{"id":455,"date":"2021-02-04T15:27:44","date_gmt":"2021-02-04T15:27:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=455"},"modified":"2021-03-29T22:23:20","modified_gmt":"2021-03-29T22:23:20","slug":"problem-set-a-preview-of-calculus","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-a-preview-of-calculus\/","title":{"raw":"Problem Set: A Preview of Calculus","rendered":"Problem Set: A Preview of Calculus"},"content":{"raw":"

For the following exercises (1-3), points [latex]P(1,2)[\/latex] and [latex]Q(x,y)[\/latex] are on the graph of the function [latex]f(x)=x^2+1[\/latex].<\/p>\r\n\r\n

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1. [T]<\/strong> Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of Q<\/em>, the point [latex]Q(x,y),[\/latex] and the slope of the secant line passing through points P<\/em> and Q<\/em>. Round your answer to eight significant digits.<\/p>\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\n[latex]y[\/latex]<\/th>\r\n[latex]Q(x,y)[\/latex]<\/th>\r\n[latex]m_{\\sec}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1.1<\/td>\r\na.<\/td>\r\ne.<\/td>\r\ni.<\/td>\r\n<\/tr>\r\n
1.01<\/td>\r\nb.<\/td>\r\nf.<\/td>\r\nj.<\/td>\r\n<\/tr>\r\n
1.001<\/td>\r\nc.<\/td>\r\ng.<\/td>\r\nk.<\/td>\r\n<\/tr>\r\n
1.0001<\/td>\r\nd.<\/td>\r\nh.<\/td>\r\nl.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1170573244874\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573244874\"]\r\n

a. 2.2100000; b. 2.0201000; c. 2.0020010; d. 2.0002000; e. (1.1000000, 2.2100000); f. (1.0100000, 2.0201000); g. (1.0010000, 2.0020010); h. (1.0001000, 2.0002000); i. 2.1000000; j. 2.0100000; k. 2.0010000; l. 2.0001000<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to [latex]f[\/latex] at [latex]x=1[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex]. Graph [latex]f(x)[\/latex] and the tangent line.<\/p>\r\n[reveal-answer q=\"fs-id1170573571230\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573571230\"]\r\n

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

For the following exercises (4-6), points [latex]P(1,1)[\/latex] and [latex]Q(x,y)[\/latex] are on the graph of the function [latex]f(x)=x^3[\/latex].<\/p>\r\n\r\n

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4. [T]<\/strong> Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of [latex]Q[\/latex], the point [latex]Q(x,y)[\/latex], and the slope of the secant line passing through points\u00a0[latex]P[\/latex] and [latex]Q[\/latex]. Round your answer to eight significant digits.<\/p>\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\n[latex]y[\/latex]<\/th>\r\n[latex]Q(x,y)[\/latex]<\/th>\r\n[latex]m_{\\sec}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1.1<\/td>\r\na.<\/td>\r\ne.<\/td>\r\ni.<\/td>\r\n<\/tr>\r\n
1.01<\/td>\r\nb.<\/td>\r\nf.<\/td>\r\nj.<\/td>\r\n<\/tr>\r\n
1.001<\/td>\r\nc.<\/td>\r\ng.<\/td>\r\nk.<\/td>\r\n<\/tr>\r\n
1.0001<\/td>\r\nd.<\/td>\r\nh.<\/td>\r\nl.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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5.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[\/latex] at [latex]x=1[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170573403971\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573403971\"]\r\n

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

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6.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex]. Graph [latex]f(x)[\/latex] and the tangent line.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (7-9), points [latex]P(4,2)[\/latex] and [latex]Q(x,y)[\/latex] are on the graph of the function [latex]f(x)=\\sqrt{x}[\/latex].<\/p>\r\n\r\n

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7. [T]<\/strong> Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of [latex]Q[\/latex], the point [latex]Q(x,y)[\/latex], and the slope of the secant line passing through points [latex]P[\/latex] and [latex]Q[\/latex]. Round your answer to eight significant digits.<\/p>\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\n[latex]y[\/latex]<\/th>\r\n[latex]Q(x,y)[\/latex]<\/th>\r\n[latex]m_{\\sec}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
4.1<\/td>\r\na.<\/td>\r\ne.<\/td>\r\ni.<\/td>\r\n<\/tr>\r\n
4.01<\/td>\r\nb.<\/td>\r\nf.<\/td>\r\nj.<\/td>\r\n<\/tr>\r\n
4.001<\/td>\r\nc.<\/td>\r\ng.<\/td>\r\nk.<\/td>\r\n<\/tr>\r\n
4.0001<\/td>\r\nd.<\/td>\r\nh.<\/td>\r\nl.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1170571119936\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571119936\"]\r\n

a. 2.0248457; b. 2.0024984; c. 2.0002500; d. 2.0000250; e. (4.1000000,2.0248457); f. (4.0100000,2.0024984); g. (4.0010000,2.0002500); h. (4.00010000,2.0000250); i. 0.24845673; j. 0.24984395; k. 0.24998438; l. 0.24999844<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[\/latex] at [latex]x=4[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170571124877\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571124877\"]\r\n

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

For the following exercises (10-12), points [latex]P(1.5,0)[\/latex] and [latex]Q(\\phi ,y)[\/latex] are on the graph of the function [latex]f(\\phi )= \\cos (\\pi \\phi )[\/latex].<\/p>\r\n\r\n

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10. [T]\u00a0<\/strong>Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of [latex]Q[\/latex], the point [latex]Q(x,y)[\/latex], and the slope of the secant line passing through points [latex]P[\/latex] and [latex]Q[\/latex]. Round your answer to eight significant digits.<\/p>\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\n[latex]y[\/latex]<\/th>\r\n[latex]Q(\\phi ,y)[\/latex]<\/th>\r\n[latex]m_{\\sec}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1.4<\/td>\r\na.<\/td>\r\ne.<\/td>\r\ni.<\/td>\r\n<\/tr>\r\n
1.49<\/td>\r\nb.<\/td>\r\nf.<\/td>\r\nj.<\/td>\r\n<\/tr>\r\n
1.499<\/td>\r\nc.<\/td>\r\ng.<\/td>\r\nk.<\/td>\r\n<\/tr>\r\n
1.4999<\/td>\r\nd.<\/td>\r\nh.<\/td>\r\nl.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
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11.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[\/latex] at [latex]x=4[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170573417203\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573417203\"]\r\n

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

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12.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (13-15), points [latex]P(-1,-1)[\/latex] and [latex]Q(x,y)[\/latex] are on the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex].<\/p>\r\n\r\n

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13. [T]<\/strong> Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of [latex]Q[\/latex], the point [latex]Q(x,y)[\/latex], and the slope of the secant line passing through points [latex]P[\/latex] and [latex]Q[\/latex]. Round your answer to eight significant digits.<\/p>\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\n[latex]y[\/latex]<\/th>\r\n[latex]Q(x,y)[\/latex]<\/th>\r\n[latex]m_{\\sec}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
\u22121.05<\/td>\r\na.<\/td>\r\ne.<\/td>\r\ni.<\/td>\r\n<\/tr>\r\n
\u22121.01<\/td>\r\nb.<\/td>\r\nf.<\/td>\r\nj.<\/td>\r\n<\/tr>\r\n
\u22121.005<\/td>\r\nc.<\/td>\r\ng.<\/td>\r\nk.<\/td>\r\n<\/tr>\r\n
\u22121.001<\/td>\r\nd.<\/td>\r\nh.<\/td>\r\nl.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"fs-id1170573595212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573595212\"]\r\n

a. \u22120.95238095; b. \u22120.99009901; c. \u22120.99502488; d. \u22120.99900100; e. (\u22121;.0500000,\u22120;.95238095); f. (\u22121;.0100000,\u22120;.9909901); g. (\u22121;.0050000,\u22120;.99502488); h. (1.0010000,\u22120;.99900100); i. \u22120.95238095; j. \u22120.99009901; k. \u22120.99502488; l. \u22120.99900100<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to [latex]f[\/latex] at [latex]x=-1[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170571033605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571033605\"]\r\n

[latex]y=\u2212x-2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (16-17), the position function of a ball dropped from the top of a 200-meter tall building is given by [latex]s(t)=200-4.9t^2[\/latex], where position [latex]s[\/latex] is measured in meters and time [latex]t[\/latex] is measured in seconds. Round your answer to eight significant digits.<\/p>\r\n\r\n

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16. [T]<\/strong> Compute the average velocity of the ball over the given time intervals.<\/p>\r\n\r\n

    \r\n \t
  1. [latex][4.99,5][\/latex]<\/li>\r\n \t
  2. [latex][5,5.01][\/latex]<\/li>\r\n \t
  3. [latex][4.999,5][\/latex]<\/li>\r\n \t
  4. [latex][5,5.001][\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
    \r\n
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    17.\u00a0<\/strong>Use the preceding exercise to guess the instantaneous velocity of the ball at [latex]t=5[\/latex] sec.<\/p>\r\n[reveal-answer q=\"fs-id1170571124871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571124871\"]\r\n

    \u221249 m\/sec (velocity of the ball is 49 m\/sec downward)<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    For the following exercises (18-19), consider a stone tossed into the air from ground level with an initial velocity of 15 m\/sec. Its height in meters at time [latex]t[\/latex] seconds is [latex]h(t)=15t-4.9t^2[\/latex].<\/p>\r\n\r\n

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    18. [T]<\/strong> Compute the average velocity of the stone over the given time intervals.<\/p>\r\n\r\n

      \r\n \t
    1. [latex][1,1.05][\/latex]<\/li>\r\n \t
    2. [latex][1,1.01][\/latex]<\/li>\r\n \t
    3. [latex][1,1.005][\/latex]<\/li>\r\n \t
    4. [latex][1,1.001][\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
      \r\n
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      19.\u00a0<\/strong>Use the preceding exercise to guess the instantaneous velocity of the stone at [latex]t=1[\/latex] sec.<\/p>\r\n[reveal-answer q=\"fs-id1170573427742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573427742\"]\r\n

      5.2 m\/sec<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

      For the following exercises (20-21), consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by [latex]h(t)=600+78.4t-4.9t^2[\/latex], where [latex]t[\/latex] is measured in seconds.<\/p>\r\n\r\n

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      20. [T]<\/strong> Compute the average velocity of the rocket over the given time intervals.<\/p>\r\n\r\n

        \r\n \t
      1. [latex][9,9.01][\/latex]<\/li>\r\n \t
      2. [latex][8.99,9][\/latex]<\/li>\r\n \t
      3. [latex][9,9.001][\/latex]<\/li>\r\n \t
      4. [latex][8.999,9][\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
        \r\n
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        21.\u00a0<\/strong>Use the preceding exercise to guess the instantaneous velocity of the rocket at [latex]t=9[\/latex] sec.<\/p>\r\n[reveal-answer q=\"fs-id1170570997255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170570997255\"]\r\n

        \u22129.8 m\/sec<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

        For the following exercises (22-23), consider an athlete running a 40-m dash. The position of the athlete is given by [latex]d(t)=\\dfrac{t^3}{6}+4t[\/latex], where [latex]d[\/latex] is the position in meters and [latex]t[\/latex] is the time elapsed, measured in seconds.<\/p>\r\n\r\n

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        22. [T]<\/strong> Compute the average velocity of the runner over the given time intervals.<\/p>\r\n\r\n

          \r\n \t
        1. [latex][1.95,2.05][\/latex]<\/li>\r\n \t
        2. [latex][1.995,2.005][\/latex]<\/li>\r\n \t
        3. [latex][1.9995,2.0005][\/latex]<\/li>\r\n \t
        4. [latex][2,2.00001][\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
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          23.\u00a0<\/strong>Use the preceding exercise to guess the instantaneous velocity of the runner at [latex]t=2[\/latex] sec.<\/p>\r\n[reveal-answer q=\"fs-id1170573429477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573429477\"]\r\n

          6 m\/sec<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

          For the following exercises (24-25), consider the function [latex]f(x)=|x|[\/latex].<\/p>\r\n\r\n

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          24.\u00a0<\/strong>Sketch the graph of [latex]f[\/latex] over the interval [latex][-1,2][\/latex] and shade the region above the [latex]x[\/latex]-axis.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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          25.\u00a0<\/strong>Use the preceding exercise to find the exact value of the area between the [latex]x[\/latex]-axis and the graph of [latex]f[\/latex] over the interval [latex][-1,2][\/latex] using rectangles. For the rectangles, use the square units, and approximate both above and below the lines. Use geometry to find the exact answer.<\/p>\r\n[reveal-answer q=\"fs-id1170573413784\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573413784\"]\r\n

          Under, 1 unit2<\/sup>; over: 4 unit2<\/sup>. The exact area of the two triangles is [latex]\\frac{1}{2}(1)(1)+\\frac{1}{2}(2)(2)=2.5 \\text{units}^2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

          For the following exercises (26-27), consider the function [latex]f(x)=\\sqrt{1-x^2}[\/latex]. (Hint<\/em>: This is the upper half of a circle of radius 1 positioned at [latex](0,0)[\/latex].)<\/p>\r\n\r\n

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          26.\u00a0<\/strong>Sketch the graph of [latex]f[\/latex] over the interval [latex][-1,1][\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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          27.\u00a0<\/strong>Use the preceding exercise to find the exact area between the [latex]x[\/latex]-axis and the graph of [latex]f[\/latex] over the interval [latex][-1,1][\/latex] using rectangles. For the rectangles, use squares 0.4 by 0.4 units, and approximate both above and below the lines. Use geometry to find the exact answer.<\/p>\r\n[reveal-answer q=\"fs-id1170573397798\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573397798\"]\r\n

          Under, 0.96 unit2<\/sup>; over, 1.92 unit2<\/sup>. The exact area of the semicircle with radius 1 is [latex]\\frac{\\pi (1)^2}{2}=\\frac{\\pi }{2}[\/latex] unit2<\/sup>.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

          For the following exercises (28-29), consider the function [latex]f(x)=\u2212x^2+1[\/latex].<\/p>\r\n\r\n

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          28.\u00a0<\/strong>Sketch the graph of [latex]f[\/latex] over the interval [latex][-1,1][\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

          \r\n
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          29.\u00a0<\/strong>Approximate the area of the region between the [latex]x[\/latex]-axis and the graph of [latex]f[\/latex] over the interval [latex][-1,1][\/latex].<\/p>\r\n[reveal-answer q=\"fs-id1170573255383\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170573255383\"]\r\n

          Approximately 1.3333333 unit2<\/sup><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

          For the following exercises (1-3), points [latex]P(1,2)[\/latex] and [latex]Q(x,y)[\/latex] are on the graph of the function [latex]f(x)=x^2+1[\/latex].<\/p>\n

          \n
          \n

          1. [T]<\/strong> Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of Q<\/em>, the point [latex]Q(x,y),[\/latex] and the slope of the secant line passing through points P<\/em> and Q<\/em>. Round your answer to eight significant digits.<\/p>\n\n\n\n\n\n\n\n\n
          [latex]x[\/latex]<\/th>\n[latex]y[\/latex]<\/th>\n[latex]Q(x,y)[\/latex]<\/th>\n[latex]m_{\\sec}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
          1.1<\/td>\na.<\/td>\ne.<\/td>\ni.<\/td>\n<\/tr>\n
          1.01<\/td>\nb.<\/td>\nf.<\/td>\nj.<\/td>\n<\/tr>\n
          1.001<\/td>\nc.<\/td>\ng.<\/td>\nk.<\/td>\n<\/tr>\n
          1.0001<\/td>\nd.<\/td>\nh.<\/td>\nl.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
          Show Solution<\/span><\/p>\n
          \n

          a. 2.2100000; b. 2.0201000; c. 2.0020010; d. 2.0002000; e. (1.1000000, 2.2100000); f. (1.0100000, 2.0201000); g. (1.0010000, 2.0020010); h. (1.0001000, 2.0002000); i. 2.1000000; j. 2.0100000; k. 2.0010000; l. 2.0001000<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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          \n

          2.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to [latex]f[\/latex] at [latex]x=1[\/latex].<\/p>\n<\/div>\n<\/div>\n

          \n
          \n

          3.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex]. Graph [latex]f(x)[\/latex] and the tangent line.<\/p>\n

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

          [latex]y=2x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          For the following exercises (4-6), points [latex]P(1,1)[\/latex] and [latex]Q(x,y)[\/latex] are on the graph of the function [latex]f(x)=x^3[\/latex].<\/p>\n

          \n
          \n

          4. [T]<\/strong> Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of [latex]Q[\/latex], the point [latex]Q(x,y)[\/latex], and the slope of the secant line passing through points\u00a0[latex]P[\/latex] and [latex]Q[\/latex]. Round your answer to eight significant digits.<\/p>\n\n\n\n\n\n\n\n\n
          [latex]x[\/latex]<\/th>\n[latex]y[\/latex]<\/th>\n[latex]Q(x,y)[\/latex]<\/th>\n[latex]m_{\\sec}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
          1.1<\/td>\na.<\/td>\ne.<\/td>\ni.<\/td>\n<\/tr>\n
          1.01<\/td>\nb.<\/td>\nf.<\/td>\nj.<\/td>\n<\/tr>\n
          1.001<\/td>\nc.<\/td>\ng.<\/td>\nk.<\/td>\n<\/tr>\n
          1.0001<\/td>\nd.<\/td>\nh.<\/td>\nl.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
          \n
          \n

          5.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[\/latex] at [latex]x=1[\/latex].<\/p>\n

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

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

          \n
          \n

          6.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex]. Graph [latex]f(x)[\/latex] and the tangent line.<\/p>\n<\/div>\n<\/div>\n

          For the following exercises (7-9), points [latex]P(4,2)[\/latex] and [latex]Q(x,y)[\/latex] are on the graph of the function [latex]f(x)=\\sqrt{x}[\/latex].<\/p>\n

          \n
          \n

          7. [T]<\/strong> Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of [latex]Q[\/latex], the point [latex]Q(x,y)[\/latex], and the slope of the secant line passing through points [latex]P[\/latex] and [latex]Q[\/latex]. Round your answer to eight significant digits.<\/p>\n\n\n\n\n\n\n\n\n
          [latex]x[\/latex]<\/th>\n[latex]y[\/latex]<\/th>\n[latex]Q(x,y)[\/latex]<\/th>\n[latex]m_{\\sec}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
          4.1<\/td>\na.<\/td>\ne.<\/td>\ni.<\/td>\n<\/tr>\n
          4.01<\/td>\nb.<\/td>\nf.<\/td>\nj.<\/td>\n<\/tr>\n
          4.001<\/td>\nc.<\/td>\ng.<\/td>\nk.<\/td>\n<\/tr>\n
          4.0001<\/td>\nd.<\/td>\nh.<\/td>\nl.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
          Show Solution<\/span><\/p>\n
          \n

          a. 2.0248457; b. 2.0024984; c. 2.0002500; d. 2.0000250; e. (4.1000000,2.0248457); f. (4.0100000,2.0024984); g. (4.0010000,2.0002500); h. (4.00010000,2.0000250); i. 0.24845673; j. 0.24984395; k. 0.24998438; l. 0.24999844<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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          \n

          8.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[\/latex] at [latex]x=4[\/latex].<\/p>\n<\/div>\n<\/div>\n

          \n
          \n

          9.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex].<\/p>\n

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

          [latex]y=\\frac{x}{4}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          For the following exercises (10-12), points [latex]P(1.5,0)[\/latex] and [latex]Q(\\phi ,y)[\/latex] are on the graph of the function [latex]f(\\phi )= \\cos (\\pi \\phi )[\/latex].<\/p>\n

          \n
          \n

          10. [T]\u00a0<\/strong>Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of [latex]Q[\/latex], the point [latex]Q(x,y)[\/latex], and the slope of the secant line passing through points [latex]P[\/latex] and [latex]Q[\/latex]. Round your answer to eight significant digits.<\/p>\n\n\n\n\n\n\n\n\n
          [latex]x[\/latex]<\/th>\n[latex]y[\/latex]<\/th>\n[latex]Q(\\phi ,y)[\/latex]<\/th>\n[latex]m_{\\sec}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
          1.4<\/td>\na.<\/td>\ne.<\/td>\ni.<\/td>\n<\/tr>\n
          1.49<\/td>\nb.<\/td>\nf.<\/td>\nj.<\/td>\n<\/tr>\n
          1.499<\/td>\nc.<\/td>\ng.<\/td>\nk.<\/td>\n<\/tr>\n
          1.4999<\/td>\nd.<\/td>\nh.<\/td>\nl.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
          \n
          \n

          11.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to [latex]f[\/latex] at [latex]x=4[\/latex].<\/p>\n

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

          [latex]\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          \n
          \n

          12.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex].<\/p>\n<\/div>\n<\/div>\n

          For the following exercises (13-15), points [latex]P(-1,-1)[\/latex] and [latex]Q(x,y)[\/latex] are on the graph of the function [latex]f(x)=\\dfrac{1}{x}[\/latex].<\/p>\n

          \n
          \n

          13. [T]<\/strong> Complete the following table with the appropriate values: [latex]y[\/latex]-coordinate of [latex]Q[\/latex], the point [latex]Q(x,y)[\/latex], and the slope of the secant line passing through points [latex]P[\/latex] and [latex]Q[\/latex]. Round your answer to eight significant digits.<\/p>\n\n\n\n\n\n\n\n\n
          [latex]x[\/latex]<\/th>\n[latex]y[\/latex]<\/th>\n[latex]Q(x,y)[\/latex]<\/th>\n[latex]m_{\\sec}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
          \u22121.05<\/td>\na.<\/td>\ne.<\/td>\ni.<\/td>\n<\/tr>\n
          \u22121.01<\/td>\nb.<\/td>\nf.<\/td>\nj.<\/td>\n<\/tr>\n
          \u22121.005<\/td>\nc.<\/td>\ng.<\/td>\nk.<\/td>\n<\/tr>\n
          \u22121.001<\/td>\nd.<\/td>\nh.<\/td>\nl.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
          Show Solution<\/span><\/p>\n
          \n

          a. \u22120.95238095; b. \u22120.99009901; c. \u22120.99502488; d. \u22120.99900100; e. (\u22121;.0500000,\u22120;.95238095); f. (\u22121;.0100000,\u22120;.9909901); g. (\u22121;.0050000,\u22120;.99502488); h. (1.0010000,\u22120;.99900100); i. \u22120.95238095; j. \u22120.99009901; k. \u22120.99502488; l. \u22120.99900100<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          \n
          \n

          14.\u00a0<\/strong>Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to [latex]f[\/latex] at [latex]x=-1[\/latex].<\/p>\n<\/div>\n<\/div>\n

          \n
          \n

          15.\u00a0<\/strong>Use the value in the preceding exercise to find the equation of the tangent line at point [latex]P[\/latex].<\/p>\n

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

          [latex]y=\u2212x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          For the following exercises (16-17), the position function of a ball dropped from the top of a 200-meter tall building is given by [latex]s(t)=200-4.9t^2[\/latex], where position [latex]s[\/latex] is measured in meters and time [latex]t[\/latex] is measured in seconds. Round your answer to eight significant digits.<\/p>\n

          \n
          \n

          16. [T]<\/strong> Compute the average velocity of the ball over the given time intervals.<\/p>\n

            \n
          1. [latex][4.99,5][\/latex]<\/li>\n
          2. [latex][5,5.01][\/latex]<\/li>\n
          3. [latex][4.999,5][\/latex]<\/li>\n
          4. [latex][5,5.001][\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
            \n
            \n

            17.\u00a0<\/strong>Use the preceding exercise to guess the instantaneous velocity of the ball at [latex]t=5[\/latex] sec.<\/p>\n

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

            \u221249 m\/sec (velocity of the ball is 49 m\/sec downward)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            For the following exercises (18-19), consider a stone tossed into the air from ground level with an initial velocity of 15 m\/sec. Its height in meters at time [latex]t[\/latex] seconds is [latex]h(t)=15t-4.9t^2[\/latex].<\/p>\n

            \n
            \n

            18. [T]<\/strong> Compute the average velocity of the stone over the given time intervals.<\/p>\n

              \n
            1. [latex][1,1.05][\/latex]<\/li>\n
            2. [latex][1,1.01][\/latex]<\/li>\n
            3. [latex][1,1.005][\/latex]<\/li>\n
            4. [latex][1,1.001][\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
              \n
              \n

              19.\u00a0<\/strong>Use the preceding exercise to guess the instantaneous velocity of the stone at [latex]t=1[\/latex] sec.<\/p>\n

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

              5.2 m\/sec<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

              For the following exercises (20-21), consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by [latex]h(t)=600+78.4t-4.9t^2[\/latex], where [latex]t[\/latex] is measured in seconds.<\/p>\n

              \n
              \n

              20. [T]<\/strong> Compute the average velocity of the rocket over the given time intervals.<\/p>\n

                \n
              1. [latex][9,9.01][\/latex]<\/li>\n
              2. [latex][8.99,9][\/latex]<\/li>\n
              3. [latex][9,9.001][\/latex]<\/li>\n
              4. [latex][8.999,9][\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                \n
                \n

                21.\u00a0<\/strong>Use the preceding exercise to guess the instantaneous velocity of the rocket at [latex]t=9[\/latex] sec.<\/p>\n

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

                \u22129.8 m\/sec<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                For the following exercises (22-23), consider an athlete running a 40-m dash. The position of the athlete is given by [latex]d(t)=\\dfrac{t^3}{6}+4t[\/latex], where [latex]d[\/latex] is the position in meters and [latex]t[\/latex] is the time elapsed, measured in seconds.<\/p>\n

                \n
                \n

                22. [T]<\/strong> Compute the average velocity of the runner over the given time intervals.<\/p>\n

                  \n
                1. [latex][1.95,2.05][\/latex]<\/li>\n
                2. [latex][1.995,2.005][\/latex]<\/li>\n
                3. [latex][1.9995,2.0005][\/latex]<\/li>\n
                4. [latex][2,2.00001][\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
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                  \n

                  23.\u00a0<\/strong>Use the preceding exercise to guess the instantaneous velocity of the runner at [latex]t=2[\/latex] sec.<\/p>\n

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

                  6 m\/sec<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  For the following exercises (24-25), consider the function [latex]f(x)=|x|[\/latex].<\/p>\n

                  \n
                  \n

                  24.\u00a0<\/strong>Sketch the graph of [latex]f[\/latex] over the interval [latex][-1,2][\/latex] and shade the region above the [latex]x[\/latex]-axis.<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  25.\u00a0<\/strong>Use the preceding exercise to find the exact value of the area between the [latex]x[\/latex]-axis and the graph of [latex]f[\/latex] over the interval [latex][-1,2][\/latex] using rectangles. For the rectangles, use the square units, and approximate both above and below the lines. Use geometry to find the exact answer.<\/p>\n

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

                  Under, 1 unit2<\/sup>; over: 4 unit2<\/sup>. The exact area of the two triangles is [latex]\\frac{1}{2}(1)(1)+\\frac{1}{2}(2)(2)=2.5 \\text{units}^2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  For the following exercises (26-27), consider the function [latex]f(x)=\\sqrt{1-x^2}[\/latex]. (Hint<\/em>: This is the upper half of a circle of radius 1 positioned at [latex](0,0)[\/latex].)<\/p>\n

                  \n
                  \n

                  26.\u00a0<\/strong>Sketch the graph of [latex]f[\/latex] over the interval [latex][-1,1][\/latex].<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  27.\u00a0<\/strong>Use the preceding exercise to find the exact area between the [latex]x[\/latex]-axis and the graph of [latex]f[\/latex] over the interval [latex][-1,1][\/latex] using rectangles. For the rectangles, use squares 0.4 by 0.4 units, and approximate both above and below the lines. Use geometry to find the exact answer.<\/p>\n

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

                  Under, 0.96 unit2<\/sup>; over, 1.92 unit2<\/sup>. The exact area of the semicircle with radius 1 is [latex]\\frac{\\pi (1)^2}{2}=\\frac{\\pi }{2}[\/latex] unit2<\/sup>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                  For the following exercises (28-29), consider the function [latex]f(x)=\u2212x^2+1[\/latex].<\/p>\n

                  \n
                  \n

                  28.\u00a0<\/strong>Sketch the graph of [latex]f[\/latex] over the interval [latex][-1,1][\/latex].<\/p>\n<\/div>\n<\/div>\n

                  \n
                  \n

                  29.\u00a0<\/strong>Approximate the area of the region between the [latex]x[\/latex]-axis and the graph of [latex]f[\/latex] over the interval [latex][-1,1][\/latex].<\/p>\n

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

                  Approximately 1.3333333 unit2<\/sup><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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                  Candela Citations<\/h3>\n\t\t\t\t\t
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