{"id":1159,"date":"2021-06-30T17:02:02","date_gmt":"2021-06-30T17:02:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/chapter-5-review-exercises\/"},"modified":"2021-11-17T01:52:02","modified_gmt":"2021-11-17T01:52:02","slug":"module-1-review-exercises","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/module-1-review-exercises\/","title":{"raw":"Module 1 Review Problems","rendered":"Module 1 Review Problems"},"content":{"raw":"

True or False.<\/em> Justify your answer with a proof or a counterexample. Assume all functions [latex]f[\/latex] and [latex]g[\/latex] are continuous over their domains (1-4).<\/p>\r\n\r\n

\r\n
\r\n

1. <\/strong>If [latex]f(x)>0,{f}^{\\prime }(x)>0[\/latex] for all [latex]x,[\/latex] then the right-hand rule underestimates the integral [latex]{\\displaystyle\\int }_{a}^{b}f(x).[\/latex] Use a graph to justify your answer.<\/p>\r\n[reveal-answer q=\"fs-id1170572451349\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572451349\"]\r\n

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

\r\n
\r\n

2.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{a}^{b}f{(x)}^{2}dx={\\displaystyle\\int }_{a}^{b}f(x)dx{\\displaystyle\\int }_{a}^{b}f(x)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

3.\u00a0<\/strong>If [latex]f(x)\\le g(x)[\/latex] for all [latex]x\\in \\left[a,b\\right],[\/latex] then [latex]{\\displaystyle\\int }_{a}^{b}f(x)\\le {\\displaystyle\\int }_{a}^{b}g(x).[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572379114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572379114\"]\r\n

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

\r\n
\r\n\r\n4.\u00a0<\/strong>All continuous functions have an antiderivative.\r\n\r\n<\/div>\r\n<\/div>\r\n

Evaluate the Riemann sums [latex]{L}_{4}\\text{ and }{R}_{4}[\/latex] for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.<\/p>\r\n\r\n

\r\n
\r\n

5.\u00a0<\/strong>[latex]y=3{x}^{2}-2x+1[\/latex] over [latex]\\left[-1,1\\right][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571613630\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571613630\"][latex]{L}_{4}=5.25,{R}_{4}=3.25,[\/latex] exact answer: 4[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

6.\u00a0<\/strong>[latex]y=\\text{ln}({x}^{2}+1)[\/latex] over [latex]\\left[0,e\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

7.\u00a0<\/strong>[latex]y={x}^{2} \\sin x[\/latex] over [latex]\\left[0,\\pi \\right][\/latex]<\/p>\r\n\r\n

\r\n\r\n[reveal-answer q=\"fs-id1170572434982\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572434982\"]\r\n

[latex]{L}_{4}=5.364,{R}_{4}=5.364,[\/latex] exact answer: 5.870<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

8.\u00a0<\/strong>[latex]y=\\sqrt{x}+\\frac{1}{x}[\/latex] over [latex]\\left[1,4\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Evaluate the following integrals.<\/p>\r\n\r\n

\r\n
\r\n\r\n9.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx[\/latex]\r\n
\r\n\r\n[reveal-answer q=\"fs-id1170572627137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572627137\"]\r\n

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

\r\n
\r\n

10.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{4}\\frac{3t}{\\sqrt{1+6{t}^{2}}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

11.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\pi \\text{\/}3}^{\\pi \\text{\/}2}2 \\sec (2\\theta ) \\tan (2\\theta )d\\theta [\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571712789\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571712789\"]\r\n

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

\r\n
\r\n

12.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi \\text{\/}4}{e}^{{ \\cos }^{2}x} \\sin x \\cos{x} dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Find the antiderivative.<\/p>\r\n\r\n

\r\n
\r\n

13.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{(x+4)}^{3}}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572379231\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572379231\"]\r\n

[latex]-\\frac{1}{2{(x+4)}^{2}}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

14.\u00a0<\/strong>[latex]\\displaystyle\\int x\\text{ln}({x}^{2})dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

15.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{4{x}^{2}}{\\sqrt{1-{x}^{6}}}dx[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572223484\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572223484\"]\r\n

[latex]\\frac{4}{3}\\phantom{\\rule{0.05em}{0ex}}{ \\sin }^{-1}({x}^{3})+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

16.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{2x}}{1+{e}^{4x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Find the derivative.<\/p>\r\n\r\n

\r\n
\r\n

17.\u00a0<\/strong>[latex]\\frac{d}{dt}{\\displaystyle\\int }_{0}^{t}\\frac{ \\sin x}{\\sqrt{1+{x}^{2}}}dx[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571569005\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571569005\"]\r\n

[latex]\\frac{ \\sin t}{\\sqrt{1+{t}^{2}}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

18.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{1}^{{x}^{3}}\\sqrt{4-{t}^{2}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

19.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{1}^{\\text{ln}(x)}(4t+{e}^{t})dt[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572582592\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572582592\"]\r\n

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

\r\n
\r\n

20.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{0}^{ \\cos x}{e}^{{t}^{2}}dt[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

The following problems consider the historic average cost per gigabyte of RAM on a computer.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Year<\/th>\r\n5-Year Change ($)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1980<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
1985<\/td>\r\n\u22125,468,750<\/td>\r\n<\/tr>\r\n
1990<\/td>\r\n\u2212<\/strong>755,495<\/td>\r\n<\/tr>\r\n
1995<\/td>\r\n\u221273,005<\/td>\r\n<\/tr>\r\n
2000<\/td>\r\n\u221229,768<\/td>\r\n<\/tr>\r\n
2005<\/td>\r\n\u2212918<\/td>\r\n<\/tr>\r\n
2010<\/td>\r\n\u2212177<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n
\r\n

21.\u00a0<\/strong>If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.<\/p>\r\n[reveal-answer q=\"fs-id1170571610268\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571610268\"]\r\n

$6,328,113<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

22.\u00a0<\/strong>The average cost per gigabyte of RAM can be approximated by the function [latex]C(t)=8,500,000{(0.65)}^{t},[\/latex] where [latex]t[\/latex] is measured in years since 1980, and [latex]C[\/latex] is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

23.\u00a0<\/strong>Find the average cost of 1GB RAM for 2005 to 2010.<\/p>\r\n[reveal-answer q=\"fs-id1170571610350\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571610350\"]\r\n

$73.36<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

24.\u00a0<\/strong>The velocity of a bullet from a rifle can be approximated by [latex]v(t)=6400{t}^{2}-6505t+2686,[\/latex] where [latex]t[\/latex] is seconds after the shot and [latex]v[\/latex] is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: [latex]0\\le t\\le 0.5.[\/latex] What is the total distance the bullet travels in 0.5 sec?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

25.\u00a0<\/strong>What is the average velocity of the bullet for the first half-second?<\/p>\r\n[reveal-answer q=\"fs-id1170572504548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572504548\"]\r\n

[latex]\\frac{19117}{12}\\text{ft\/sec},\\text{or}1593\\text{ft\/sec}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

True or False.<\/em> Justify your answer with a proof or a counterexample. Assume all functions [latex]f[\/latex] and [latex]g[\/latex] are continuous over their domains (1-4).<\/p>\n

\n
\n

1. <\/strong>If [latex]f(x)>0,{f}^{\\prime }(x)>0[\/latex] for all [latex]x,[\/latex] then the right-hand rule underestimates the integral [latex]{\\displaystyle\\int }_{a}^{b}f(x).[\/latex] Use a graph to justify your answer.<\/p>\n

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

False<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

2.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{a}^{b}f{(x)}^{2}dx={\\displaystyle\\int }_{a}^{b}f(x)dx{\\displaystyle\\int }_{a}^{b}f(x)dx[\/latex]<\/p>\n<\/div>\n<\/div>\n

\n
\n

3.\u00a0<\/strong>If [latex]f(x)\\le g(x)[\/latex] for all [latex]x\\in \\left[a,b\\right],[\/latex] then [latex]{\\displaystyle\\int }_{a}^{b}f(x)\\le {\\displaystyle\\int }_{a}^{b}g(x).[\/latex]<\/p>\n

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

True<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

4.\u00a0<\/strong>All continuous functions have an antiderivative.<\/p>\n<\/div>\n<\/div>\n

Evaluate the Riemann sums [latex]{L}_{4}\\text{ and }{R}_{4}[\/latex] for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer.<\/p>\n

\n
\n

5.\u00a0<\/strong>[latex]y=3{x}^{2}-2x+1[\/latex] over [latex]\\left[-1,1\\right][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
[latex]{L}_{4}=5.25,{R}_{4}=3.25,[\/latex] exact answer: 4<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n

6.\u00a0<\/strong>[latex]y=\\text{ln}({x}^{2}+1)[\/latex] over [latex]\\left[0,e\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n

\n
\n

7.\u00a0<\/strong>[latex]y={x}^{2} \\sin x[\/latex] over [latex]\\left[0,\\pi \\right][\/latex]<\/p>\n

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

[latex]{L}_{4}=5.364,{R}_{4}=5.364,[\/latex] exact answer: 5.870<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

8.\u00a0<\/strong>[latex]y=\\sqrt{x}+\\frac{1}{x}[\/latex] over [latex]\\left[1,4\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n

Evaluate the following integrals.<\/p>\n

\n
\n

9.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-1}^{1}({x}^{3}-2{x}^{2}+4x)dx[\/latex]<\/p>\n

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

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

\n
\n

10.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{4}\\frac{3t}{\\sqrt{1+6{t}^{2}}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n

\n
\n

11.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\pi \\text{\/}3}^{\\pi \\text{\/}2}2 \\sec (2\\theta ) \\tan (2\\theta )d\\theta[\/latex]<\/p>\n

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

1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

12.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi \\text{\/}4}{e}^{{ \\cos }^{2}x} \\sin x \\cos{x} dx[\/latex]<\/p>\n<\/div>\n<\/div>\n

Find the antiderivative.<\/p>\n

\n
\n

13.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{(x+4)}^{3}}[\/latex]<\/p>\n

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

[latex]-\\frac{1}{2{(x+4)}^{2}}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

14.\u00a0<\/strong>[latex]\\displaystyle\\int x\\text{ln}({x}^{2})dx[\/latex]<\/p>\n<\/div>\n<\/div>\n

\n
\n

15.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{4{x}^{2}}{\\sqrt{1-{x}^{6}}}dx[\/latex]<\/p>\n

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

[latex]\\frac{4}{3}\\phantom{\\rule{0.05em}{0ex}}{ \\sin }^{-1}({x}^{3})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

16.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{2x}}{1+{e}^{4x}}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n

Find the derivative.<\/p>\n

\n
\n

17.\u00a0<\/strong>[latex]\\frac{d}{dt}{\\displaystyle\\int }_{0}^{t}\\frac{ \\sin x}{\\sqrt{1+{x}^{2}}}dx[\/latex]<\/p>\n

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

[latex]\\frac{ \\sin t}{\\sqrt{1+{t}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

18.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{1}^{{x}^{3}}\\sqrt{4-{t}^{2}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n

\n
\n

19.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{1}^{\\text{ln}(x)}(4t+{e}^{t})dt[\/latex]<\/p>\n

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

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

\n
\n

20.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{0}^{ \\cos x}{e}^{{t}^{2}}dt[\/latex]<\/p>\n<\/div>\n<\/div>\n

The following problems consider the historic average cost per gigabyte of RAM on a computer.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Year<\/th>\n5-Year Change ($)<\/th>\n<\/tr>\n<\/thead>\n
1980<\/td>\n0<\/td>\n<\/tr>\n
1985<\/td>\n\u22125,468,750<\/td>\n<\/tr>\n
1990<\/td>\n\u2212<\/strong>755,495<\/td>\n<\/tr>\n
1995<\/td>\n\u221273,005<\/td>\n<\/tr>\n
2000<\/td>\n\u221229,768<\/td>\n<\/tr>\n
2005<\/td>\n\u2212918<\/td>\n<\/tr>\n
2010<\/td>\n\u2212177<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

21.\u00a0<\/strong>If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.<\/p>\n

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

$6,328,113<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

22.\u00a0<\/strong>The average cost per gigabyte of RAM can be approximated by the function [latex]C(t)=8,500,000{(0.65)}^{t},[\/latex] where [latex]t[\/latex] is measured in years since 1980, and [latex]C[\/latex] is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.<\/p>\n<\/div>\n<\/div>\n

\n
\n

23.\u00a0<\/strong>Find the average cost of 1GB RAM for 2005 to 2010.<\/p>\n

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

$73.36<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n
\n

24.\u00a0<\/strong>The velocity of a bullet from a rifle can be approximated by [latex]v(t)=6400{t}^{2}-6505t+2686,[\/latex] where [latex]t[\/latex] is seconds after the shot and [latex]v[\/latex] is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: [latex]0\\le t\\le 0.5.[\/latex] What is the total distance the bullet travels in 0.5 sec?<\/p>\n<\/div>\n<\/div>\n

\n
\n

25.\u00a0<\/strong>What is the average velocity of the bullet for the first half-second?<\/p>\n

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

[latex]\\frac{19117}{12}\\text{ft\/sec},\\text{or}1593\\text{ft\/sec}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

\n\t\t\t

Candela Citations<\/h3>\n\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
CC licensed content, Shared previously<\/div>