{"id":486,"date":"2021-02-04T15:30:52","date_gmt":"2021-02-04T15:30:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=486"},"modified":"2021-03-24T17:53:52","modified_gmt":"2021-03-24T17:53:52","slug":"problem-set-the-mean-value-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-the-mean-value-theorem\/","title":{"raw":"Problem Set: The Mean Value Theorem","rendered":"Problem Set: The Mean Value Theorem"},"content":{"raw":"
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1.<\/strong> Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.<\/strong> Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.<\/p>\r\n[reveal-answer q=\"fs-id1165042617583\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042617583\"]\r\n

One example is [latex]f(x)=|x|+3, \\, -2 \\le x \\le 2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.<\/strong> When are Rolle\u2019s theorem and the Mean Value Theorem equivalent?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.<\/strong> If you have a function with a discontinuity, is it still possible to have [latex]f^{\\prime}(c)(b-a)=f(b)-f(a)[\/latex]? Draw such an example or prove why not.<\/p>\r\n[reveal-answer q=\"fs-id1165042617737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042617737\"]\r\n

Yes, but the Mean Value Theorem still does not apply<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (5-9), determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.<\/p>\r\n\r\n

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5.<\/strong> [latex]y= \\sin (\\pi x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.<\/strong> [latex]y=\\frac{1}{x^3}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043382959\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043382959\"]\r\n

[latex](\u2212\\infty,0), \\, (0,\\infty)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.<\/strong> [latex]y=\\sqrt{4-x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.<\/strong> [latex]y=\\sqrt{x^2-4}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043383070\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043383070\"]\r\n

[latex](\u2212\\infty,-2), \\, (2,\\infty)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.<\/strong> [latex]y=\\ln (3x-5)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (10-13), graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points [latex]c[\/latex] such that [latex]f^{\\prime}(c)(b-a)=f(b)-f(a)[\/latex].<\/p>\r\n\r\n

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10. [T] <\/strong>[latex]y=3x^3+2x+1[\/latex] over [latex][-1,1][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042525415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042525415\"]\r\n

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

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11. [T] <\/strong>[latex]y= \\tan (\\frac{\\pi}{4}x)[\/latex] over [latex][-\\frac{3}{2},\\frac{3}{2}][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. [T] <\/strong>[latex]y=x^2 \\cos (\\pi x)[\/latex] over [latex][-2,2][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042525541\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042525541\"]\r\n

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

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13. [T] <\/strong>[latex]y=x^6-\\frac{3}{4}x^5-\\frac{9}{8}x^4+\\frac{15}{16}x^3+\\frac{3}{32}x^2+\\frac{3}{16}x+\\frac{1}{32}[\/latex] over [latex][-1,1][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (14-19), use the Mean Value Theorem and find all points [latex]0<c<2[\/latex] such that [latex]f(2)-f(0)=f^{\\prime}(c)(2-0)[\/latex].<\/p>\r\n\r\n

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14.<\/strong> [latex]f(x)=x^3[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042710364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042710364\"]\r\n

[latex]c=\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.<\/strong> [latex]f(x)= \\sin (\\pi x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.<\/strong> [latex]f(x)= \\cos (2\\pi x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042471131\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042471131\"]\r\n

[latex]c=\\frac{1}{2}, \\, 1, \\, \\frac{3}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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17.<\/strong> [latex]f(x)=1+x+x^2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.<\/strong> [latex]f(x)=(x-1)^{10}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042471253\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042471253\"]\r\n

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

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19.<\/strong> [latex]f(x)=(x-1)^9[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (20-23), show there is no [latex]c[\/latex] such that [latex]f(1)-f(-1)=f^{\\prime}(c)(2)[\/latex]. Explain why the Mean Value Theorem does not apply over the interval [latex][-1,1][\/latex]<\/p>\r\n\r\n

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20.<\/strong> [latex]f(x)=|x-\\frac{1}{2}|[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042709965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042709965\"]\r\n

Not differentiable<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.<\/strong> [latex]f(x)=\\frac{1}{x^2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.<\/strong> [latex]f(x)=\\sqrt{|x|}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042710044\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042710044\"]\r\n

Not differentiable<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.<\/strong> [latex]f(x)=\u230ax\u230b[\/latex] (Hint<\/em>: This is called the floor function<\/em> and it is defined so that [latex]f(x)[\/latex] is the largest integer less than or equal to [latex]x[\/latex].)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (24-34), determine whether the Mean Value Theorem applies for the functions over the given interval [latex][a,b][\/latex]. Justify your answer.<\/p>\r\n\r\n

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24.<\/strong> [latex]y=e^x[\/latex] over [latex][0,1][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042407397\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042407397\"]\r\n

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

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25.<\/strong> [latex]y=\\ln (2x+3)[\/latex] over [latex][-\\frac{3}{2},0][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26.<\/strong> [latex]f(x)= \\tan (2\\pi x)[\/latex] over [latex][0,2][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042407531\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042407531\"]\r\n

The Mean Value Theorem does not apply since the function is discontinuous at [latex]x=\\frac{1}{4}, \\, \\frac{3}{4}, \\, \\frac{5}{4}, \\, \\frac{7}{4}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27.<\/strong> [latex]y=\\sqrt{9-x^2}[\/latex] over [latex][-3,3][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.<\/strong> [latex]y=\\dfrac{1}{|x+1|}[\/latex] over [latex][0,3][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042407671\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042407671\"]\r\n

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

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29.<\/strong> [latex]y=x^3+2x+1[\/latex] over [latex][0,6][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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30.<\/strong> [latex]y=\\dfrac{x^2+3x+2}{x}[\/latex] over [latex][-1,1][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042491088\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042491088\"]\r\n

The Mean Value Theorem does not apply; discontinuous at [latex]x=0[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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31.<\/strong> [latex]y=\\dfrac{x}{ \\sin (\\pi x)+1}[\/latex] over [latex][0,1][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.<\/strong> [latex]y=\\ln (x+1)[\/latex] over [latex][0,e-1][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042491214\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042491214\"]\r\n

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

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33.<\/strong> [latex]y=x \\sin (\\pi x)[\/latex] over [latex][0,2][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.<\/strong> [latex]y=5+|x|[\/latex] over [latex][-1,1][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043262237\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043262237\"]\r\n

The Mean Value Theorem does not apply; not differentiable at [latex]x=0[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (35-37), consider the roots of the equation.<\/p>\r\n\r\n

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35.<\/strong> Show that the equation [latex]y=x^3+3x^2+16[\/latex] has exactly one real root. What is it?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36.<\/strong> Find the conditions for exactly one root (double root) for the equation [latex]y=x^2+bx+c[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043262339\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043262339\"]\r\n

[latex]b=\\pm 2\\sqrt{c}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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37.<\/strong> Find the conditions for [latex]y=e^x-b[\/latex] to have one root. Is it possible to have more than one root?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use a calculator to graph the function over the interval [latex][a,b][\/latex] and graph the secant line from [latex]a[\/latex] to [latex]b[\/latex]. Use the calculator to estimate all values of [latex]c[\/latex] as guaranteed by the Mean Value Theorem. Then, find the exact value of [latex]c[\/latex], if possible, or write the final equation and use a calculator to estimate to four digits.<\/p>\r\n\r\n

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38. [T] <\/strong>[latex]y= \\tan (\\pi x)[\/latex] over [latex][-\\frac{1}{4},\\frac{1}{4}][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043341438\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043341438\"]\r\n

[latex]c=\\pm \\frac{1}{\\pi} \\cos^{-1}(\\frac{\\sqrt{\\pi}}{2})[\/latex]; [latex]c=\\pm 0.1533[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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39. [T] <\/strong>[latex]y=\\frac{1}{\\sqrt{x+1}}[\/latex] over [latex][0,3][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. [T] <\/strong>[latex]y=|x^2+2x-4|[\/latex] over [latex][-4,0][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165043341624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043341624\"]\r\n

The Mean Value Theorem does not apply.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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41. [T] <\/strong>[latex]y=x+\\frac{1}{x}[\/latex] over [latex][\\frac{1}{2},4][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. [T] <\/strong>[latex]y=\\sqrt{x+1}+\\frac{1}{x^2}[\/latex] over [latex][3,8][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1165042595208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042595208\"]\r\n

[latex]\\frac{1}{2\\sqrt{c+1}}-\\frac{2}{c^3}=\\frac{521}{2880}[\/latex]; [latex]c=3.133,5.867[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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43.<\/strong> At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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44.<\/strong> Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.<\/p>\r\n[reveal-answer q=\"fs-id1165042595298\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042595298\"]\r\n

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

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45.<\/strong> Show that [latex]y= \\sec^2 x[\/latex] and [latex]y= \\tan^2 x[\/latex] have the same derivative. What can you say about [latex]y= \\sec^2 x - \\tan^2 x[\/latex]?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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46.<\/strong> Show that [latex]y= \\csc^2 x[\/latex] and [latex]y= \\cot^2 x[\/latex] have the same derivative. What can you say about [latex]y= \\csc^2 x - \\cot^2 x[\/latex]?<\/p>\r\n[reveal-answer q=\"fs-id1165042595449\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042595449\"]\r\n

It is constant.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>","rendered":"

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1.<\/strong> Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.<\/p>\n<\/div>\n<\/div>\n

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2.<\/strong> Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.<\/p>\n

Show Solution<\/span><\/p>\n
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One example is [latex]f(x)=|x|+3, \\, -2 \\le x \\le 2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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3.<\/strong> When are Rolle\u2019s theorem and the Mean Value Theorem equivalent?<\/p>\n<\/div>\n<\/div>\n

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4.<\/strong> If you have a function with a discontinuity, is it still possible to have [latex]f^{\\prime}(c)(b-a)=f(b)-f(a)[\/latex]? Draw such an example or prove why not.<\/p>\n

Show Solution<\/span><\/p>\n
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Yes, but the Mean Value Theorem still does not apply<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (5-9), determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.<\/p>\n

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5.<\/strong> [latex]y= \\sin (\\pi x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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6.<\/strong> [latex]y=\\frac{1}{x^3}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](\u2212\\infty,0), \\, (0,\\infty)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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7.<\/strong> [latex]y=\\sqrt{4-x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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8.<\/strong> [latex]y=\\sqrt{x^2-4}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](\u2212\\infty,-2), \\, (2,\\infty)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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9.<\/strong> [latex]y=\\ln (3x-5)[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (10-13), graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points [latex]c[\/latex] such that [latex]f^{\\prime}(c)(b-a)=f(b)-f(a)[\/latex].<\/p>\n

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10. [T] <\/strong>[latex]y=3x^3+2x+1[\/latex] over [latex][-1,1][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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2 points<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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11. [T] <\/strong>[latex]y= \\tan (\\frac{\\pi}{4}x)[\/latex] over [latex][-\\frac{3}{2},\\frac{3}{2}][\/latex]<\/p>\n<\/div>\n<\/div>\n

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12. [T] <\/strong>[latex]y=x^2 \\cos (\\pi x)[\/latex] over [latex][-2,2][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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5 points<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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13. [T] <\/strong>[latex]y=x^6-\\frac{3}{4}x^5-\\frac{9}{8}x^4+\\frac{15}{16}x^3+\\frac{3}{32}x^2+\\frac{3}{16}x+\\frac{1}{32}[\/latex] over [latex][-1,1][\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (14-19), use the Mean Value Theorem and find all points [latex]0\n

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14.<\/strong> [latex]f(x)=x^3[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]c=\\frac{2\\sqrt{3}}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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15.<\/strong> [latex]f(x)= \\sin (\\pi x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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16.<\/strong> [latex]f(x)= \\cos (2\\pi x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]c=\\frac{1}{2}, \\, 1, \\, \\frac{3}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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17.<\/strong> [latex]f(x)=1+x+x^2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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18.<\/strong> [latex]f(x)=(x-1)^{10}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]c=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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19.<\/strong> [latex]f(x)=(x-1)^9[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (20-23), show there is no [latex]c[\/latex] such that [latex]f(1)-f(-1)=f^{\\prime}(c)(2)[\/latex]. Explain why the Mean Value Theorem does not apply over the interval [latex][-1,1][\/latex]<\/p>\n

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20.<\/strong> [latex]f(x)=|x-\\frac{1}{2}|[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Not differentiable<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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21.<\/strong> [latex]f(x)=\\frac{1}{x^2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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22.<\/strong> [latex]f(x)=\\sqrt{|x|}[\/latex]<\/p>\n

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Not differentiable<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23.<\/strong> [latex]f(x)=\u230ax\u230b[\/latex] (Hint<\/em>: This is called the floor function<\/em> and it is defined so that [latex]f(x)[\/latex] is the largest integer less than or equal to [latex]x[\/latex].)<\/p>\n<\/div>\n<\/div>\n

For the following exercises (24-34), determine whether the Mean Value Theorem applies for the functions over the given interval [latex][a,b][\/latex]. Justify your answer.<\/p>\n

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24.<\/strong> [latex]y=e^x[\/latex] over [latex][0,1][\/latex]<\/p>\n

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Yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25.<\/strong> [latex]y=\\ln (2x+3)[\/latex] over [latex][-\\frac{3}{2},0][\/latex]<\/p>\n<\/div>\n<\/div>\n

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26.<\/strong> [latex]f(x)= \\tan (2\\pi x)[\/latex] over [latex][0,2][\/latex]<\/p>\n

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The Mean Value Theorem does not apply since the function is discontinuous at [latex]x=\\frac{1}{4}, \\, \\frac{3}{4}, \\, \\frac{5}{4}, \\, \\frac{7}{4}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27.<\/strong> [latex]y=\\sqrt{9-x^2}[\/latex] over [latex][-3,3][\/latex]<\/p>\n<\/div>\n<\/div>\n

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28.<\/strong> [latex]y=\\dfrac{1}{|x+1|}[\/latex] over [latex][0,3][\/latex]<\/p>\n

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Yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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29.<\/strong> [latex]y=x^3+2x+1[\/latex] over [latex][0,6][\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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The Mean Value Theorem does not apply; discontinuous at [latex]x=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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31.<\/strong> [latex]y=\\dfrac{x}{ \\sin (\\pi x)+1}[\/latex] over [latex][0,1][\/latex]<\/p>\n<\/div>\n<\/div>\n

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32.<\/strong> [latex]y=\\ln (x+1)[\/latex] over [latex][0,e-1][\/latex]<\/p>\n

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Yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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33.<\/strong> [latex]y=x \\sin (\\pi x)[\/latex] over [latex][0,2][\/latex]<\/p>\n<\/div>\n<\/div>\n

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34.<\/strong> [latex]y=5+|x|[\/latex] over [latex][-1,1][\/latex]<\/p>\n

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The Mean Value Theorem does not apply; not differentiable at [latex]x=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (35-37), consider the roots of the equation.<\/p>\n

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35.<\/strong> Show that the equation [latex]y=x^3+3x^2+16[\/latex] has exactly one real root. What is it?<\/p>\n<\/div>\n<\/div>\n

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36.<\/strong> Find the conditions for exactly one root (double root) for the equation [latex]y=x^2+bx+c[\/latex]<\/p>\n

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[latex]b=\\pm 2\\sqrt{c}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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37.<\/strong> Find the conditions for [latex]y=e^x-b[\/latex] to have one root. Is it possible to have more than one root?<\/p>\n<\/div>\n<\/div>\n

For the following exercises, use a calculator to graph the function over the interval [latex][a,b][\/latex] and graph the secant line from [latex]a[\/latex] to [latex]b[\/latex]. Use the calculator to estimate all values of [latex]c[\/latex] as guaranteed by the Mean Value Theorem. Then, find the exact value of [latex]c[\/latex], if possible, or write the final equation and use a calculator to estimate to four digits.<\/p>\n

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38. [T] <\/strong>[latex]y= \\tan (\\pi x)[\/latex] over [latex][-\\frac{1}{4},\\frac{1}{4}][\/latex]<\/p>\n

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[latex]c=\\pm \\frac{1}{\\pi} \\cos^{-1}(\\frac{\\sqrt{\\pi}}{2})[\/latex]; [latex]c=\\pm 0.1533[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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39. [T] <\/strong>[latex]y=\\frac{1}{\\sqrt{x+1}}[\/latex] over [latex][0,3][\/latex]<\/p>\n<\/div>\n<\/div>\n

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40. [T] <\/strong>[latex]y=|x^2+2x-4|[\/latex] over [latex][-4,0][\/latex]<\/p>\n

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The Mean Value Theorem does not apply.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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41. [T] <\/strong>[latex]y=x+\\frac{1}{x}[\/latex] over [latex][\\frac{1}{2},4][\/latex]<\/p>\n<\/div>\n<\/div>\n

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42. [T] <\/strong>[latex]y=\\sqrt{x+1}+\\frac{1}{x^2}[\/latex] over [latex][3,8][\/latex]<\/p>\n

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[latex]\\frac{1}{2\\sqrt{c+1}}-\\frac{2}{c^3}=\\frac{521}{2880}[\/latex]; [latex]c=3.133,5.867[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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43.<\/strong> At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?<\/p>\n<\/div>\n<\/div>\n

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44.<\/strong> Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.<\/p>\n

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Yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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45.<\/strong> Show that [latex]y= \\sec^2 x[\/latex] and [latex]y= \\tan^2 x[\/latex] have the same derivative. What can you say about [latex]y= \\sec^2 x - \\tan^2 x[\/latex]?<\/p>\n<\/div>\n<\/div>\n

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46.<\/strong> Show that [latex]y= \\csc^2 x[\/latex] and [latex]y= \\cot^2 x[\/latex] have the same derivative. What can you say about [latex]y= \\csc^2 x - \\cot^2 x[\/latex]?<\/p>\n

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It is constant.<\/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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