{"id":339,"date":"2021-03-25T15:53:18","date_gmt":"2021-03-25T15:53:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&p=339"},"modified":"2021-11-17T02:39:30","modified_gmt":"2021-11-17T02:39:30","slug":"module-3-review-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/module-3-review-problems\/","title":{"raw":"Module 3 Review Problems","rendered":"Module 3 Review Problems"},"content":{"raw":"

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]\\displaystyle\\int {e}^{x}\\sin\\left(x\\right)dx[\/latex] cannot be integrated by parts.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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2.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{4}+1}dx[\/latex] cannot be integrated using partial fractions.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"425440\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"425440\"]False[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>In numerical integration, increasing the number of points decreases the error.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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4.\u00a0<\/strong>Integration by parts can always yield the integral.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"3612\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"3612\"]False[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, evaluate the integral using the specified method.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]\\displaystyle\\int {x}^{2}\\sin\\left(4x\\right)dx[\/latex] using integration by parts<\/div>\r\n<\/div>\r\n<\/div>\r\n
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6.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{2}\\sqrt{{x}^{2}+16}}dx[\/latex] using trigonometric substitution<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"537918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"537918\"][latex]-\\frac{\\sqrt{{x}^{2}+16}}{16x}+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{x}\\text{ln}\\left(x\\right)dx[\/latex] using integration by parts<\/div>\r\n<\/div>\r\n<\/div>\r\n
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8.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x}{{x}^{3}+2{x}^{2}-5x - 6}dx[\/latex] using partial fractions<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"449235\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"449235\"][latex]\\frac{1}{10}\\left(4\\text{ln}\\left(2-x\\right)+5\\text{ln}\\left(x+1\\right)-9\\text{ln}\\left(x+3\\right)\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{5}}{{\\left(4{x}^{2}+4\\right)}^{\\frac{5}{2}}}dx[\/latex] using trigonometric substitution<\/div>\r\n<\/div>\r\n<\/div>\r\n
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10.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{4-{\\sin}^{2}\\left(x\\right)}}{{\\sin}^{2}\\left(x\\right)}\\cos\\left(x\\right)dx[\/latex] using a table of integrals or a CAS<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"488903\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"488903\"][latex]-\\frac{\\sqrt{4-{\\sin}^{2}\\left(x\\right)}}{\\sin\\left(x\\right)}-\\frac{x}{2}+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, integrate using whatever method you choose.<\/p>\r\n\r\n

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11.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}\\left(x\\right){\\cos}^{2}\\left(x\\right)dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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12.\u00a0<\/strong>[latex]\\displaystyle\\int {x}^{3}\\sqrt{{x}^{2}+2}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"996516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"996516\"][latex]\\frac{1}{15}{\\left({x}^{2}+2\\right)}^{\\frac{3}{2}}\\left(3{x}^{2}-4\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

13.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3{x}^{2}+1}{{x}^{4}-2{x}^{3}-{x}^{2}+2x}dx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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14.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{4}+4}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"139195\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"139195\"][latex]\\frac{1}{16}\\text{ln}\\left(\\frac{{x}^{2}+2x+2}{{x}^{2}-2x+2}\\right)-\\frac{1}{8}{\\tan}^{-1}\\left(1-x\\right)+\\frac{1}{8}{\\tan}^{-1}\\left(x+1\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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15.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{3+16{x}^{4}}}{{x}^{4}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson\u2019s rule using four subintervals, rounding to three decimals.<\/p>\r\n\r\n

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16. [T]<\/strong> [latex]{\\displaystyle\\int }_{1}^{2}\\sqrt{{x}^{5}+2}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"295311\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"295311\"][latex]{\\text{M}}_{4}=3.312,{\\text{T}}_{4}=3.354,{S}_{4}=3.326[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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17. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{\\sqrt{\\pi }}{e}^{\\text{-}\\sin\\left({x}^{2}\\right)}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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18. [T]<\/strong> [latex]{\\displaystyle\\int }_{1}^{4}\\frac{\\text{ln}\\frac{1}{x}}{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"169235\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"169235\"][latex]{\\text{M}}_{4}=-0.982,{\\text{T}}_{4}=-0.917,{S}_{4}=-0.952[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, evaluate the integrals, if possible.<\/p>\r\n\r\n

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19.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{1}{{x}^{n}}dx[\/latex], for what values of [latex]n[\/latex] does this integral converge or diverge?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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20.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{{e}^{\\text{-}x}}{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"356144\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"356144\"]approximately 0.2194[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, consider the gamma function given by [latex]\\Gamma\\left(a\\right)={\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}y}{y}^{a - 1}dy[\/latex].<\/p>\r\n\r\n

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21.\u00a0<\/strong>Show that [latex]\\Gamma\\left(a\\right)=\\left(a - 1\\right)\\Gamma\\left(a - 1\\right)[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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22.\u00a0<\/strong>Extend to show that [latex]\\Gamma\\left(a\\right)=\\left(a - 1\\right)\\text{!}[\/latex], assuming [latex]a[\/latex] is a positive integer.<\/div>\r\n

The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km\/h. The graph represents its velocity.<\/p>\r\n\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [T]<\/strong> Use the graph to estimate the velocity every 20 sec and fit to a graph of the form [latex]v\\left(t\\right)=a{\\text{exp}}^{bx}\\sin\\left(cx\\right)+d[\/latex]. (Hint:<\/em> Consider the time units.)<\/div>\r\n<\/div>\r\n<\/div>\r\n
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24. [T]<\/strong> Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"115920\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"115920\"]Answers may vary. Ex: [latex]9.405[\/latex] km[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.<\/p>\n

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\n
1.\u00a0<\/strong>[latex]\\displaystyle\\int {e}^{x}\\sin\\left(x\\right)dx[\/latex] cannot be integrated by parts.<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{4}+1}dx[\/latex] cannot be integrated using partial fractions.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
False<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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3.\u00a0<\/strong>In numerical integration, increasing the number of points decreases the error.<\/div>\n<\/div>\n<\/div>\n
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4.\u00a0<\/strong>Integration by parts can always yield the integral.<\/p>\n<\/div>\n

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

For the following exercises, evaluate the integral using the specified method.<\/span><\/p>\n<\/div>\n<\/div>\n

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5.\u00a0<\/strong>[latex]\\displaystyle\\int {x}^{2}\\sin\\left(4x\\right)dx[\/latex] using integration by parts<\/div>\n<\/div>\n<\/div>\n
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6.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{2}\\sqrt{{x}^{2}+16}}dx[\/latex] using trigonometric substitution<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{\\sqrt{{x}^{2}+16}}{16x}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{x}\\text{ln}\\left(x\\right)dx[\/latex] using integration by parts<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x}{{x}^{3}+2{x}^{2}-5x - 6}dx[\/latex] using partial fractions<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{10}\\left(4\\text{ln}\\left(2-x\\right)+5\\text{ln}\\left(x+1\\right)-9\\text{ln}\\left(x+3\\right)\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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9.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{5}}{{\\left(4{x}^{2}+4\\right)}^{\\frac{5}{2}}}dx[\/latex] using trigonometric substitution<\/div>\n<\/div>\n<\/div>\n
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10.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{4-{\\sin}^{2}\\left(x\\right)}}{{\\sin}^{2}\\left(x\\right)}\\cos\\left(x\\right)dx[\/latex] using a table of integrals or a CAS<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{\\sqrt{4-{\\sin}^{2}\\left(x\\right)}}{\\sin\\left(x\\right)}-\\frac{x}{2}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, integrate using whatever method you choose.<\/p>\n

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\n
11.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}\\left(x\\right){\\cos}^{2}\\left(x\\right)dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]\\displaystyle\\int {x}^{3}\\sqrt{{x}^{2}+2}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{15}{\\left({x}^{2}+2\\right)}^{\\frac{3}{2}}\\left(3{x}^{2}-4\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
13.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3{x}^{2}+1}{{x}^{4}-2{x}^{3}-{x}^{2}+2x}dx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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14.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{4}+4}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{16}\\text{ln}\\left(\\frac{{x}^{2}+2x+2}{{x}^{2}-2x+2}\\right)-\\frac{1}{8}{\\tan}^{-1}\\left(1-x\\right)+\\frac{1}{8}{\\tan}^{-1}\\left(x+1\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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15.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{3+16{x}^{4}}}{{x}^{4}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson\u2019s rule using four subintervals, rounding to three decimals.<\/p>\n

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16. [T]<\/strong> [latex]{\\displaystyle\\int }_{1}^{2}\\sqrt{{x}^{5}+2}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]{\\text{M}}_{4}=3.312,{\\text{T}}_{4}=3.354,{S}_{4}=3.326[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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17. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{\\sqrt{\\pi }}{e}^{\\text{-}\\sin\\left({x}^{2}\\right)}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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18. [T]<\/strong> [latex]{\\displaystyle\\int }_{1}^{4}\\frac{\\text{ln}\\frac{1}{x}}{x}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]{\\text{M}}_{4}=-0.982,{\\text{T}}_{4}=-0.917,{S}_{4}=-0.952[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, evaluate the integrals, if possible.<\/p>\n

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\n
19.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{1}{{x}^{n}}dx[\/latex], for what values of [latex]n[\/latex] does this integral converge or diverge?<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{{e}^{\\text{-}x}}{x}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
approximately 0.2194<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, consider the gamma function given by [latex]\\Gamma\\left(a\\right)={\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}y}{y}^{a - 1}dy[\/latex].<\/p>\n

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\n
21.\u00a0<\/strong>Show that [latex]\\Gamma\\left(a\\right)=\\left(a - 1\\right)\\Gamma\\left(a - 1\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n
\n
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22.\u00a0<\/strong>Extend to show that [latex]\\Gamma\\left(a\\right)=\\left(a - 1\\right)\\text{!}[\/latex], assuming [latex]a[\/latex] is a positive integer.<\/div>\n

The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km\/h. The graph represents its velocity.<\/p>\n


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

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\n
23. [T]<\/strong> Use the graph to estimate the velocity every 20 sec and fit to a graph of the form [latex]v\\left(t\\right)=a{\\text{exp}}^{bx}\\sin\\left(cx\\right)+d[\/latex]. (Hint:<\/em> Consider the time units.)<\/div>\n<\/div>\n<\/div>\n
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24. [T]<\/strong> Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Answers may vary. Ex: [latex]9.405[\/latex] km<\/div>\n<\/div>\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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CC licensed content, Shared previously<\/div>