{"id":467,"date":"2021-02-04T15:29:08","date_gmt":"2021-02-04T15:29:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=467"},"modified":"2021-04-08T15:09:56","modified_gmt":"2021-04-08T15:09:56","slug":"problem-set-differentiation-rules","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-differentiation-rules\/","title":{"raw":"Problem Set: Differentiation Rules","rendered":"Problem Set: Differentiation Rules"},"content":{"raw":"

For the following exercises (1-12), find [latex]f^{\\prime}(x)[\/latex] for each function.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]f(x)=x^7+10[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex]f(x)=5x^3-x+1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739293719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739293719\"]\r\n

[latex]f^{\\prime}(x)=15x^2-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]f(x)=4x^2-7x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>[latex]f(x)=8x^4+9x^2-1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736596010\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736596010\"]\r\n

[latex]f^{\\prime}(x)=32x^3+18x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]f(x)=x^4+\\dfrac{2}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]f(x)=3x\\left(18x^4+\\dfrac{13}{x+1}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739282715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739282715\"]\r\n

[latex]f^{\\prime}(x)=270x^4+\\frac{39}{(x+1)^2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>[latex]f(x)=(x+2)(2x^2-3)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>[latex]f(x)=x^2\\left(\\dfrac{2}{x^2}+\\dfrac{5}{x^3}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736662686\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662686\"]\r\n

[latex]f^{\\prime}(x)=\\frac{-5}{x^2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]f(x)=\\dfrac{x^3+2x^2-4}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>[latex]f(x)=\\dfrac{4x^3-2x+1}{x^2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736659021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736659021\"]\r\n

[latex]f^{\\prime}(x)=\\frac{4x^4+2x^2-2x}{x^4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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11.\u00a0<\/strong>[latex]f(x)=\\dfrac{x^2+4}{x^2-4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>[latex]f(x)=\\dfrac{x+9}{x^2-7x+1}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739284960\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739284960\"]\r\n

[latex]f^{\\prime}(x)=\\frac{\u2212x^2-18x+64}{(x^2-7x+1)^2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (13-16), find the equation of the tangent line [latex]T(x)[\/latex] to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.<\/p>\r\n\r\n

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13. [T]\u00a0<\/strong>[latex]y=3x^2+4x+1[\/latex]\u00a0 at\u00a0 [latex](0,1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [T]\u00a0<\/strong>[latex]y=2\\sqrt{x}+1[\/latex]\u00a0 at\u00a0 [latex](4,5)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739376158\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739376158\"]\"The<\/span>\r\n[latex]T(x)=\\frac{1}{2}x+3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [T]\u00a0<\/strong>[latex]y=\\dfrac{2x}{x-1}[\/latex]\u00a0 at\u00a0 [latex](-1,1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [T]\u00a0<\/strong>[latex]y=\\dfrac{2}{x}-\\dfrac{3}{x^2}[\/latex]\u00a0 at\u00a0 [latex](1,-1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739345906\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739345906\"]\"The<\/span>\r\n[latex]T(x)=4x-5[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (17-20), assume that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are both differentiable functions for all [latex]x[\/latex]. Find the derivative of each of the functions [latex]h(x)[\/latex].<\/p>\r\n\r\n

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17.\u00a0<\/strong>[latex]h(x)=4f(x)+\\dfrac{g(x)}{7}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]h(x)=x^3f(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739325536\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739325536\"]\r\n

[latex]h^{\\prime}(x)=3x^2f(x)+x^3f^{\\prime}(x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>[latex]h(x)=\\dfrac{f(x)g(x)}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]h(x)=\\dfrac{3f(x)}{g(x)+2}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739304972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739304972\"]\r\n

[latex]h^{\\prime}(x)=\\frac{3f^{\\prime}(x)(g(x)+2)-3f(x)g^{\\prime}(x)}{(g(x)+2)^2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (21-24), assume that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n1<\/td>\r\n2<\/td>\r\n3<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
[latex]f(x)[\/latex]<\/strong><\/td>\r\n3<\/td>\r\n5<\/td>\r\n-2<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
[latex]g(x)[\/latex]<\/strong><\/td>\r\n2<\/td>\r\n3<\/td>\r\n-4<\/td>\r\n6<\/td>\r\n<\/tr>\r\n
[latex]f^{\\prime}(x)[\/latex]<\/strong><\/td>\r\n-1<\/td>\r\n7<\/td>\r\n8<\/td>\r\n-3<\/td>\r\n<\/tr>\r\n
[latex]g^{\\prime}(x)[\/latex]<\/strong><\/td>\r\n4<\/td>\r\n1<\/td>\r\n2<\/td>\r\n9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
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21.\u00a0<\/strong>Find [latex]h^{\\prime}(1)[\/latex] if [latex]h(x)=xf(x)+4g(x)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>Find [latex]h^{\\prime}(2)[\/latex] if [latex]h(x)=\\dfrac{f(x)}{g(x)}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739303632\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739303632\"]\r\n

[latex]\\frac{16}{9}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>Find [latex]h^{\\prime}(3)[\/latex] if [latex]h(x)=2x+f(x)g(x)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>Find [latex]h^{\\prime}(4)[\/latex] if [latex]h(x)=\\dfrac{1}{x}+\\dfrac{g(x)}{f(x)}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739350740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739350740\"]\r\n

Undefined<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (25-27), use the following figure to find the indicated derivatives, if they exist.<\/p>\r\n\"Two<\/span>\r\n

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25.\u00a0<\/strong>Let [latex]h(x)=f(x)+g(x)[\/latex]. Find<\/p>\r\n\r\n

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  1. [latex]h^{\\prime}(1)[\/latex]<\/li>\r\n \t
  2. [latex]h^{\\prime}(3)[\/latex]<\/li>\r\n \t
  3. [latex]h^{\\prime}(4)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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    26.\u00a0<\/strong>Let [latex]h(x)=f(x)g(x)[\/latex]. Find<\/p>\r\n\r\n

      \r\n \t
    1. [latex]h^{\\prime}(1)[\/latex]<\/li>\r\n \t
    2. [latex]h^{\\prime}(3)[\/latex]<\/li>\r\n \t
    3. [latex]h^{\\prime}(4)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169736593622\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736593622\"]\r\n

      a. 2\r\nb. does not exist\r\nc. 2.5<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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      27.\u00a0<\/strong>Let [latex]h(x)=\\dfrac{f(x)}{g(x)}[\/latex]. Find<\/p>\r\n\r\n

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      1. [latex]h^{\\prime}(1)[\/latex]<\/li>\r\n \t
      2. [latex]h^{\\prime}(3)[\/latex]<\/li>\r\n \t
      3. [latex]h^{\\prime}(4)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

        For the following exercises (28-31),<\/p>\r\n\r\n

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        1. Evaluate [latex]f^{\\prime}(a)[\/latex], and<\/li>\r\n \t
        2. Graph the function [latex]f(x)[\/latex] and the tangent line at [latex]x=a[\/latex].<\/li>\r\n<\/ol>\r\n
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          28. [T]\u00a0<\/strong>[latex]f(x)=2x^3+3x-x^2, \\,\\,\\, a=2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736655171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736655171\"]\r\n

          a. 23\r\nb. [latex]y=23x-28[\/latex]<\/p>\r\n\"The<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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          29. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x}-x^2, \\,\\,\\, a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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          30. [T]\u00a0<\/strong>[latex]f(x)=x^2-x^{12}+3x+2, \\,\\,\\, a=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739305462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739305462\"]\r\n

          a. 3\r\nb. [latex]y=3x+2[\/latex]<\/p>\r\n\"The<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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          31. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x}-x^{\\frac{2}{3}}, \\,\\,\\, a=-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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          32.\u00a0<\/strong>Find the equation of the tangent line to the graph of [latex]f(x)=2x^3+4x^2-5x-3[\/latex] at [latex]x=-1[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736662292\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736662292\"]\r\n

          [latex]y=-7x-3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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          33.\u00a0<\/strong>Find the equation of the tangent line to the graph of [latex]f(x)=x^2+\\dfrac{4}{x}-10[\/latex] at [latex]x=8[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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          34.\u00a0<\/strong>Find the equation of the tangent line to the graph of [latex]f(x)=(3x-x^2)(3-x-x^2)[\/latex] at [latex]x=1[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739303404\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739303404\"]\r\n

          [latex]y=-5x+7[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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          35.\u00a0<\/strong>Find the point on the graph of [latex]f(x)=x^3[\/latex] such that the tangent line at that point has an [latex]x[\/latex] intercept of 6.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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          36.\u00a0<\/strong>Find the equation of the line passing through the point [latex]P(3,3)[\/latex] and tangent to the graph of [latex]f(x)=\\dfrac{6}{x-1}[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169739303530\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739303530\"]\r\n

          [latex]y=-\\frac{3}{2}x+\\frac{15}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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          37.\u00a0<\/strong>Determine all points on the graph of [latex]f(x)=x^3+x^2-x-1[\/latex] for which<\/p>\r\n\r\n

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          1. the tangent line is horizontal<\/li>\r\n \t
          2. the tangent line has a slope of [latex]-1[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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            38.\u00a0<\/strong>Find a quadratic polynomial such that [latex]f(1)=5, \\, f^{\\prime}(1)=3[\/latex], and [latex]f''(1)=-6[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1169736613846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736613846\"][latex]y=-3x^2+9x-1[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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            39.\u00a0<\/strong>A car driving along a freeway with traffic has traveled [latex]s(t)=t^3-6t^2+9t[\/latex] meters in [latex]t[\/latex] seconds.<\/p>\r\n\r\n

              \r\n \t
            1. Determine the time in seconds when the velocity of the car is 0.<\/li>\r\n \t
            2. Determine the acceleration of the car when the velocity is 0.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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              40. [T]<\/strong> A herring swimming along a straight line has traveled [latex]s(t)=\\dfrac{t^2}{t^2+2}[\/latex] feet in [latex]t[\/latex] seconds.<\/p>\r\n

              Determine the velocity of the herring when it has traveled 3 seconds.<\/p>\r\n[reveal-answer q=\"fs-id1169739341306\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739341306\"]\r\n

              [latex]\\frac{12}{121}[\/latex] or 0.0992 ft\/s<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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              41.\u00a0<\/strong>The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function [latex]P(t)=\\dfrac{8t+3}{0.2t^2+1}[\/latex], where [latex]t[\/latex] is measured in years.<\/p>\r\n\r\n

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              1. Determine the initial flounder population.<\/li>\r\n \t
              2. Determine [latex]P^{\\prime}(10)[\/latex] and briefly interpret the result.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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                42. [T]<\/strong> The concentration of antibiotic in the bloodstream [latex]t[\/latex] hours after being injected is given by the function [latex]C(t)=\\dfrac{2t^2+t}{t^3+50}[\/latex], where [latex]C[\/latex] is measured in milligrams per liter of blood.<\/p>\r\n\r\n

                  \r\n \t
                1. Find the rate of change of [latex]C(t)[\/latex].<\/li>\r\n \t
                2. Determine the rate of change for [latex]t=8, \\, 12, \\, 24[\/latex], and [latex]36[\/latex].<\/li>\r\n \t
                3. Briefly describe what seems to be occurring as the number of hours increases.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169739353279\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739353279\"]\r\n

                  a. [latex]\\frac{-2t^4-2t^3+200t+50}{(t^3+50)^2}[\/latex]\r\nb. -0.02395 mg\/L-hr, \u22120.01344 mg\/L-hr, \u22120.003566 mg\/L-hr, \u22120.001579 mg\/L-hr\r\nc. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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                  43.\u00a0<\/strong>A book publisher has a cost function given by [latex]C(x)=\\dfrac{x^3+2x+3}{x^2}[\/latex], where [latex]x[\/latex] is the number of copies of a book in thousands and [latex]C[\/latex] is the cost, per book, measured in dollars. Evaluate [latex]C^{\\prime}(2)[\/latex] and explain its meaning.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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                  44. [T]<\/strong> According to Newton\u2019s law of universal gravitation, the force [latex]F[\/latex] between two bodies of constant mass [latex]m_1[\/latex] and [latex]m_2[\/latex] is given by the formula [latex]F=\\dfrac{G m_1 m_2}{d^2}[\/latex], where [latex]G[\/latex] is the gravitational constant and [latex]d[\/latex] is the distance between the bodies.<\/p>\r\n\r\n

                    \r\n \t
                  1. Suppose that [latex]G, \\, m_1[\/latex], and [latex]m_2[\/latex] are constants. Find the rate of change of force [latex]F[\/latex] with respect to distance [latex]d[\/latex].<\/li>\r\n \t
                  2. Find the rate of change of force [latex]F[\/latex] with gravitational constant [latex]G=6.67 \\times 10^{-11} \\, \\text{Nm}^2\/\\text{kg}^2[\/latex], on two bodies 10 meters apart, each with a mass of 1000 kilograms.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169739307960\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739307960\"]\r\n

                    a. [latex]F^{\\prime}(d)=\\frac{-2Gm_1 m_2}{d^3}[\/latex]\r\nb. [latex]-1.33 \\times 10^{-7}[\/latex] N\/m<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

                    For the following exercises (1-12), find [latex]f^{\\prime}(x)[\/latex] for each function.<\/p>\n

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                    1.\u00a0<\/strong>[latex]f(x)=x^7+10[\/latex]<\/p>\n<\/div>\n<\/div>\n

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                    2.\u00a0<\/strong>[latex]f(x)=5x^3-x+1[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
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                    [latex]f^{\\prime}(x)=15x^2-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    3.\u00a0<\/strong>[latex]f(x)=4x^2-7x[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    4.\u00a0<\/strong>[latex]f(x)=8x^4+9x^2-1[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    [latex]f^{\\prime}(x)=32x^3+18x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    5.\u00a0<\/strong>[latex]f(x)=x^4+\\dfrac{2}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    6.\u00a0<\/strong>[latex]f(x)=3x\\left(18x^4+\\dfrac{13}{x+1}\\right)[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    [latex]f^{\\prime}(x)=270x^4+\\frac{39}{(x+1)^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    7.\u00a0<\/strong>[latex]f(x)=(x+2)(2x^2-3)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    8.\u00a0<\/strong>[latex]f(x)=x^2\\left(\\dfrac{2}{x^2}+\\dfrac{5}{x^3}\\right)[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    [latex]f^{\\prime}(x)=\\frac{-5}{x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    9.\u00a0<\/strong>[latex]f(x)=\\dfrac{x^3+2x^2-4}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    10.\u00a0<\/strong>[latex]f(x)=\\dfrac{4x^3-2x+1}{x^2}[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    [latex]f^{\\prime}(x)=\\frac{4x^4+2x^2-2x}{x^4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    11.\u00a0<\/strong>[latex]f(x)=\\dfrac{x^2+4}{x^2-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    12.\u00a0<\/strong>[latex]f(x)=\\dfrac{x+9}{x^2-7x+1}[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    [latex]f^{\\prime}(x)=\\frac{\u2212x^2-18x+64}{(x^2-7x+1)^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    For the following exercises (13-16), find the equation of the tangent line [latex]T(x)[\/latex] to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.<\/p>\n

                    \n
                    \n

                    13. [T]\u00a0<\/strong>[latex]y=3x^2+4x+1[\/latex]\u00a0 at\u00a0 [latex](0,1)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    14. [T]\u00a0<\/strong>[latex]y=2\\sqrt{x}+1[\/latex]\u00a0 at\u00a0 [latex](4,5)[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \"The<\/span>
                    \n[latex]T(x)=\\frac{1}{2}x+3[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
                    \n
                    \n

                    15. [T]\u00a0<\/strong>[latex]y=\\dfrac{2x}{x-1}[\/latex]\u00a0 at\u00a0 [latex](-1,1)[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    16. [T]\u00a0<\/strong>[latex]y=\\dfrac{2}{x}-\\dfrac{3}{x^2}[\/latex]\u00a0 at\u00a0 [latex](1,-1)[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \"The<\/span>
                    \n[latex]T(x)=4x-5[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    For the following exercises (17-20), assume that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are both differentiable functions for all [latex]x[\/latex]. Find the derivative of each of the functions [latex]h(x)[\/latex].<\/p>\n

                    \n
                    \n

                    17.\u00a0<\/strong>[latex]h(x)=4f(x)+\\dfrac{g(x)}{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    18.\u00a0<\/strong>[latex]h(x)=x^3f(x)[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    [latex]h^{\\prime}(x)=3x^2f(x)+x^3f^{\\prime}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    19.\u00a0<\/strong>[latex]h(x)=\\dfrac{f(x)g(x)}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    20.\u00a0<\/strong>[latex]h(x)=\\dfrac{3f(x)}{g(x)+2}[\/latex]<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    [latex]h^{\\prime}(x)=\\frac{3f^{\\prime}(x)(g(x)+2)-3f(x)g^{\\prime}(x)}{(g(x)+2)^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    For the following exercises (21-24), assume that [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.<\/p>\n\n\n\n\n\n\n\n
                    [latex]x[\/latex]<\/strong><\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n4<\/td>\n<\/tr>\n
                    [latex]f(x)[\/latex]<\/strong><\/td>\n3<\/td>\n5<\/td>\n-2<\/td>\n0<\/td>\n<\/tr>\n
                    [latex]g(x)[\/latex]<\/strong><\/td>\n2<\/td>\n3<\/td>\n-4<\/td>\n6<\/td>\n<\/tr>\n
                    [latex]f^{\\prime}(x)[\/latex]<\/strong><\/td>\n-1<\/td>\n7<\/td>\n8<\/td>\n-3<\/td>\n<\/tr>\n
                    [latex]g^{\\prime}(x)[\/latex]<\/strong><\/td>\n4<\/td>\n1<\/td>\n2<\/td>\n9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                    \n
                    \n

                    21.\u00a0<\/strong>Find [latex]h^{\\prime}(1)[\/latex] if [latex]h(x)=xf(x)+4g(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    22.\u00a0<\/strong>Find [latex]h^{\\prime}(2)[\/latex] if [latex]h(x)=\\dfrac{f(x)}{g(x)}[\/latex].<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    [latex]\\frac{16}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    23.\u00a0<\/strong>Find [latex]h^{\\prime}(3)[\/latex] if [latex]h(x)=2x+f(x)g(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n

                    \n
                    \n

                    24.\u00a0<\/strong>Find [latex]h^{\\prime}(4)[\/latex] if [latex]h(x)=\\dfrac{1}{x}+\\dfrac{g(x)}{f(x)}[\/latex].<\/p>\n

                    Show Solution<\/span><\/p>\n
                    \n

                    Undefined<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                    For the following exercises (25-27), use the following figure to find the indicated derivatives, if they exist.<\/p>\n

                    \"Two<\/span><\/p>\n

                    \n
                    \n

                    25.\u00a0<\/strong>Let [latex]h(x)=f(x)+g(x)[\/latex]. Find<\/p>\n

                      \n
                    1. [latex]h^{\\prime}(1)[\/latex]<\/li>\n
                    2. [latex]h^{\\prime}(3)[\/latex]<\/li>\n
                    3. [latex]h^{\\prime}(4)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                      \n
                      \n

                      26.\u00a0<\/strong>Let [latex]h(x)=f(x)g(x)[\/latex]. Find<\/p>\n

                        \n
                      1. [latex]h^{\\prime}(1)[\/latex]<\/li>\n
                      2. [latex]h^{\\prime}(3)[\/latex]<\/li>\n
                      3. [latex]h^{\\prime}(4)[\/latex]<\/li>\n<\/ol>\n
                        Show Solution<\/span><\/p>\n
                        \n

                        a. 2
                        \nb. does not exist
                        \nc. 2.5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                        \n
                        \n

                        27.\u00a0<\/strong>Let [latex]h(x)=\\dfrac{f(x)}{g(x)}[\/latex]. Find<\/p>\n

                          \n
                        1. [latex]h^{\\prime}(1)[\/latex]<\/li>\n
                        2. [latex]h^{\\prime}(3)[\/latex]<\/li>\n
                        3. [latex]h^{\\prime}(4)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                          For the following exercises (28-31),<\/p>\n

                            \n
                          1. Evaluate [latex]f^{\\prime}(a)[\/latex], and<\/li>\n
                          2. Graph the function [latex]f(x)[\/latex] and the tangent line at [latex]x=a[\/latex].<\/li>\n<\/ol>\n
                            \n
                            \n

                            28. [T]\u00a0<\/strong>[latex]f(x)=2x^3+3x-x^2, \\,\\,\\, a=2[\/latex]<\/p>\n

                            Show Solution<\/span><\/p>\n
                            \n

                            a. 23
                            \nb. [latex]y=23x-28[\/latex]<\/p>\n

                            \"The<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            29. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x}-x^2, \\,\\,\\, a=1[\/latex]<\/p>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            30. [T]\u00a0<\/strong>[latex]f(x)=x^2-x^{12}+3x+2, \\,\\,\\, a=0[\/latex]<\/p>\n

                            Show Solution<\/span><\/p>\n
                            \n

                            a. 3
                            \nb. [latex]y=3x+2[\/latex]<\/p>\n

                            \"The<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            31. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x}-x^{\\frac{2}{3}}, \\,\\,\\, a=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            32.\u00a0<\/strong>Find the equation of the tangent line to the graph of [latex]f(x)=2x^3+4x^2-5x-3[\/latex] at [latex]x=-1[\/latex].<\/p>\n

                            Show Solution<\/span><\/p>\n
                            \n

                            [latex]y=-7x-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            33.\u00a0<\/strong>Find the equation of the tangent line to the graph of [latex]f(x)=x^2+\\dfrac{4}{x}-10[\/latex] at [latex]x=8[\/latex].<\/p>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            34.\u00a0<\/strong>Find the equation of the tangent line to the graph of [latex]f(x)=(3x-x^2)(3-x-x^2)[\/latex] at [latex]x=1[\/latex].<\/p>\n

                            Show Solution<\/span><\/p>\n
                            \n

                            [latex]y=-5x+7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            35.\u00a0<\/strong>Find the point on the graph of [latex]f(x)=x^3[\/latex] such that the tangent line at that point has an [latex]x[\/latex] intercept of 6.<\/p>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            36.\u00a0<\/strong>Find the equation of the line passing through the point [latex]P(3,3)[\/latex] and tangent to the graph of [latex]f(x)=\\dfrac{6}{x-1}[\/latex].<\/p>\n

                            Show Solution<\/span><\/p>\n
                            \n

                            [latex]y=-\\frac{3}{2}x+\\frac{15}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                            \n
                            \n

                            37.\u00a0<\/strong>Determine all points on the graph of [latex]f(x)=x^3+x^2-x-1[\/latex] for which<\/p>\n

                              \n
                            1. the tangent line is horizontal<\/li>\n
                            2. the tangent line has a slope of [latex]-1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                              \n
                              \n

                              38.\u00a0<\/strong>Find a quadratic polynomial such that [latex]f(1)=5, \\, f^{\\prime}(1)=3[\/latex], and [latex]f''(1)=-6[\/latex].<\/p>\n

                              Show Solution<\/span><\/p>\n
                              [latex]y=-3x^2+9x-1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
                              \n
                              \n

                              39.\u00a0<\/strong>A car driving along a freeway with traffic has traveled [latex]s(t)=t^3-6t^2+9t[\/latex] meters in [latex]t[\/latex] seconds.<\/p>\n

                                \n
                              1. Determine the time in seconds when the velocity of the car is 0.<\/li>\n
                              2. Determine the acceleration of the car when the velocity is 0.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                \n
                                \n

                                40. [T]<\/strong> A herring swimming along a straight line has traveled [latex]s(t)=\\dfrac{t^2}{t^2+2}[\/latex] feet in [latex]t[\/latex] seconds.<\/p>\n

                                Determine the velocity of the herring when it has traveled 3 seconds.<\/p>\n

                                Show Solution<\/span><\/p>\n
                                \n

                                [latex]\\frac{12}{121}[\/latex] or 0.0992 ft\/s<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                41.\u00a0<\/strong>The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function [latex]P(t)=\\dfrac{8t+3}{0.2t^2+1}[\/latex], where [latex]t[\/latex] is measured in years.<\/p>\n

                                  \n
                                1. Determine the initial flounder population.<\/li>\n
                                2. Determine [latex]P^{\\prime}(10)[\/latex] and briefly interpret the result.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                  \n
                                  \n

                                  42. [T]<\/strong> The concentration of antibiotic in the bloodstream [latex]t[\/latex] hours after being injected is given by the function [latex]C(t)=\\dfrac{2t^2+t}{t^3+50}[\/latex], where [latex]C[\/latex] is measured in milligrams per liter of blood.<\/p>\n

                                    \n
                                  1. Find the rate of change of [latex]C(t)[\/latex].<\/li>\n
                                  2. Determine the rate of change for [latex]t=8, \\, 12, \\, 24[\/latex], and [latex]36[\/latex].<\/li>\n
                                  3. Briefly describe what seems to be occurring as the number of hours increases.<\/li>\n<\/ol>\n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    a. [latex]\\frac{-2t^4-2t^3+200t+50}{(t^3+50)^2}[\/latex]
                                    \nb. -0.02395 mg\/L-hr, \u22120.01344 mg\/L-hr, \u22120.003566 mg\/L-hr, \u22120.001579 mg\/L-hr
                                    \nc. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    43.\u00a0<\/strong>A book publisher has a cost function given by [latex]C(x)=\\dfrac{x^3+2x+3}{x^2}[\/latex], where [latex]x[\/latex] is the number of copies of a book in thousands and [latex]C[\/latex] is the cost, per book, measured in dollars. Evaluate [latex]C^{\\prime}(2)[\/latex] and explain its meaning.<\/p>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    44. [T]<\/strong> According to Newton\u2019s law of universal gravitation, the force [latex]F[\/latex] between two bodies of constant mass [latex]m_1[\/latex] and [latex]m_2[\/latex] is given by the formula [latex]F=\\dfrac{G m_1 m_2}{d^2}[\/latex], where [latex]G[\/latex] is the gravitational constant and [latex]d[\/latex] is the distance between the bodies.<\/p>\n

                                      \n
                                    1. Suppose that [latex]G, \\, m_1[\/latex], and [latex]m_2[\/latex] are constants. Find the rate of change of force [latex]F[\/latex] with respect to distance [latex]d[\/latex].<\/li>\n
                                    2. Find the rate of change of force [latex]F[\/latex] with gravitational constant [latex]G=6.67 \\times 10^{-11} \\, \\text{Nm}^2\/\\text{kg}^2[\/latex], on two bodies 10 meters apart, each with a mass of 1000 kilograms.<\/li>\n<\/ol>\n
                                      Show Solution<\/span><\/p>\n
                                      \n

                                      a. [latex]F^{\\prime}(d)=\\frac{-2Gm_1 m_2}{d^3}[\/latex]
                                      \nb. [latex]-1.33 \\times 10^{-7}[\/latex] N\/m<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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