{"id":15792,"date":"2019-09-05T19:53:57","date_gmt":"2019-09-05T19:53:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=15792"},"modified":"2025-02-05T05:25:28","modified_gmt":"2025-02-05T05:25:28","slug":"problem-set-73-derivatives","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-73-derivatives\/","title":{"raw":"Problem Set 73: Derivatives","rendered":"Problem Set 73: Derivatives"},"content":{"raw":"1. How is the slope of a linear function similar to the derivative?\r\n\r\n2.\u00a0What is the difference between the average rate of change of a function on the interval [latex]\\left[x,x+h\\right][\/latex] and the derivative of the function at [latex]x?[\/latex]\r\n\r\n3. A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car's average velocity? At exactly 2:30 P.M., the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30 P.M.? Why does this speed differ from the average velocity?\r\n\r\n4. Explain the concept of the slope of a curve at point [latex]x[\/latex].\r\n\r\n5. Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics.\r\n\r\nFor the following exercises, use the definition of derivative [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\dfrac{f\\left(x+h\\right)-f\\left(x\\right)}{h}[\/latex] to calculate the derivative of each function.\r\n\r\n6. [latex]f\\left(x\\right)=3x - 4[\/latex]\r\n\r\n7. [latex]f\\left(x\\right)=-2x+1[\/latex]\r\n\r\n8.\u00a0[latex]f\\left(x\\right)={x}^{2}-2x+1[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)=2{x}^{2}+x - 3[\/latex]\r\n\r\n10.\u00a0[latex]f\\left(x\\right)=2{x}^{2}+5[\/latex]\r\n\r\n11. [latex]f\\left(x\\right)=\\frac{-1}{x - 2}[\/latex]\r\n\r\n12.\u00a0[latex]f\\left(x\\right)=\\frac{2+x}{1-x}[\/latex]\r\n\r\n13. [latex]f\\left(x\\right)=\\frac{5 - 2x}{3+2x}[\/latex]\r\n\r\n14.\u00a0[latex]f\\left(x\\right)=\\sqrt{1+3x}[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)=3{x}^{3}-{x}^{2}+2x+5[\/latex]\r\n\r\n16.\u00a0[latex]f\\left(x\\right)=5[\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=5\\pi [\/latex]\r\n\r\nFor the following exercises, find the average rate of change between the two points.\r\n\r\n18. [latex]\\left(-2,0\\right)[\/latex] and [latex]\\left(-4,5\\right)[\/latex]\r\n\r\n19. [latex]\\left(4,-3\\right)[\/latex] and [latex]\\left(-2,-1\\right)[\/latex]\r\n\r\n20.\u00a0[latex]\\left(0,5\\right)[\/latex] and [latex]\\left(6,5\\right)[\/latex]\r\n\r\n21. [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(7,10\\right)[\/latex]\r\n\r\nFor the following polynomial functions, find the derivatives.\r\n\r\n22. [latex]f\\left(x\\right)={x}^{3}+1[\/latex]\r\n\r\n23. [latex]f\\left(x\\right)=-3{x}^{2}-7x=6[\/latex]\r\n\r\n24.\u00a0[latex]f\\left(x\\right)=7{x}^{2}[\/latex]\r\n\r\n25. [latex]f\\left(x\\right)=3{x}^{3}+2{x}^{2}+x - 26[\/latex]\r\n\r\nFor the following functions, find the equation of the tangent line to the curve at the given point [latex]x[\/latex] on the curve.\r\n\r\n26. [latex]\\begin{array}{ll}f\\left(x\\right)=2{x}^{2}-3x\\hfill &amp; x=3\\hfill \\end{array}[\/latex]\r\n\r\n27. [latex]\\begin{array}{ll}f\\left(x\\right)={x}^{3}+1\\hfill &amp; x=2\\hfill \\end{array}[\/latex]\r\n\r\n28.\u00a0[latex]\\begin{array}{ll}f\\left(x\\right)=\\sqrt{x}\\hfill &amp; x=9\\hfill \\end{array}[\/latex]\r\n\r\nFor the following exercise, find [latex]k[\/latex]\u00a0such that the given line is tangent to the graph of the function.\r\n\r\n29. [latex]\\begin{array}{ll}f\\left(x\\right)={x}^{2}-kx,\\hfill &amp; y=4x - 9\\hfill \\end{array}[\/latex]\r\n\r\nFor the following exercises, consider the graph of the function [latex]f[\/latex] and determine where the function is continuous\/discontinuous and differentiable\/not differentiable.\r\n\r\n30.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185439\/CNX_Precalc_Figure_12_04_201.jpg\" alt=\"Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-3, -2), an open point; the second segment goes from (-3, 1) to (2, 3), which are both open points; the final segment goes from (2, 2), an open point, to positive infinity.\" \/>\r\n\r\n31.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185442\/CNX_Precalc_Figure_12_04_202.jpg\" alt=\"Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.\" \/>\r\n\r\n32.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185444\/CNX_Precalc_Figure_12_04_203.jpg\" alt=\"Graph of a piecewise function with two segments and an asymptote at x = 3. The first segment, which has a removable discontinuity at x = -2, goes from negative infinity to the asymptote, and the final segment goes from the asymptote to positive infinity.\" \/>\r\n\r\n33.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185446\/CNX_Precalc_Figure_12_04_204.jpg\" alt=\"Graph of a piecewise function with two segments. The first segment goes from (-4, 0), an open point to (5, -2), and the final segment goes from (5, 3), an open point, to positive infinity.\" \/>\r\n\r\nFor the following exercises, use Figure 20\u00a0to estimate either the function at a given value of [latex]x[\/latex] or the derivative at a given value of [latex]x[\/latex], as indicated.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185448\/CNX_Precalc_Figure_12_04_205.jpg\" alt=\"Graph of an odd function with multiplicity of 2 with a turning point at (0, -2) and (2, -6).\" width=\"487\" height=\"644\" \/> <b>Figure 20<\/b>[\/caption]\r\n\r\n34. [latex]f\\left(-1\\right)[\/latex]\r\n\r\n35. [latex]f\\left(0\\right)[\/latex]\r\n\r\n36.\u00a0[latex]f\\left(1\\right)[\/latex]\r\n\r\n37. [latex]f\\left(2\\right)[\/latex]\r\n\r\n38.\u00a0[latex]f\\left(3\\right)[\/latex]\r\n\r\n39. [latex]\\begin{align}{f}^{\\prime }\\left(-1\\right)\\end{align}[\/latex]\r\n\r\n40.\u00a0[latex]\\begin{align}{f}^{\\prime }\\left(0\\right)\\end{align}[\/latex]\r\n\r\n41. [latex]\\begin{align}{f}^{\\prime }\\left(1\\right)\\end{align}[\/latex]\r\n\r\n42.\u00a0[latex]\\begin{align}{f}^{\\prime }\\left(2\\right)\\end{align}[\/latex]\r\n\r\n43. [latex]\\begin{align}{f}^{\\prime }\\left(3\\right)\\end{align}[\/latex]\r\n\r\n44. Sketch the function based on the information below:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}{f}^{\\prime }\\left(x\\right)=2x, f\\left(2\\right)=4\\end{align}[\/latex]<\/p>\r\n45.\u00a0Numerically evaluate the derivative. Explore the behavior of the graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] around [latex]x=1[\/latex] by graphing the function on the following domains: [latex]\\left[0.9,1.1\\right][\/latex] , [latex]\\left[0.99,1.01\\right][\/latex] , [latex]\\left[0.999,1.001\\right][\/latex], and [latex]\\left[0.9999,1.0001\\right][\/latex]. We can use the feature on our calculator that automatically sets Ymin and Ymax to the Xmin and Xmax values we preset. (On some of the commonly used graphing calculators, this feature may be called ZOOM FIT or ZOOM AUTO). By examining the corresponding range values for this viewing window, approximate how the curve changes at [latex]x=1[\/latex], that is, approximate the derivative at [latex]x=1[\/latex].\r\n\r\nFor the following exercises, explain the notation in words. The volume [latex]f\\left(t\\right)[\/latex] of a tank of gasoline, in gallons, [latex]t[\/latex] minutes after noon.\r\n\r\n46. [latex]f\\left(0\\right)=600[\/latex]\r\n\r\n47. [latex]\\begin{align}f^{\\prime}\\left(30\\right)=-20\\end{align}[\/latex]\r\n\r\n48.\u00a0[latex]f\\left(30\\right)=0[\/latex]\r\n\r\n49. [latex]\\begin{align}f^{\\prime}\\left(200\\right)=30\\end{align}[\/latex]\r\n\r\n50.\u00a0[latex]f\\left(240\\right)=500[\/latex]\r\n\r\nFor the following exercises, explain the functions in words. The height, [latex]s[\/latex], of a projectile after [latex]t[\/latex] seconds is given by [latex]s\\left(t\\right)=-16{t}^{2}+80t[\/latex].\r\n\r\n51. [latex]s\\left(2\\right)=96[\/latex]\r\n\r\n52.\u00a0[latex]\\begin{align}s^{\\prime}\\left(2\\right)=16\\end{align}[\/latex]\r\n\r\n53. [latex]s\\left(3\\right)=96[\/latex]\r\n\r\n54.\u00a0[latex]\\begin{align}s^{\\prime}\\left(3\\right)=-16\\end{align}[\/latex]\r\n\r\n55. [latex]s\\left(0\\right)=0,s\\left(5\\right)=0[\/latex].\r\n\r\nFor the following exercises, the volume [latex]V[\/latex] of a sphere with respect to its radius [latex]r[\/latex] is given by [latex]V=\\frac{4}{3}\\pi {r}^{3}[\/latex].\r\n\r\n56. Find the average rate of change of [latex]V[\/latex] as [latex]r[\/latex] changes from 1 cm to 2 cm.\r\n\r\n57. Find the instantaneous rate of change of [latex]V[\/latex] when [latex]r=3\\text{ cm}\\text{.}[\/latex]\r\n\r\nFor the following exercises, the revenue generated by selling [latex]x[\/latex] items is given by [latex]R\\left(x\\right)=2{x}^{2}+10x[\/latex].\r\n\r\n58. Find the average change of the revenue function as [latex]x[\/latex] changes from [latex]x=10[\/latex] to [latex]x=20[\/latex].\r\n\r\n59. Find [latex]\\begin{align}R^{\\prime}\\left(10\\right)\\end{align}[\/latex] and interpret.\r\n\r\n60.\u00a0Find [latex]\\begin{align}R^{\\prime}\\left(15\\right)\\end{align}[\/latex] and interpret. Compare [latex]\\begin{align}R^{\\prime}\\left(15\\right)\\end{align}[\/latex] to [latex]\\begin{align}]R^{\\prime}\\left(10\\right)\\end{align}[\/latex], and explain the difference.\r\n\r\nFor the following exercises, the cost of producing [latex]x[\/latex] cellphones is described by the function [latex]C\\left(x\\right)={x}^{2}-4x+1000[\/latex].\r\n\r\n61. Find the average rate of change in the total cost as [latex]x[\/latex] changes from [latex]x=10\\text{ to }x=15[\/latex].\r\n\r\n62.\u00a0Find the approximate marginal cost, when 15 cellphones have been produced, of producing the 16<sup>th<\/sup> cellphone.\r\n\r\n63. Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21<sup>st<\/sup> cellphone.\r\n\r\nFor the following exercises, use the definition for the derivative at a point [latex]x=a[\/latex], [latex]\\underset{x\\to a}{\\mathrm{lim}}\\dfrac{f\\left(x\\right)-f\\left(a\\right)}{x-a}[\/latex], to find the derivative of the functions.\r\n\r\n64. [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]\r\n\r\n65. [latex]f\\left(x\\right)=5{x}^{2}-x+4[\/latex]\r\n\r\n66.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+4x+7[\/latex]\r\n\r\n67. [latex]f\\left(x\\right)=\\frac{-4}{3-{x}^{2}}[\/latex]","rendered":"<p>1. How is the slope of a linear function similar to the derivative?<\/p>\n<p>2.\u00a0What is the difference between the average rate of change of a function on the interval [latex]\\left[x,x+h\\right][\/latex] and the derivative of the function at [latex]x?[\/latex]<\/p>\n<p>3. A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car&#8217;s average velocity? At exactly 2:30 P.M., the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30 P.M.? Why does this speed differ from the average velocity?<\/p>\n<p>4. Explain the concept of the slope of a curve at point [latex]x[\/latex].<\/p>\n<p>5. Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics.<\/p>\n<p>For the following exercises, use the definition of derivative [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\dfrac{f\\left(x+h\\right)-f\\left(x\\right)}{h}[\/latex] to calculate the derivative of each function.<\/p>\n<p>6. [latex]f\\left(x\\right)=3x - 4[\/latex]<\/p>\n<p>7. [latex]f\\left(x\\right)=-2x+1[\/latex]<\/p>\n<p>8.\u00a0[latex]f\\left(x\\right)={x}^{2}-2x+1[\/latex]<\/p>\n<p>9. [latex]f\\left(x\\right)=2{x}^{2}+x - 3[\/latex]<\/p>\n<p>10.\u00a0[latex]f\\left(x\\right)=2{x}^{2}+5[\/latex]<\/p>\n<p>11. [latex]f\\left(x\\right)=\\frac{-1}{x - 2}[\/latex]<\/p>\n<p>12.\u00a0[latex]f\\left(x\\right)=\\frac{2+x}{1-x}[\/latex]<\/p>\n<p>13. [latex]f\\left(x\\right)=\\frac{5 - 2x}{3+2x}[\/latex]<\/p>\n<p>14.\u00a0[latex]f\\left(x\\right)=\\sqrt{1+3x}[\/latex]<\/p>\n<p>15. [latex]f\\left(x\\right)=3{x}^{3}-{x}^{2}+2x+5[\/latex]<\/p>\n<p>16.\u00a0[latex]f\\left(x\\right)=5[\/latex]<\/p>\n<p>17. [latex]f\\left(x\\right)=5\\pi[\/latex]<\/p>\n<p>For the following exercises, find the average rate of change between the two points.<\/p>\n<p>18. [latex]\\left(-2,0\\right)[\/latex] and [latex]\\left(-4,5\\right)[\/latex]<\/p>\n<p>19. [latex]\\left(4,-3\\right)[\/latex] and [latex]\\left(-2,-1\\right)[\/latex]<\/p>\n<p>20.\u00a0[latex]\\left(0,5\\right)[\/latex] and [latex]\\left(6,5\\right)[\/latex]<\/p>\n<p>21. [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(7,10\\right)[\/latex]<\/p>\n<p>For the following polynomial functions, find the derivatives.<\/p>\n<p>22. [latex]f\\left(x\\right)={x}^{3}+1[\/latex]<\/p>\n<p>23. [latex]f\\left(x\\right)=-3{x}^{2}-7x=6[\/latex]<\/p>\n<p>24.\u00a0[latex]f\\left(x\\right)=7{x}^{2}[\/latex]<\/p>\n<p>25. [latex]f\\left(x\\right)=3{x}^{3}+2{x}^{2}+x - 26[\/latex]<\/p>\n<p>For the following functions, find the equation of the tangent line to the curve at the given point [latex]x[\/latex] on the curve.<\/p>\n<p>26. [latex]\\begin{array}{ll}f\\left(x\\right)=2{x}^{2}-3x\\hfill & x=3\\hfill \\end{array}[\/latex]<\/p>\n<p>27. [latex]\\begin{array}{ll}f\\left(x\\right)={x}^{3}+1\\hfill & x=2\\hfill \\end{array}[\/latex]<\/p>\n<p>28.\u00a0[latex]\\begin{array}{ll}f\\left(x\\right)=\\sqrt{x}\\hfill & x=9\\hfill \\end{array}[\/latex]<\/p>\n<p>For the following exercise, find [latex]k[\/latex]\u00a0such that the given line is tangent to the graph of the function.<\/p>\n<p>29. [latex]\\begin{array}{ll}f\\left(x\\right)={x}^{2}-kx,\\hfill & y=4x - 9\\hfill \\end{array}[\/latex]<\/p>\n<p>For the following exercises, consider the graph of the function [latex]f[\/latex] and determine where the function is continuous\/discontinuous and differentiable\/not differentiable.<\/p>\n<p>30.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185439\/CNX_Precalc_Figure_12_04_201.jpg\" alt=\"Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-3, -2), an open point; the second segment goes from (-3, 1) to (2, 3), which are both open points; the final segment goes from (2, 2), an open point, to positive infinity.\" \/><\/p>\n<p>31.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185442\/CNX_Precalc_Figure_12_04_202.jpg\" alt=\"Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.\" \/><\/p>\n<p>32.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185444\/CNX_Precalc_Figure_12_04_203.jpg\" alt=\"Graph of a piecewise function with two segments and an asymptote at x = 3. The first segment, which has a removable discontinuity at x = -2, goes from negative infinity to the asymptote, and the final segment goes from the asymptote to positive infinity.\" \/><\/p>\n<p>33.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185446\/CNX_Precalc_Figure_12_04_204.jpg\" alt=\"Graph of a piecewise function with two segments. The first segment goes from (-4, 0), an open point to (5, -2), and the final segment goes from (5, 3), an open point, to positive infinity.\" \/><\/p>\n<p>For the following exercises, use Figure 20\u00a0to estimate either the function at a given value of [latex]x[\/latex] or the derivative at a given value of [latex]x[\/latex], as indicated.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185448\/CNX_Precalc_Figure_12_04_205.jpg\" alt=\"Graph of an odd function with multiplicity of 2 with a turning point at (0, -2) and (2, -6).\" width=\"487\" height=\"644\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 20<\/b><\/p>\n<\/div>\n<p>34. [latex]f\\left(-1\\right)[\/latex]<\/p>\n<p>35. [latex]f\\left(0\\right)[\/latex]<\/p>\n<p>36.\u00a0[latex]f\\left(1\\right)[\/latex]<\/p>\n<p>37. [latex]f\\left(2\\right)[\/latex]<\/p>\n<p>38.\u00a0[latex]f\\left(3\\right)[\/latex]<\/p>\n<p>39. [latex]\\begin{align}{f}^{\\prime }\\left(-1\\right)\\end{align}[\/latex]<\/p>\n<p>40.\u00a0[latex]\\begin{align}{f}^{\\prime }\\left(0\\right)\\end{align}[\/latex]<\/p>\n<p>41. [latex]\\begin{align}{f}^{\\prime }\\left(1\\right)\\end{align}[\/latex]<\/p>\n<p>42.\u00a0[latex]\\begin{align}{f}^{\\prime }\\left(2\\right)\\end{align}[\/latex]<\/p>\n<p>43. [latex]\\begin{align}{f}^{\\prime }\\left(3\\right)\\end{align}[\/latex]<\/p>\n<p>44. Sketch the function based on the information below:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}{f}^{\\prime }\\left(x\\right)=2x, f\\left(2\\right)=4\\end{align}[\/latex]<\/p>\n<p>45.\u00a0Numerically evaluate the derivative. Explore the behavior of the graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] around [latex]x=1[\/latex] by graphing the function on the following domains: [latex]\\left[0.9,1.1\\right][\/latex] , [latex]\\left[0.99,1.01\\right][\/latex] , [latex]\\left[0.999,1.001\\right][\/latex], and [latex]\\left[0.9999,1.0001\\right][\/latex]. We can use the feature on our calculator that automatically sets Ymin and Ymax to the Xmin and Xmax values we preset. (On some of the commonly used graphing calculators, this feature may be called ZOOM FIT or ZOOM AUTO). By examining the corresponding range values for this viewing window, approximate how the curve changes at [latex]x=1[\/latex], that is, approximate the derivative at [latex]x=1[\/latex].<\/p>\n<p>For the following exercises, explain the notation in words. The volume [latex]f\\left(t\\right)[\/latex] of a tank of gasoline, in gallons, [latex]t[\/latex] minutes after noon.<\/p>\n<p>46. [latex]f\\left(0\\right)=600[\/latex]<\/p>\n<p>47. [latex]\\begin{align}f^{\\prime}\\left(30\\right)=-20\\end{align}[\/latex]<\/p>\n<p>48.\u00a0[latex]f\\left(30\\right)=0[\/latex]<\/p>\n<p>49. [latex]\\begin{align}f^{\\prime}\\left(200\\right)=30\\end{align}[\/latex]<\/p>\n<p>50.\u00a0[latex]f\\left(240\\right)=500[\/latex]<\/p>\n<p>For the following exercises, explain the functions in words. The height, [latex]s[\/latex], of a projectile after [latex]t[\/latex] seconds is given by [latex]s\\left(t\\right)=-16{t}^{2}+80t[\/latex].<\/p>\n<p>51. [latex]s\\left(2\\right)=96[\/latex]<\/p>\n<p>52.\u00a0[latex]\\begin{align}s^{\\prime}\\left(2\\right)=16\\end{align}[\/latex]<\/p>\n<p>53. [latex]s\\left(3\\right)=96[\/latex]<\/p>\n<p>54.\u00a0[latex]\\begin{align}s^{\\prime}\\left(3\\right)=-16\\end{align}[\/latex]<\/p>\n<p>55. [latex]s\\left(0\\right)=0,s\\left(5\\right)=0[\/latex].<\/p>\n<p>For the following exercises, the volume [latex]V[\/latex] of a sphere with respect to its radius [latex]r[\/latex] is given by [latex]V=\\frac{4}{3}\\pi {r}^{3}[\/latex].<\/p>\n<p>56. Find the average rate of change of [latex]V[\/latex] as [latex]r[\/latex] changes from 1 cm to 2 cm.<\/p>\n<p>57. Find the instantaneous rate of change of [latex]V[\/latex] when [latex]r=3\\text{ cm}\\text{.}[\/latex]<\/p>\n<p>For the following exercises, the revenue generated by selling [latex]x[\/latex] items is given by [latex]R\\left(x\\right)=2{x}^{2}+10x[\/latex].<\/p>\n<p>58. Find the average change of the revenue function as [latex]x[\/latex] changes from [latex]x=10[\/latex] to [latex]x=20[\/latex].<\/p>\n<p>59. Find [latex]\\begin{align}R^{\\prime}\\left(10\\right)\\end{align}[\/latex] and interpret.<\/p>\n<p>60.\u00a0Find [latex]\\begin{align}R^{\\prime}\\left(15\\right)\\end{align}[\/latex] and interpret. Compare [latex]\\begin{align}R^{\\prime}\\left(15\\right)\\end{align}[\/latex] to [latex]\\begin{align}]R^{\\prime}\\left(10\\right)\\end{align}[\/latex], and explain the difference.<\/p>\n<p>For the following exercises, the cost of producing [latex]x[\/latex] cellphones is described by the function [latex]C\\left(x\\right)={x}^{2}-4x+1000[\/latex].<\/p>\n<p>61. Find the average rate of change in the total cost as [latex]x[\/latex] changes from [latex]x=10\\text{ to }x=15[\/latex].<\/p>\n<p>62.\u00a0Find the approximate marginal cost, when 15 cellphones have been produced, of producing the 16<sup>th<\/sup> cellphone.<\/p>\n<p>63. Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21<sup>st<\/sup> cellphone.<\/p>\n<p>For the following exercises, use the definition for the derivative at a point [latex]x=a[\/latex], [latex]\\underset{x\\to a}{\\mathrm{lim}}\\dfrac{f\\left(x\\right)-f\\left(a\\right)}{x-a}[\/latex], to find the derivative of the functions.<\/p>\n<p>64. [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/p>\n<p>65. [latex]f\\left(x\\right)=5{x}^{2}-x+4[\/latex]<\/p>\n<p>66.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+4x+7[\/latex]<\/p>\n<p>67. [latex]f\\left(x\\right)=\\frac{-4}{3-{x}^{2}}[\/latex]<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15792\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15792","chapter","type-chapter","status-publish","hentry"],"part":14921,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15792","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15792\/revisions"}],"predecessor-version":[{"id":15794,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15792\/revisions\/15794"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/14921"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/15792\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=15792"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=15792"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=15792"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=15792"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}