-3, -2)
    Increasing \((-2, \infty)\)
    Concave up \((-3, \infty)\)

     

     

     

    41. Local minimums at (-3.152, -47.626) and (2.041, -32.041)Screen Shot 2019-10-01 at 8.54.26 AM.png
    Local maximum at (-0.389, 5.979)
    Inflection points at (-2, -24) and (1, -15)
    Increasing \((-3.152, -0.389) \cup (2.041, \infty)\)
    Decreasing \((-\infty, -3.152) \cup (-0.389, 2.041)\)
    Concave up \((-\infty, -2) \cup (1, \infty)\)
    Concave down (-2, 1