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\n65. [latex]f\\left(x\\right)=5{x}^{2}-x+4[\/latex]\r\n\r\n67. [latex]f\\left(x\\right)=\\frac{-4}{3-{x}^{2}}[\/latex]","rendered":"Finding Limits: Numerical and Graphical Approaches<\/h1>\n
1. Explain the difference between a value at [latex]x=a[\/latex] and the limit as [latex]x[\/latex] approaches [latex]a[\/latex].<\/p>\n
For the following exercises, estimate the functional values and the limits from the graph of the function [latex]f[\/latex].<\/p>\n
![\"A](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185247\/CNX_Precalc_Figure_12_01_2012.jpg\")
<\/p>\n
3. [latex]\\underset{x\\to -{2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/p>\n
5. [latex]\\underset{x\\to -2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/p>\n
7. [latex]\\underset{x\\to -{1}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/p>\n
9. [latex]\\underset{x\\to 1}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/p>\n
11. [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/p>\n
13. [latex]\\underset{x\\to 4}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/p>\n
For the following exercises, draw the graph of a function from the functional values and limits provided.<\/p>\n
15. [latex]\\underset{x\\to {0}^{-}}{\\mathrm{lim}}f\\left(x\\right)=2,\\underset{x\\to {0}^{+}}{\\mathrm{lim}}f\\left(x\\right)=-3,\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)=2,f\\left(0\\right)=4,f\\left(2\\right)=-1,f\\left(-3\\right)\\text{ does not exist}[\/latex].<\/p>\n
17.\u00a0[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=2,\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)=-3,\\underset{x\\to 0}{\\mathrm{lim}}f\\left(x\\right)=5,f\\left(0\\right)=1,f\\left(1\\right)=0[\/latex]<\/p>\n
19.\u00a0[latex]\\underset{x\\to 4}{\\mathrm{lim}}f\\left(x\\right)=6,\\underset{x\\to {6}^{+}}{\\mathrm{lim}}f\\left(x\\right)=-1,\\underset{x\\to 0}{\\mathrm{lim}}f\\left(x\\right)=5,f\\left(4\\right)=6,f\\left(2\\right)=6[\/latex]<\/p>\n
21.\u00a0[latex]\\underset{x\\to \\pi }{\\mathrm{lim}}f\\left(x\\right)={\\pi }^{2},\\underset{x\\to -\\pi }{\\mathrm{lim}}f\\left(x\\right)=\\frac{\\pi }{2},\\underset{x\\to {1}^{-}}{\\mathrm{lim}}f\\left(x\\right)=0,f\\left(\\pi \\right)=\\sqrt{2},f\\left(0\\right)\\text{ does not exist}[\/latex].<\/p>\n
For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as [latex]x[\/latex] approaches 0.<\/p>\n
23. [latex]g\\left(x\\right)={\\left(1+x\\right)}^{\\frac{2}{x}}[\/latex]<\/p>\n
25. [latex]i\\left(x\\right)={\\left(1+x\\right)}^{\\frac{4}{x}}[\/latex]<\/p>\n
27. Based on the pattern you observed in the exercises above, make a conjecture as to the limit of [latex]f\\left(x\\right)={\\left(1+x\\right)}^{\\frac{6}{x}}[\/latex], [latex]g\\left(x\\right)={\\left(1+x\\right)}^{\\frac{7}{x}}[\/latex], [latex]\\text{and }h\\left(x\\right)={\\left(1+x\\right)}^{\\frac{n}{x}}[\/latex].<\/p>\n
For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as [latex]x[\/latex] approaches [latex]a[\/latex]. If the function has a limit as [latex]x[\/latex] approaches [latex]a[\/latex], state it. If not, discuss why there is no limit.<\/p>\n
29. [latex]f\\left(x\\right)=\\begin{cases}\\dfrac{1}{x+1},\\hfill& \\text{if }x=\u22122 \\\\ \\left(x+1\\right)^{2},\\hfill& \\text{if }x\\ne\u22122\\end{cases};\\text{ }a=\u22122[\/latex]<\/p>\n
For the following exercises, use numerical evidence to determine whether the limit exists at [latex]x=a[\/latex]. If not, describe the behavior of the graph of the function near [latex]x=a[\/latex]. Round answers to two decimal places.<\/p>\n
31. [latex]f\\left(x\\right)=\\frac{{x}^{2}-x - 6}{{x}^{2}-9};a=3[\/latex]<\/p>\n
33. [latex]f\\left(x\\right)=\\frac{{x}^{2}-1}{{x}^{2}-3x+2};a=1[\/latex]<\/p>\n
35. [latex]f\\left(x\\right)=\\frac{10 - 10{x}^{2}}{{x}^{2}-3x+2};a=1[\/latex]<\/p>\n
37. [latex]f\\left(x\\right)=\\frac{x}{4{x}^{2}+4x+1};a=-\\frac{1}{2}[\/latex]<\/p>\n
For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as [latex]x[\/latex] approaches the given value.<\/p>\n
39. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\dfrac{7\\tan x}{3x}[\/latex]<\/p>\n
41. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\dfrac{2\\sin x}{4\\tan x}[\/latex]<\/p>\n
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as [latex]x[\/latex] approaches [latex]a[\/latex]. If the function has a limit as [latex]x[\/latex] approaches [latex]a[\/latex], state it. If not, discuss why there is no limit.<\/p>\n
43. [latex]\\underset{x\\to 0}{\\mathrm{lim}}{e}^{{e}^{-\\frac{1}{{x}^{2}}}}[\/latex]<\/p>\n
45. [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\dfrac{|x+1|}{x+1}[\/latex]<\/p>\n
47. [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\dfrac{1}{{\\left(x+1\\right)}^{2}}[\/latex]<\/p>\n
49. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\dfrac{5}{1-{e}^{\\frac{2}{x}}}[\/latex]<\/p>\n
51.\u00a0According to the Theory of Relativity, the mass [latex]m[\/latex] of a particle depends on its velocity [latex]v[\/latex] . That is<\/p>\n
[latex]m=\\frac{{m}_{o}}{\\sqrt{1-\\left({v}^{2}\/{c}^{2}\\right)}}[\/latex]<\/p>\n
where [latex]{m}_{o}[\/latex] is the mass when the particle is at rest and [latex]c[\/latex] is the speed of light. Find the limit of the mass, [latex]m[\/latex], as [latex]v[\/latex] approaches [latex]{c}^{-}[\/latex].<\/p>\n
Finding Limits: Properties of Limits<\/h1>\n
1. Give an example of a type of function [latex]f[\/latex] whose limit, as [latex]x[\/latex] approaches [latex]a[\/latex], is [latex]f\\left(a\\right)[\/latex].<\/p>\n
3. What does it mean to say the limit of [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]c[\/latex], is undefined?<\/p>\n
For the following exercises, evaluate the limits algebraically.<\/p>\n
5. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{-5x}{{x}^{2}-1}\\right)[\/latex]<\/p>\n
7. [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-9}{x - 3}\\right)[\/latex]<\/p>\n
9. [latex]\\underset{x\\to \\frac{3}{2}}{\\mathrm{lim}}\\left(\\dfrac{6{x}^{2}-17x+12}{2x - 3}\\right)[\/latex]<\/p>\n
11. [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-9}{x - 5x+6}\\right)[\/latex]<\/p>\n
13. [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}+2x - 3}{x - 3}\\right)[\/latex]<\/p>\n
15. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{{\\left(2-h\\right)}^{3}-8}{h}\\right)[\/latex]<\/p>\n
17. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{5-h}-\\sqrt{5}}{h}\\right)[\/latex]<\/p>\n
19. [latex]\\underset{x\\to 9}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-81}{3-\\sqrt{x}}\\right)[\/latex]<\/p>\n
21. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{x}{\\sqrt{1+2x}-1}\\right)[\/latex]<\/p>\n
23. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{{x}^{3}-64}{{x}^{2}-16}\\right)[\/latex]<\/p>\n
25. [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}\\left(\\dfrac{|x - 2|}{x - 2}\\right)[\/latex]<\/p>\n
27. [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}\\left(\\dfrac{|x - 4|}{4-x}\\right)[\/latex]<\/p>\n
29. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{|x - 4|}{4-x}\\right)[\/latex]<\/p>\n
For the following exercise, use the given information to evaluate the limits: [latex]\\underset{x\\to c}{\\mathrm{lim}}f\\left(x\\right)=3[\/latex], [latex]\\underset{x\\to c}{\\mathrm{lim}}g\\left(x\\right)=5[\/latex]<\/p>\n
31. [latex]\\underset{x\\to c}{\\mathrm{lim}}\\left[2f\\left(x\\right)+\\sqrt{g\\left(x\\right)}\\right][\/latex]<\/p>\n
33. [latex]\\underset{x\\to c}{\\mathrm{lim}}\\frac{f\\left(x\\right)}{g\\left(x\\right)}[\/latex]<\/p>\n
For the following exercises, evaluate the following limits.<\/p>\n
35. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\sin \\left(\\pi x\\right)[\/latex]<\/p>\n
37. [latex]{f}\\left(x\\right)=\\begin{cases}2x^{2}+2x+1, \\hfill& x\\leq0 \\\\ x-3, \\hfill& x>0\\end{cases};\\underset{x\\to 0^{+}}{\\mathrm{lim}}f \\left(x\\right)[\/latex]<\/p>\n
39. [latex]{f}\\left(x\\right)=\\begin{cases}2x^{2}+2x+1, \\hfill& x\\leq0 \\\\ x-3, \\hfill& x>0\\end{cases};\\underset{x\\to 0}{\\mathrm{lim}}f \\left(x\\right)[\/latex]<\/p>\n
41.\u00a0[latex]\\underset{x\\to {3}^{+}}{\\mathrm{lim}}\\dfrac{{x}^{2}}{{x}^{2}-9}[\/latex]<\/p>\n
For the following exercises, find the average rate of change [latex]\\frac{f\\left(x+h\\right)-f\\left(x\\right)}{h}[\/latex].<\/p>\n
43. [latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]<\/p>\n
45. [latex]f\\left(x\\right)={x}^{2}+4x - 100[\/latex]<\/p>\n
47. [latex]f\\left(x\\right)=\\cos \\left(x\\right)[\/latex]<\/p>\n
49. [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/p>\n
51. [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n
53. Find an equation that could be represented by the graph.<\/p>\n
![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185304\/CNX_Precalc_Figure_12_02_2022.jpg\")
<\/p>\n
For the following exercises, refer to the graph.<\/p>\n
![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185307\/CNX_Precalc_Figure_12_02_203F2.jpg\")
<\/p>\n
55. What is the left-hand limit of the function as [latex]x[\/latex] approaches 0?<\/p>\n
57. The height of a projectile is given by [latex]s\\left(t\\right)=-64{t}^{2}+192t[\/latex] Find the average rate of change of the height from [latex]t=1[\/latex] second to [latex]t=1.5[\/latex] seconds.<\/p>\n
Continuity<\/h1>\n
1. State in your own words what it means for a function [latex]f[\/latex] to be continuous at [latex]x=c[\/latex].<\/p>\n
For the following exercises, determine why the function [latex]f[\/latex] is discontinuous at a given point [latex]a[\/latex] on the graph. State which condition fails.<\/p>\n
3. [latex]f\\left(x\\right)=\\mathrm{ln}\\text{ }|\\text{ }x+3\\text{ }|,a=-3[\/latex]<\/p>\n
5. [latex]f\\left(x\\right)=\\frac{{x}^{2}-16}{x+4},a=-4[\/latex]<\/p>\n
7. [latex]f\\left(x\\right)=\\begin{cases}x,\\hfill& x\\neq 3 \\\\ 2x, \\hfill& x=3\\end{cases}a=3[\/latex]<\/p>\n
9.\u00a0[latex]f\\left(x\\right)=\\begin{cases}\\frac{1}{2-x}, \\hfill& x\\neq 2 \\\\ 3, \\hfill& x=2\\end{cases}a=2[\/latex]<\/p>\n
11. [latex]f\\left(x\\right)=\\begin{cases}3+x, \\hfill& x<1 \\\\ x, \\hfill& x=1 \\\\ x^{2}, \\hfill& x>1\\end{cases}a=1[\/latex]<\/p>\n
13. [latex]f\\left(x\\right)=\\begin{cases}3+2x, \\hfill& x<1 \\\\ x, \\hfill& x=1 \\\\ -x^{2}, \\hfill& x>1\\end{cases}a=1[\/latex]<\/p>\n
15. [latex]f\\left(x\\right)=\\begin{cases}\\frac{x^{2}-9}{x+3}, \\hfill& x<-3 \\\\ x-9, \\hfill& x=-3 \\\\ \\frac{1}{x}, \\hfill& x>-3\\end{cases}a=-3[\/latex]<\/p>\n
17. [latex]f\\left(x\\right)=\\frac{{x}^{2}-4}{x - 2},\\text{ }a=2[\/latex]<\/p>\n
19. [latex]f\\left(x\\right)=\\frac{{x}^{3}-9x}{{x}^{2}+11x+24},\\text{ }a=-3[\/latex]<\/p>\n
21. [latex]f\\left(x\\right)=\\frac{x}{|x|},\\text{ }a=0[\/latex]<\/p>\n
For the following exercises, determine whether or not the given function [latex]f[\/latex] is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.<\/p>\n
23. [latex]f\\left(x\\right)={x}^{3}-2x - 15[\/latex]<\/p>\n
25. [latex]f\\left(x\\right)=2\\cdot {3}^{x+4}[\/latex]<\/p>\n
27. [latex]f\\left(x\\right)=\\frac{|x - 2|}{{x}^{2}-2x}[\/latex]<\/p>\n
29. [latex]f\\left(x\\right)=2x+\\frac{5}{x}[\/latex]<\/p>\n
31. [latex]f\\left(x\\right)=\\mathrm{ln}\\text{ }{x}^{2}[\/latex]<\/p>\n
33. [latex]f\\left(x\\right)=\\sqrt{x - 4}[\/latex]<\/p>\n
35. [latex]f\\left(x\\right)={x}^{2}+\\sin \\left(x\\right)[\/latex]<\/p>\n
For the following exercises, refer to the graph. Each square represents one square unit. For each value of [latex]a[\/latex], determine which of the three conditions of continuity are satisfied at [latex]x=a[\/latex] and which are not.<\/p>\n
![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185341\/CNX_Precalc_Figure_12_03_201F2.jpg\")
<\/p>\n
37. [latex]x=-3[\/latex]<\/p>\n
39. [latex]x=4[\/latex]<\/p>\n
For the following exercises, consider the function shown in the graph.<\/p>\n
![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185346\/CNX_Precalc_Figure_12_03_203.jpg\")
<\/p>\n
45. What condition of continuity is violated at each of the points where the graph is discontinuous?<\/p>\n
47. Construct a function that passes through the origin with a constant slope of 1, with r<\/p>\n
51. The graph of [latex]f\\left(x\\right)=\\frac{\\sin \\left(2x\\right)}{x}[\/latex] is shown. Is the function [latex]f\\left(x\\right)[\/latex] continuous at [latex]x=0?[\/latex] Why or why not?<\/p>\n
![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185352\/CNX_Precalc_Figure_12_03_206.jpg\")
<\/p>\n
Derivatives<\/h1>\n
1. How is the slope of a linear function similar to the derivative?<\/p>\n
3. 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?<\/p>\n
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
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
7. [latex]f\\left(x\\right)=-2x+1[\/latex]<\/p>\n
9. [latex]f\\left(x\\right)=2{x}^{2}+x - 3[\/latex]<\/p>\n
11. [latex]f\\left(x\\right)=\\frac{-1}{x - 2}[\/latex]<\/p>\n
13. [latex]f\\left(x\\right)=\\frac{5 - 2x}{3+2x}[\/latex]<\/p>\n
15. [latex]f\\left(x\\right)=3{x}^{3}-{x}^{2}+2x+5[\/latex]<\/p>\n
17. [latex]f\\left(x\\right)=5\\pi[\/latex]<\/p>\n
For the following exercises, find the average rate of change between the two points.<\/p>\n
19. [latex]\\left(4,-3\\right)[\/latex] and [latex]\\left(-2,-1\\right)[\/latex]<\/p>\n
21. [latex]\\left(7,-2\\right)[\/latex] and [latex]\\left(7,10\\right)[\/latex]<\/p>\n
For the following polynomial functions, find the derivatives.<\/p>\n
23. [latex]f\\left(x\\right)=-3{x}^{2}-7x=6[\/latex]<\/p>\n
25. [latex]f\\left(x\\right)=3{x}^{3}+2{x}^{2}+x - 26[\/latex]<\/p>\n
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
27. [latex]\\begin{array}{ll}f\\left(x\\right)={x}^{3}+1\\hfill & x=2\\hfill \\end{array}[\/latex]<\/p>\n
For the following exercise, find [latex]k[\/latex]\u00a0such that the given line is tangent to the graph of the function.<\/p>\n
29. [latex]\\begin{array}{ll}f\\left(x\\right)={x}^{2}-kx,\\hfill & y=4x - 9\\hfill \\end{array}[\/latex]<\/p>\n
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
31.
\n![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185442\/CNX_Precalc_Figure_12_04_202.jpg\")
<\/p>\n
33.
\n![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185446\/CNX_Precalc_Figure_12_04_204.jpg\")
<\/p>\n
For the following exercises, use the graph to 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
![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185448\/CNX_Precalc_Figure_12_04_205.jpg\")
<\/p>\n
35. [latex]f\\left(0\\right)[\/latex]<\/p>\n
37. [latex]f\\left(2\\right)[\/latex]<\/p>\n
39. [latex]\\begin{align}{f}^{\\prime }\\left(-1\\right)\\end{align}[\/latex]<\/p>\n
41. [latex]\\begin{align}{f}^{\\prime }\\left(1\\right)\\end{align}[\/latex]<\/p>\n
43. [latex]\\begin{align}{f}^{\\prime }\\left(3\\right)\\end{align}[\/latex]<\/p>\n
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
47. [latex]\\begin{align}f^{\\prime}\\left(30\\right)=-20\\end{align}[\/latex]<\/p>\n
49. [latex]\\begin{align}f^{\\prime}\\left(200\\right)=30\\end{align}[\/latex]<\/p>\n
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
51. [latex]s\\left(2\\right)=96[\/latex]<\/p>\n
53. [latex]s\\left(3\\right)=96[\/latex]<\/p>\n
55. [latex]s\\left(0\\right)=0,s\\left(5\\right)=0[\/latex].<\/p>\n
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
57. Find the instantaneous rate of change of [latex]V[\/latex] when [latex]r=3\\text{ cm}\\text{.}[\/latex]<\/p>\n
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
59. Find [latex]\\begin{align}R^{\\prime}\\left(10\\right)\\end{align}[\/latex] and interpret.<\/p>\n
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
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
63. Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21st<\/sup> cellphone.<\/p>\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.<\/p>\n
65. [latex]f\\left(x\\right)=5{x}^{2}-x+4[\/latex]<\/p>\n
67. [latex]f\\left(x\\right)=\\frac{-4}{3-{x}^{2}}[\/latex]<\/p>\n","protected":false},"author":67,"menu_order":31,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":263,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2379"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2379\/revisions"}],"predecessor-version":[{"id":5724,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2379\/revisions\/5724"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/263"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2379\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2379"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2379"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2379"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}