{"id":465,"date":"2021-02-04T15:28:58","date_gmt":"2021-02-04T15:28:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=465"},"modified":"2021-04-08T18:11:37","modified_gmt":"2021-04-08T18:11:37","slug":"problem-set-defining-the-derivative","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-defining-the-derivative\/","title":{"raw":"Problem Set: Defining the Derivative","rendered":"Problem Set: Defining the Derivative"},"content":{"raw":"

For the following exercises (1-10), use the definition of a derivative to find the slope of the secant line between the values [latex]x_1[\/latex] and [latex]x_2[\/latex] for each function [latex]y=f(x)[\/latex].<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]f(x)=4x+7; \\,\\,\\, x_1=2, \\,\\,\\, x_2=5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739301851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739301851\"]\r\n

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

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2.\u00a0<\/strong>[latex]f(x)=8x-3; \\,\\,\\, x_1=-1, \\,\\,\\, x_2=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]f(x)=x^2+2x+1; \\,\\,\\, x_1=3, \\,\\,\\, x_2=3.5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739302801\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739302801\"]\r\n

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

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4.\u00a0<\/strong>[latex]f(x)=\\text{\u2212}{x}^{2}+x+2;{x}_{1}=0.5,{x}_{2}=1.5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]f(x)=\\dfrac{4}{3x-1}; \\,\\,\\, x_1=1, \\,\\,\\, x_2=3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739304630\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739304630\"]\r\n

[latex]-\\frac{3}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]f(x)=\\dfrac{x-7}{2x+1}; \\,\\,\\, x_1=0, \\,\\,\\, x_2=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>[latex]f(x)=\\sqrt{x}; \\,\\,\\, x_1=1, \\,\\,\\, x_2=16[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736613358\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736613358\"]\r\n

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

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8.\u00a0<\/strong>[latex]f(x)=\\sqrt{x-9}; \\,\\,\\, x_1=10, \\,\\,\\, x_2=13[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]f(x)=x^{\\frac{1}{3}}+1; \\,\\,\\, x_1=0, \\,\\,\\, x_2=8[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736594914\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736594914\"]\r\n

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

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10.\u00a0<\/strong>[latex]f(x)=6x^{\\frac{2}{3}}+2x^{\\frac{1}{3}}; \\,\\,\\, x_1=1, \\,\\,\\, x_2=27[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following functions (11-20),<\/p>\r\n\r\n

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  1. Use\u00a0[latex]m_{\\tan}=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex] to find the slope of the tangent line [latex]m_{\\tan}=f^{\\prime}(a)[\/latex], and<\/li>\r\n \t
  2. find the equation of the tangent line to [latex]f[\/latex] at [latex]x=a[\/latex].<\/li>\r\n<\/ol>\r\n
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    11.\u00a0<\/strong>[latex]f(x)=3-4x, \\,\\,\\, a=2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736610087\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736610087\"]\r\n

    a. -4 b. [latex]y=3-4x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    12.\u00a0<\/strong>[latex]f(x)=\\dfrac{x}{5}+6, \\,\\,\\, a=-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    13.\u00a0<\/strong>[latex]f(x)=x^2+x, \\,\\,\\, a=1[\/latex]<\/p>\r\n\r\n

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    a. 3 b. [latex]y=3x-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    14.\u00a0<\/strong>[latex]f(x)=1-x-x^2, \\,\\,\\, a=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    15.\u00a0<\/strong>[latex]f(x)=\\dfrac{7}{x}, \\,\\,\\, a=3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739302451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739302451\"]\r\n

    a. [latex]\\frac{-7}{9}[\/latex] b. [latex]y=\\frac{-7}{9}x+\\frac{14}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    16.\u00a0<\/strong>[latex]f(x)=\\sqrt{x+8}, \\,\\,\\, a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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    a. 12 b. [latex]y=12x+14[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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    18.\u00a0<\/strong>[latex]f(x)=\\dfrac{-3}{x-1}, \\,\\,\\, a=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    19.\u00a0<\/strong>[latex]f(x)=\\dfrac{2}{x+3}, \\,\\,\\, a=-4[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739327268\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739327268\"]\r\n

    a. -2 b. [latex]y=-2x-10[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

    For the following functions [latex]y=f(x)[\/latex] (21-30), find [latex]f^{\\prime}(a)[\/latex] using\u00a0[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex].<\/p>\r\n\r\n

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    21.\u00a0<\/strong>[latex]f(x)=5x+4, \\,\\,\\, a=-1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739270690\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739270690\"]\r\n

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

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    22.\u00a0<\/strong>[latex]f(x)=-7x+1, \\,\\,\\, a=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    23.\u00a0<\/strong>[latex]f(x)=x^2+9x, \\,\\,\\, a=2[\/latex]<\/p>\r\n\r\n

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

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    24.\u00a0<\/strong>[latex]f(x)=3x^2-x+2, \\,\\,\\, a=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    25.\u00a0<\/strong>[latex]f(x)=\\sqrt{x}, \\,\\,\\, a=4[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739194786\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739194786\"]\r\n

    [latex]\\frac{1}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    26.\u00a0<\/strong>[latex]f(x)=\\sqrt{x-2}, \\,\\,\\, a=6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    27.\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x}, \\,\\,\\, a=2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739304208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739304208\"]\r\n

    [latex]-\\frac{1}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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    29.\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x^3}, \\,\\,\\, a=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736611421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736611421\"]\r\n

    -3<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    30.\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{\\sqrt{x}}, \\,\\,\\, a=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (31-34), given the function [latex]y=f(x)[\/latex],<\/p>\r\n\r\n

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    1. find the slope of the secant line [latex]PQ[\/latex] for each point [latex]Q(x,f(x))[\/latex] with [latex]x[\/latex] value given in the table.<\/li>\r\n \t
    2. Use the answers from a. to estimate the value of the slope of the tangent line at [latex]P[\/latex].<\/li>\r\n \t
    3. Use the answer from b. to find the equation of the tangent line to [latex]f[\/latex] at point [latex]P[\/latex].<\/li>\r\n<\/ol>\r\n
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      31. [T]\u00a0<\/strong>[latex]f(x)=x^2+3x+4, \\,\\,\\, P(1,8)[\/latex] (Round to 6 decimal places.)<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      [latex]x[\/latex]<\/th>\r\nSlope [latex]m_{PQ}[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\nSlope [latex]m_{PQ}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      1.1<\/td>\r\n(i)<\/td>\r\n0.9<\/td>\r\n(vii)<\/td>\r\n<\/tr>\r\n
      1.01<\/td>\r\n(ii)<\/td>\r\n0.99<\/td>\r\n(viii)<\/td>\r\n<\/tr>\r\n
      1.001<\/td>\r\n(iii)<\/td>\r\n0.999<\/td>\r\n(ix)<\/td>\r\n<\/tr>\r\n
      1.0001<\/td>\r\n(iv)<\/td>\r\n0.9999<\/td>\r\n(x)<\/td>\r\n<\/tr>\r\n
      1.00001<\/td>\r\n(v)<\/td>\r\n0.99999<\/td>\r\n(xi)<\/td>\r\n<\/tr>\r\n
      1.000001<\/td>\r\n(vi)<\/td>\r\n0.999999<\/td>\r\n(xii)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1169739269416\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739269416\"]\r\n

      a. (i) [latex]5.100000[\/latex], (ii) [latex]5.010000[\/latex], (iii) [latex]5.001000[\/latex], (iv) [latex]5.000100[\/latex], (v) [latex]5.000010[\/latex], (vi) [latex]5.000001[\/latex],\r\n(vii) [latex]4.900000[\/latex], (viii) [latex]4.990000[\/latex], (ix) [latex]4.999000[\/latex], (x) [latex]4.999900[\/latex], (xi) [latex]4.999990[\/latex], (xii) [latex]4.999999[\/latex]<\/p>\r\nb. [latex]m_{\\tan}=5[\/latex]\r\nc. [latex]y=5x+3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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      32. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{x+1}{x^2-1}, \\,\\,\\, P(0,-1)[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      [latex]x[\/latex]<\/th>\r\nSlope [latex]m_{PQ}[\/latex]<\/th>\r\n[latex]x[\/latex]<\/th>\r\nSlope [latex]m_{PQ}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      0.1<\/td>\r\n(i)<\/td>\r\n-0.1<\/td>\r\n(vii)<\/td>\r\n<\/tr>\r\n
      0.01<\/td>\r\n(ii)<\/td>\r\n-0.01<\/td>\r\n(viii)<\/td>\r\n<\/tr>\r\n
      0.001<\/td>\r\n(iii)<\/td>\r\n-0.001<\/td>\r\n(ix)<\/td>\r\n<\/tr>\r\n
      0.0001<\/td>\r\n(iv)<\/td>\r\n-0.0001<\/td>\r\n(x)<\/td>\r\n<\/tr>\r\n
      0.00001<\/td>\r\n(v)<\/td>\r\n-0.00001<\/td>\r\n(xi)<\/td>\r\n<\/tr>\r\n
      0.000001<\/td>\r\n(vi)<\/td>\r\n-0.000001<\/td>\r\n(xii)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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      33. [T]\u00a0<\/strong>[latex]f(x)=10e^{0.5x}, \\,\\,\\, P(0,10)[\/latex] (Round to 4 decimal places.)<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      [latex]x[\/latex]<\/th>\r\nSlope [latex]m_{PQ}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      -0.1<\/td>\r\n(i)<\/td>\r\n<\/tr>\r\n
      -0.01<\/td>\r\n(ii)<\/td>\r\n<\/tr>\r\n
      -0.001<\/td>\r\n(iii)<\/td>\r\n<\/tr>\r\n
      -0.0001<\/td>\r\n(iv)<\/td>\r\n<\/tr>\r\n
      -0.00001<\/td>\r\n(v)<\/td>\r\n<\/tr>\r\n
      \u22120.000001<\/td>\r\n(vi)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1169739305302\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739305302\"]\r\n

      a. (i) [latex]4.8771[\/latex], (ii) [latex]4.9875[\/latex], (iii) [latex]4.9988[\/latex], (iv) [latex]4.9999[\/latex], (v) [latex]4.9999[\/latex], (vi) [latex]4.9999[\/latex]\r\nb. [latex]m_{\\tan}=5[\/latex]\r\nc. [latex]y=5x+10[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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      34. [T]\u00a0<\/strong>[latex]f(x)= \\tan (x), \\,\\,\\, P(\\pi,0)[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      [latex]x[\/latex]<\/th>\r\nSlope [latex]m_{PQ}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      3.1<\/td>\r\n(i)<\/td>\r\n<\/tr>\r\n
      3.14<\/td>\r\n(ii)<\/td>\r\n<\/tr>\r\n
      3.141<\/td>\r\n(iii)<\/td>\r\n<\/tr>\r\n
      3.1415<\/td>\r\n(iv)<\/td>\r\n<\/tr>\r\n
      3.14159<\/td>\r\n(v)<\/td>\r\n<\/tr>\r\n
      3.141592<\/td>\r\n(vi)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n

      For the following position functions [latex]y=s(t)[\/latex], an object is moving along a straight line, where [latex]t[\/latex] is in seconds and [latex]s[\/latex] is in meters. Find<\/p>\r\n\r\n

        \r\n \t
      1. the simplified expression for the average velocity from [latex]t=2[\/latex] to [latex]t=2+h[\/latex];<\/li>\r\n \t
      2. the average velocity between [latex]t=2[\/latex] and [latex]t=2+h[\/latex], where (i) [latex]h=0.1[\/latex], (ii) [latex]h=0.01[\/latex], (iii) [latex]h=0.001[\/latex], and (iv) [latex]h=0.0001[\/latex]; and<\/li>\r\n \t
      3. use the answer from a. to estimate the instantaneous velocity at [latex]t=2[\/latex] seconds.<\/li>\r\n<\/ol>\r\n
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        35. [T]\u00a0<\/strong>[latex]s(t)=\\frac{1}{3}t+5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736613850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736613850\"]\r\n

        a. [latex]\\frac{1}{3}[\/latex];\r\nb. (i) [latex]0.\\bar{3}[\/latex] m\/s, (ii) [latex]0.\\bar{3}[\/latex] m\/s, (iii) [latex]0.\\bar{3}[\/latex] m\/s, (iv) [latex]0.\\bar{3}[\/latex] m\/s;\r\nc. [latex]0.\\bar{3}=\\frac{1}{3}[\/latex] m\/s<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        36. [T]\u00a0<\/strong>[latex]s(t)=t^2-2t[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        37. [T]\u00a0<\/strong>[latex]s(t)=2t^3+3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736617696\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736617696\"]\r\n

        a. [latex]2(h^2+6h+12)[\/latex];\r\nb. (i) 25.22 m\/s, (ii) 24.12 m\/s, (iii) 24.01 m\/s, (iv) 24 m\/s;\r\nc. 24 m\/s<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        38. [T]\u00a0<\/strong>[latex]s(t)=\\dfrac{16}{t^2}-\\dfrac{4}{t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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        39.\u00a0<\/strong>Use the following graph to evaluate a. [latex]f^{\\prime}(1)[\/latex] and b. [latex]f^{\\prime}(6)[\/latex].<\/p>\r\n\"This<\/span>\r\n\r\n[reveal-answer q=\"fs-id1169739300016\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739300016\"]\r\n\r\na. [latex]1.25[\/latex]; b. 0.5\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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        40.\u00a0<\/strong>Use the following graph to evaluate a. [latex]f^{\\prime}(-3)[\/latex] and b. [latex]f^{\\prime}(1.5)[\/latex].<\/p>\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

        For the following exercises, use the limit definition of derivative to show that the derivative does not exist at [latex]x=a[\/latex] for each of the given functions.<\/p>\r\n\r\n

        \r\n
        \r\n

        41.\u00a0<\/strong>[latex]f(x)=x^{\\frac{1}{3}}, \\, x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739251994\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739251994\"]\r\n

        [latex]\\underset{x\\to 0^-}{\\lim}\\frac{x^{1\/3}-0}{x-0}=\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^{2\/3}}=\\infty [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

        \r\n
        \r\n

        42.\u00a0<\/strong>[latex]f(x)=x^{\\frac{2}{3}}, \\, x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

        \r\n
        \r\n

        43.\u00a0<\/strong>[latex]f(x)=\\begin{cases} 1 & \\text{ if } \\, x<1 \\\\ x & \\text{ if } \\, x \\ge 1 \\end{cases}, \\, x=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739274908\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739274908\"]\r\n

        [latex]\\underset{x\\to 1^-}{\\lim}\\frac{1-1}{x-1}=0\\ne 1=\\underset{x\\to 1^+}{\\lim}\\frac{x-1}{x-1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

        \r\n
        \r\n

        44.\u00a0<\/strong>[latex]f(x)=\\dfrac{|x|}{x}, \\, x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

        \r\n
        \r\n

        45. [T]<\/strong> The position in feet of a race car along a straight track after [latex]t[\/latex] seconds is modeled by the function [latex]s(t)=8t^2-\\frac{1}{16}t^3[\/latex].<\/p>\r\n\r\n

          \r\n \t
        1. Find the average velocity of the vehicle over the following time intervals to four decimal places:\r\n
            \r\n \t
          1. [4, 4.1]<\/li>\r\n \t
          2. [4, 4.01]<\/li>\r\n \t
          3. [4, 4.001]<\/li>\r\n \t
          4. [4, 4.0001]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
          5. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at [latex]t=4[\/latex] seconds.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169739340476\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739340476\"]\r\n

            a. (i) 61.7244 ft\/s, (ii) 61.0725 ft\/s, (iii) 61.0072 ft\/s, (iv) 61.0007 ft\/s\r\nb. At 4 seconds the race car is traveling at a rate\/velocity of 61 ft\/s.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

            \r\n
            \r\n

            46. [T]<\/strong> The distance in feet that a ball rolls down an incline is modeled by the function [latex]s(t)=14t^2[\/latex], where [latex]t[\/latex] is seconds after the ball begins rolling.<\/p>\r\n\r\n

              \r\n \t
            1. Find the average velocity of the ball over the following time intervals:\r\n
                \r\n \t
              1. [5, 5.1]<\/li>\r\n \t
              2. [5, 5.01]<\/li>\r\n \t
              3. [5, 5.001]<\/li>\r\n \t
              4. [5, 5.0001]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
              5. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at [latex]t=5[\/latex] seconds.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
                \r\n
                \r\n

                47.\u00a0<\/strong>Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by [latex]s=f(t)[\/latex] and [latex]s=g(t)[\/latex], where [latex]s[\/latex] is measured in feet and [latex]t[\/latex] is measured in seconds.<\/p>\r\n\"Two<\/span>\r\n

                  \r\n \t
                1. Which vehicle has traveled farther at [latex]t=2[\/latex] seconds?<\/li>\r\n \t
                2. What is the approximate velocity of each vehicle at [latex]t=3[\/latex] seconds?<\/li>\r\n \t
                3. Which vehicle is traveling faster at [latex]t=4[\/latex] seconds?<\/li>\r\n \t
                4. What is true about the positions of the vehicles at [latex]t=4[\/latex] seconds?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169739333859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739333859\"]\r\n

                  a. The vehicle represented by [latex]f(t)[\/latex], because it has traveled 2 feet, whereas [latex]g(t)[\/latex] has traveled 1 foot.\r\nb. The velocity of [latex]f(t)[\/latex] is constant at 1 ft\/s, while the velocity of [latex]g(t)[\/latex] is approximately 2 ft\/s.\r\nc. The vehicle represented by [latex]g(t)[\/latex], with a velocity of approximately 4 ft\/s.\r\nd. Both have traveled 4 feet in 4 seconds.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

                  \r\n
                  \r\n

                  48. [T]<\/strong> The total cost [latex]C(x)[\/latex], in hundreds of dollars, to produce [latex]x[\/latex] jars of mayonnaise is given by [latex]C(x)=0.000003x^3+4x+300[\/latex].<\/p>\r\n\r\n

                    \r\n \t
                  1. Calculate the average cost per jar over the following intervals:\r\n
                      \r\n \t
                    1. [100, 100.1]<\/li>\r\n \t
                    2. [100, 100.01]<\/li>\r\n \t
                    3. [100, 100.001]<\/li>\r\n \t
                    4. [100, 100.0001]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                    5. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
                      \r\n
                      \r\n

                      49. [T]<\/strong> For the function [latex]f(x)=x^3-2x^2-11x+12[\/latex], do the following.<\/p>\r\n\r\n

                        \r\n \t
                      1. Use a graphing calculator to graph [latex]f[\/latex] in an appropriate viewing window.<\/li>\r\n \t
                      2. Use the ZOOM feature on the calculator to approximate the two values of [latex]x=a[\/latex] for which [latex]m_{\\tan}=f^{\\prime}(a)=0[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169739351705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739351705\"]\r\n

                        a.<\/p>\r\n\"The<\/span>\r\nb. [latex]a\\approx -1.361, \\, 2.694[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

                        \r\n
                        \r\n

                        50. [T]<\/strong> For the function [latex]f(x)=\\dfrac{x}{1+x^2}[\/latex], do the following.<\/p>\r\n\r\n

                          \r\n \t
                        1. Use a graphing calculator to graph [latex]f[\/latex] in an appropriate viewing window.<\/li>\r\n \t
                        2. Use the ZOOM feature on the calculator to approximate the values of [latex]x=a[\/latex] for which [latex]m_{\\tan}=f^{\\prime}(a)=0[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
                          \r\n
                          \r\n

                          51.\u00a0<\/strong>Suppose that [latex]N(x)[\/latex] computes the number of gallons of gas used by a vehicle traveling [latex]x[\/latex] miles. Suppose the vehicle gets 30 mpg.<\/p>\r\n\r\n

                            \r\n \t
                          1. Find a mathematical expression for [latex]N(x)[\/latex].<\/li>\r\n \t
                          2. What is [latex]N(100)[\/latex]? Explain the physical meaning.<\/li>\r\n \t
                          3. What is [latex]N^{\\prime}(100)[\/latex]? Explain the physical meaning.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169739188126\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739188126\"]\r\n

                            a. [latex]N(x)=\\frac{x}{30}[\/latex]\r\nb. [latex]\\sim 3.3[\/latex] gallons. When the vehicle travels 100 miles, it has used 3.3 gallons of gas.\r\nc. [latex]\\frac{1}{30}[\/latex]. The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled 100 miles.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

                            \r\n
                            \r\n

                            52. [T]<\/strong> For the function [latex]f(x)=x^4-5x^2+4[\/latex], do the following.<\/p>\r\n\r\n

                              \r\n \t
                            1. Use a graphing calculator to graph [latex]f[\/latex] in an appropriate viewing window.<\/li>\r\n \t
                            2. Use the [latex]\\text{nDeriv}[\/latex] function, which numerically finds the derivative, on a graphing calculator to estimate [latex]f^{\\prime}(-2), \\, f^{\\prime}(-0.5), \\, f^{\\prime}(1.7)[\/latex], and [latex]f^{\\prime}(2.718)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
                              \r\n
                              \r\n

                              53. [T]<\/strong> For the function [latex]f(x)=\\dfrac{x^2}{x^2+1}[\/latex], do the following.<\/p>\r\n\r\n

                                \r\n \t
                              1. Use a graphing calculator to graph [latex]f[\/latex] in an appropriate viewing window.<\/li>\r\n \t
                              2. Use the [latex]\\text{nDeriv}[\/latex] function on a graphing calculator to find [latex]f^{\\prime}(-4), \\, f^{\\prime}(-2), \\, f^{\\prime}(2)[\/latex], and [latex]f^{\\prime}(4)[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1169736610337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736610337\"]\r\n

                                a.<\/p>\r\n\"The<\/span>\r\nb. [latex]-0.028, \\, -0.16, \\, 0.16, \\, 0.028[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

                                For the following exercises (1-10), use the definition of a derivative to find the slope of the secant line between the values [latex]x_1[\/latex] and [latex]x_2[\/latex] for each function [latex]y=f(x)[\/latex].<\/p>\n

                                \n
                                \n

                                1.\u00a0<\/strong>[latex]f(x)=4x+7; \\,\\,\\, x_1=2, \\,\\,\\, x_2=5[\/latex]<\/p>\n

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

                                4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                2.\u00a0<\/strong>[latex]f(x)=8x-3; \\,\\,\\, x_1=-1, \\,\\,\\, x_2=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                3.\u00a0<\/strong>[latex]f(x)=x^2+2x+1; \\,\\,\\, x_1=3, \\,\\,\\, x_2=3.5[\/latex]<\/p>\n

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

                                8.5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                4.\u00a0<\/strong>[latex]f(x)=\\text{\u2212}{x}^{2}+x+2;{x}_{1}=0.5,{x}_{2}=1.5[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                5.\u00a0<\/strong>[latex]f(x)=\\dfrac{4}{3x-1}; \\,\\,\\, x_1=1, \\,\\,\\, x_2=3[\/latex]<\/p>\n

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

                                [latex]-\\frac{3}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                6.\u00a0<\/strong>[latex]f(x)=\\dfrac{x-7}{2x+1}; \\,\\,\\, x_1=0, \\,\\,\\, x_2=2[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                7.\u00a0<\/strong>[latex]f(x)=\\sqrt{x}; \\,\\,\\, x_1=1, \\,\\,\\, x_2=16[\/latex]<\/p>\n

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

                                0.2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                8.\u00a0<\/strong>[latex]f(x)=\\sqrt{x-9}; \\,\\,\\, x_1=10, \\,\\,\\, x_2=13[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                9.\u00a0<\/strong>[latex]f(x)=x^{\\frac{1}{3}}+1; \\,\\,\\, x_1=0, \\,\\,\\, x_2=8[\/latex]<\/p>\n

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

                                0.25<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                \n
                                \n

                                10.\u00a0<\/strong>[latex]f(x)=6x^{\\frac{2}{3}}+2x^{\\frac{1}{3}}; \\,\\,\\, x_1=1, \\,\\,\\, x_2=27[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                For the following functions (11-20),<\/p>\n

                                  \n
                                1. Use\u00a0[latex]m_{\\tan}=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex] to find the slope of the tangent line [latex]m_{\\tan}=f^{\\prime}(a)[\/latex], and<\/li>\n
                                2. find the equation of the tangent line to [latex]f[\/latex] at [latex]x=a[\/latex].<\/li>\n<\/ol>\n
                                  \n
                                  \n

                                  11.\u00a0<\/strong>[latex]f(x)=3-4x, \\,\\,\\, a=2[\/latex]<\/p>\n

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

                                  a. -4 b. [latex]y=3-4x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  12.\u00a0<\/strong>[latex]f(x)=\\dfrac{x}{5}+6, \\,\\,\\, a=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  13.\u00a0<\/strong>[latex]f(x)=x^2+x, \\,\\,\\, a=1[\/latex]<\/p>\n

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

                                  a. 3 b. [latex]y=3x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  14.\u00a0<\/strong>[latex]f(x)=1-x-x^2, \\,\\,\\, a=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  15.\u00a0<\/strong>[latex]f(x)=\\dfrac{7}{x}, \\,\\,\\, a=3[\/latex]<\/p>\n

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

                                  a. [latex]\\frac{-7}{9}[\/latex] b. [latex]y=\\frac{-7}{9}x+\\frac{14}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  16.\u00a0<\/strong>[latex]f(x)=\\sqrt{x+8}, \\,\\,\\, a=1[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

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

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

                                  a. 12 b. [latex]y=12x+14[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  18.\u00a0<\/strong>[latex]f(x)=\\dfrac{-3}{x-1}, \\,\\,\\, a=4[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

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

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

                                  a. -2 b. [latex]y=-2x-10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

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

                                  For the following functions [latex]y=f(x)[\/latex] (21-30), find [latex]f^{\\prime}(a)[\/latex] using\u00a0[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex].<\/p>\n

                                  \n
                                  \n

                                  21.\u00a0<\/strong>[latex]f(x)=5x+4, \\,\\,\\, a=-1[\/latex]<\/p>\n

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

                                  5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  22.\u00a0<\/strong>[latex]f(x)=-7x+1, \\,\\,\\, a=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  23.\u00a0<\/strong>[latex]f(x)=x^2+9x, \\,\\,\\, a=2[\/latex]<\/p>\n

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

                                  13<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

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

                                  \n
                                  \n

                                  25.\u00a0<\/strong>[latex]f(x)=\\sqrt{x}, \\,\\,\\, a=4[\/latex]<\/p>\n

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

                                  [latex]\\frac{1}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  26.\u00a0<\/strong>[latex]f(x)=\\sqrt{x-2}, \\,\\,\\, a=6[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

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

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

                                  [latex]-\\frac{1}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

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

                                  \n
                                  \n

                                  29.\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{x^3}, \\,\\,\\, a=1[\/latex]<\/p>\n

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

                                  -3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                  \n
                                  \n

                                  30.\u00a0<\/strong>[latex]f(x)=\\dfrac{1}{\\sqrt{x}}, \\,\\,\\, a=4[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                  For the following exercises (31-34), given the function [latex]y=f(x)[\/latex],<\/p>\n

                                    \n
                                  1. find the slope of the secant line [latex]PQ[\/latex] for each point [latex]Q(x,f(x))[\/latex] with [latex]x[\/latex] value given in the table.<\/li>\n
                                  2. Use the answers from a. to estimate the value of the slope of the tangent line at [latex]P[\/latex].<\/li>\n
                                  3. Use the answer from b. to find the equation of the tangent line to [latex]f[\/latex] at point [latex]P[\/latex].<\/li>\n<\/ol>\n
                                    \n
                                    \n

                                    31. [T]\u00a0<\/strong>[latex]f(x)=x^2+3x+4, \\,\\,\\, P(1,8)[\/latex] (Round to 6 decimal places.)<\/p>\n\n\n\n\n\n\n\n\n\n\n
                                    [latex]x[\/latex]<\/th>\nSlope [latex]m_{PQ}[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\nSlope [latex]m_{PQ}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
                                    1.1<\/td>\n(i)<\/td>\n0.9<\/td>\n(vii)<\/td>\n<\/tr>\n
                                    1.01<\/td>\n(ii)<\/td>\n0.99<\/td>\n(viii)<\/td>\n<\/tr>\n
                                    1.001<\/td>\n(iii)<\/td>\n0.999<\/td>\n(ix)<\/td>\n<\/tr>\n
                                    1.0001<\/td>\n(iv)<\/td>\n0.9999<\/td>\n(x)<\/td>\n<\/tr>\n
                                    1.00001<\/td>\n(v)<\/td>\n0.99999<\/td>\n(xi)<\/td>\n<\/tr>\n
                                    1.000001<\/td>\n(vi)<\/td>\n0.999999<\/td>\n(xii)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    a. (i) [latex]5.100000[\/latex], (ii) [latex]5.010000[\/latex], (iii) [latex]5.001000[\/latex], (iv) [latex]5.000100[\/latex], (v) [latex]5.000010[\/latex], (vi) [latex]5.000001[\/latex],
                                    \n(vii) [latex]4.900000[\/latex], (viii) [latex]4.990000[\/latex], (ix) [latex]4.999000[\/latex], (x) [latex]4.999900[\/latex], (xi) [latex]4.999990[\/latex], (xii) [latex]4.999999[\/latex]<\/p>\n

                                    b. [latex]m_{\\tan}=5[\/latex]
                                    \nc. [latex]y=5x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    32. [T]\u00a0<\/strong>[latex]f(x)=\\dfrac{x+1}{x^2-1}, \\,\\,\\, P(0,-1)[\/latex]<\/p>\n\n\n\n\n\n\n\n\n\n\n
                                    [latex]x[\/latex]<\/th>\nSlope [latex]m_{PQ}[\/latex]<\/th>\n[latex]x[\/latex]<\/th>\nSlope [latex]m_{PQ}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
                                    0.1<\/td>\n(i)<\/td>\n-0.1<\/td>\n(vii)<\/td>\n<\/tr>\n
                                    0.01<\/td>\n(ii)<\/td>\n-0.01<\/td>\n(viii)<\/td>\n<\/tr>\n
                                    0.001<\/td>\n(iii)<\/td>\n-0.001<\/td>\n(ix)<\/td>\n<\/tr>\n
                                    0.0001<\/td>\n(iv)<\/td>\n-0.0001<\/td>\n(x)<\/td>\n<\/tr>\n
                                    0.00001<\/td>\n(v)<\/td>\n-0.00001<\/td>\n(xi)<\/td>\n<\/tr>\n
                                    0.000001<\/td>\n(vi)<\/td>\n-0.000001<\/td>\n(xii)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
                                    \n
                                    \n

                                    33. [T]\u00a0<\/strong>[latex]f(x)=10e^{0.5x}, \\,\\,\\, P(0,10)[\/latex] (Round to 4 decimal places.)<\/p>\n\n\n\n\n\n\n\n\n\n\n
                                    [latex]x[\/latex]<\/th>\nSlope [latex]m_{PQ}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
                                    -0.1<\/td>\n(i)<\/td>\n<\/tr>\n
                                    -0.01<\/td>\n(ii)<\/td>\n<\/tr>\n
                                    -0.001<\/td>\n(iii)<\/td>\n<\/tr>\n
                                    -0.0001<\/td>\n(iv)<\/td>\n<\/tr>\n
                                    -0.00001<\/td>\n(v)<\/td>\n<\/tr>\n
                                    \u22120.000001<\/td>\n(vi)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                    Show Solution<\/span><\/p>\n
                                    \n

                                    a. (i) [latex]4.8771[\/latex], (ii) [latex]4.9875[\/latex], (iii) [latex]4.9988[\/latex], (iv) [latex]4.9999[\/latex], (v) [latex]4.9999[\/latex], (vi) [latex]4.9999[\/latex]
                                    \nb. [latex]m_{\\tan}=5[\/latex]
                                    \nc. [latex]y=5x+10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                    \n
                                    \n

                                    34. [T]\u00a0<\/strong>[latex]f(x)= \\tan (x), \\,\\,\\, P(\\pi,0)[\/latex]<\/p>\n\n\n\n\n\n\n\n\n\n\n
                                    [latex]x[\/latex]<\/th>\nSlope [latex]m_{PQ}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
                                    3.1<\/td>\n(i)<\/td>\n<\/tr>\n
                                    3.14<\/td>\n(ii)<\/td>\n<\/tr>\n
                                    3.141<\/td>\n(iii)<\/td>\n<\/tr>\n
                                    3.1415<\/td>\n(iv)<\/td>\n<\/tr>\n
                                    3.14159<\/td>\n(v)<\/td>\n<\/tr>\n
                                    3.141592<\/td>\n(vi)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                                    For the following position functions [latex]y=s(t)[\/latex], an object is moving along a straight line, where [latex]t[\/latex] is in seconds and [latex]s[\/latex] is in meters. Find<\/p>\n

                                      \n
                                    1. the simplified expression for the average velocity from [latex]t=2[\/latex] to [latex]t=2+h[\/latex];<\/li>\n
                                    2. the average velocity between [latex]t=2[\/latex] and [latex]t=2+h[\/latex], where (i) [latex]h=0.1[\/latex], (ii) [latex]h=0.01[\/latex], (iii) [latex]h=0.001[\/latex], and (iv) [latex]h=0.0001[\/latex]; and<\/li>\n
                                    3. use the answer from a. to estimate the instantaneous velocity at [latex]t=2[\/latex] seconds.<\/li>\n<\/ol>\n
                                      \n
                                      \n

                                      35. [T]\u00a0<\/strong>[latex]s(t)=\\frac{1}{3}t+5[\/latex]<\/p>\n

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

                                      a. [latex]\\frac{1}{3}[\/latex];
                                      \nb. (i) [latex]0.\\bar{3}[\/latex] m\/s, (ii) [latex]0.\\bar{3}[\/latex] m\/s, (iii) [latex]0.\\bar{3}[\/latex] m\/s, (iv) [latex]0.\\bar{3}[\/latex] m\/s;
                                      \nc. [latex]0.\\bar{3}=\\frac{1}{3}[\/latex] m\/s<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      36. [T]\u00a0<\/strong>[latex]s(t)=t^2-2t[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      37. [T]\u00a0<\/strong>[latex]s(t)=2t^3+3[\/latex]<\/p>\n

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

                                      a. [latex]2(h^2+6h+12)[\/latex];
                                      \nb. (i) 25.22 m\/s, (ii) 24.12 m\/s, (iii) 24.01 m\/s, (iv) 24 m\/s;
                                      \nc. 24 m\/s<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      38. [T]\u00a0<\/strong>[latex]s(t)=\\dfrac{16}{t^2}-\\dfrac{4}{t}[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      39.\u00a0<\/strong>Use the following graph to evaluate a. [latex]f^{\\prime}(1)[\/latex] and b. [latex]f^{\\prime}(6)[\/latex].<\/p>\n

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

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

                                      a. [latex]1.25[\/latex]; b. 0.5<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      40.\u00a0<\/strong>Use the following graph to evaluate a. [latex]f^{\\prime}(-3)[\/latex] and b. [latex]f^{\\prime}(1.5)[\/latex].<\/p>\n

                                      \"This<\/span><\/p>\n<\/div>\n<\/div>\n

                                      For the following exercises, use the limit definition of derivative to show that the derivative does not exist at [latex]x=a[\/latex] for each of the given functions.<\/p>\n

                                      \n
                                      \n

                                      41.\u00a0<\/strong>[latex]f(x)=x^{\\frac{1}{3}}, \\, x=0[\/latex]<\/p>\n

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

                                      [latex]\\underset{x\\to 0^-}{\\lim}\\frac{x^{1\/3}-0}{x-0}=\\underset{x\\to 0^-}{\\lim}\\frac{1}{x^{2\/3}}=\\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      42.\u00a0<\/strong>[latex]f(x)=x^{\\frac{2}{3}}, \\, x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      43.\u00a0<\/strong>[latex]f(x)=\\begin{cases} 1 & \\text{ if } \\, x<1 \\\\ x & \\text{ if } \\, x \\ge 1 \\end{cases}, \\, x=1[\/latex]<\/p>\n

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

                                      [latex]\\underset{x\\to 1^-}{\\lim}\\frac{1-1}{x-1}=0\\ne 1=\\underset{x\\to 1^+}{\\lim}\\frac{x-1}{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      44.\u00a0<\/strong>[latex]f(x)=\\dfrac{|x|}{x}, \\, x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

                                      \n
                                      \n

                                      45. [T]<\/strong> The position in feet of a race car along a straight track after [latex]t[\/latex] seconds is modeled by the function [latex]s(t)=8t^2-\\frac{1}{16}t^3[\/latex].<\/p>\n

                                        \n
                                      1. Find the average velocity of the vehicle over the following time intervals to four decimal places:\n
                                          \n
                                        1. [4, 4.1]<\/li>\n
                                        2. [4, 4.01]<\/li>\n
                                        3. [4, 4.001]<\/li>\n
                                        4. [4, 4.0001]<\/li>\n<\/ol>\n<\/li>\n
                                        5. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at [latex]t=4[\/latex] seconds.<\/li>\n<\/ol>\n
                                          Show Solution<\/span><\/p>\n
                                          \n

                                          a. (i) 61.7244 ft\/s, (ii) 61.0725 ft\/s, (iii) 61.0072 ft\/s, (iv) 61.0007 ft\/s
                                          \nb. At 4 seconds the race car is traveling at a rate\/velocity of 61 ft\/s.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                          \n
                                          \n

                                          46. [T]<\/strong> The distance in feet that a ball rolls down an incline is modeled by the function [latex]s(t)=14t^2[\/latex], where [latex]t[\/latex] is seconds after the ball begins rolling.<\/p>\n

                                            \n
                                          1. Find the average velocity of the ball over the following time intervals:\n
                                              \n
                                            1. [5, 5.1]<\/li>\n
                                            2. [5, 5.01]<\/li>\n
                                            3. [5, 5.001]<\/li>\n
                                            4. [5, 5.0001]<\/li>\n<\/ol>\n<\/li>\n
                                            5. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at [latex]t=5[\/latex] seconds.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                              \n
                                              \n

                                              47.\u00a0<\/strong>Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by [latex]s=f(t)[\/latex] and [latex]s=g(t)[\/latex], where [latex]s[\/latex] is measured in feet and [latex]t[\/latex] is measured in seconds.<\/p>\n

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

                                                \n
                                              1. Which vehicle has traveled farther at [latex]t=2[\/latex] seconds?<\/li>\n
                                              2. What is the approximate velocity of each vehicle at [latex]t=3[\/latex] seconds?<\/li>\n
                                              3. Which vehicle is traveling faster at [latex]t=4[\/latex] seconds?<\/li>\n
                                              4. What is true about the positions of the vehicles at [latex]t=4[\/latex] seconds?<\/li>\n<\/ol>\n
                                                Show Solution<\/span><\/p>\n
                                                \n

                                                a. The vehicle represented by [latex]f(t)[\/latex], because it has traveled 2 feet, whereas [latex]g(t)[\/latex] has traveled 1 foot.
                                                \nb. The velocity of [latex]f(t)[\/latex] is constant at 1 ft\/s, while the velocity of [latex]g(t)[\/latex] is approximately 2 ft\/s.
                                                \nc. The vehicle represented by [latex]g(t)[\/latex], with a velocity of approximately 4 ft\/s.
                                                \nd. Both have traveled 4 feet in 4 seconds.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                                \n
                                                \n

                                                48. [T]<\/strong> The total cost [latex]C(x)[\/latex], in hundreds of dollars, to produce [latex]x[\/latex] jars of mayonnaise is given by [latex]C(x)=0.000003x^3+4x+300[\/latex].<\/p>\n

                                                  \n
                                                1. Calculate the average cost per jar over the following intervals:\n
                                                    \n
                                                  1. [100, 100.1]<\/li>\n
                                                  2. [100, 100.01]<\/li>\n
                                                  3. [100, 100.001]<\/li>\n
                                                  4. [100, 100.0001]<\/li>\n<\/ol>\n<\/li>\n
                                                  5. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                                    \n
                                                    \n

                                                    49. [T]<\/strong> For the function [latex]f(x)=x^3-2x^2-11x+12[\/latex], do the following.<\/p>\n

                                                      \n
                                                    1. Use a graphing calculator to graph [latex]f[\/latex] in an appropriate viewing window.<\/li>\n
                                                    2. Use the ZOOM feature on the calculator to approximate the two values of [latex]x=a[\/latex] for which [latex]m_{\\tan}=f^{\\prime}(a)=0[\/latex].<\/li>\n<\/ol>\n
                                                      Show Solution<\/span><\/p>\n
                                                      \n

                                                      a.<\/p>\n

                                                      \"The<\/span>
                                                      \nb. [latex]a\\approx -1.361, \\, 2.694[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                                      \n
                                                      \n

                                                      50. [T]<\/strong> For the function [latex]f(x)=\\dfrac{x}{1+x^2}[\/latex], do the following.<\/p>\n

                                                        \n
                                                      1. Use a graphing calculator to graph [latex]f[\/latex] in an appropriate viewing window.<\/li>\n
                                                      2. Use the ZOOM feature on the calculator to approximate the values of [latex]x=a[\/latex] for which [latex]m_{\\tan}=f^{\\prime}(a)=0[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                                        \n
                                                        \n

                                                        51.\u00a0<\/strong>Suppose that [latex]N(x)[\/latex] computes the number of gallons of gas used by a vehicle traveling [latex]x[\/latex] miles. Suppose the vehicle gets 30 mpg.<\/p>\n

                                                          \n
                                                        1. Find a mathematical expression for [latex]N(x)[\/latex].<\/li>\n
                                                        2. What is [latex]N(100)[\/latex]? Explain the physical meaning.<\/li>\n
                                                        3. What is [latex]N^{\\prime}(100)[\/latex]? Explain the physical meaning.<\/li>\n<\/ol>\n
                                                          Show Solution<\/span><\/p>\n
                                                          \n

                                                          a. [latex]N(x)=\\frac{x}{30}[\/latex]
                                                          \nb. [latex]\\sim 3.3[\/latex] gallons. When the vehicle travels 100 miles, it has used 3.3 gallons of gas.
                                                          \nc. [latex]\\frac{1}{30}[\/latex]. The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled 100 miles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                                                          \n
                                                          \n

                                                          52. [T]<\/strong> For the function [latex]f(x)=x^4-5x^2+4[\/latex], do the following.<\/p>\n

                                                            \n
                                                          1. Use a graphing calculator to graph [latex]f[\/latex] in an appropriate viewing window.<\/li>\n
                                                          2. Use the [latex]\\text{nDeriv}[\/latex] function, which numerically finds the derivative, on a graphing calculator to estimate [latex]f^{\\prime}(-2), \\, f^{\\prime}(-0.5), \\, f^{\\prime}(1.7)[\/latex], and [latex]f^{\\prime}(2.718)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                                            \n
                                                            \n

                                                            53. [T]<\/strong> For the function [latex]f(x)=\\dfrac{x^2}{x^2+1}[\/latex], do the following.<\/p>\n

                                                              \n
                                                            1. Use a graphing calculator to graph [latex]f[\/latex] in an appropriate viewing window.<\/li>\n
                                                            2. Use the [latex]\\text{nDeriv}[\/latex] function on a graphing calculator to find [latex]f^{\\prime}(-4), \\, f^{\\prime}(-2), \\, f^{\\prime}(2)[\/latex], and [latex]f^{\\prime}(4)[\/latex].<\/li>\n<\/ol>\n
                                                              Show Solution<\/span><\/p>\n
                                                              \n

                                                              a.<\/p>\n

                                                              \"The<\/span>
                                                              \nb. [latex]-0.028, \\, -0.16, \\, 0.16, \\, 0.028[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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