Difference between revisions of "009B Sample Midterm 3"

Latest revision as of 18:18, 23 November 2017

This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar.

Click on the boxed problem numbers to go to a solution.

Problem 1

Divide the interval $[0,\pi ]$ into four subintervals of equal length ${\frac {\pi }{4}}$ and compute the right-endpoint Riemann sum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sin (x).}

Problem 2

State the fundamental theorem of calculus, and use this theorem to find the derivative of

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=\int_{\cos (x)}^5 \frac{1}{1+u^{10}}~du.}

Problem 3

Find a curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f(x)} with the following properties:

(i) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(x)=6x}

(ii) Its graph passes through the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,1)} and has a horizontal tangent there.

Problem 4

Compute the following integrals:

(a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^2\sin (x^3) ~dx}

(b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2(x)\sin (x)~dx}

Problem 5

Evaluate the indefinite and definite integrals.

(a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x\ln x ~dx}

(b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^\pi \sin^2x~dx}

Contributions to this page were made by Kayla Murray

@@ Line 5: / Line 5: @@
 == [[009B_Sample Midterm 3,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
-<span class="exam"> Divide the interval <math style="vertical-align: -5px">[0,\pi]</math> into four subintervals of equal length <math>\frac{\pi}{4}</math> and compute the right-endpoint Riemann sum of <math style="vertical-align: -5px">y=\sin (x).</math>
+<span class="exam"> Divide the interval &nbsp;<math style="vertical-align: -5px">[0,\pi]</math>&nbsp; into four subintervals of equal length &nbsp; <math>\frac{\pi}{4}</math>&nbsp; and compute the right-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">y=\sin (x).</math>
 == [[009B_Sample Midterm 3,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
@@ Line 13: / Line 13: @@
 == [[009B_Sample Midterm 3,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam"> Compute the following integrals:
+<span class="exam"> Find a curve &nbsp;<math style="vertical-align: -5px">y=f(x)</math>&nbsp; with the following properties:
+<span class="exam">(i) &nbsp; <math style="vertical-align: -5px">f''(x)=6x</math>
-::<span class="exam">a) <math>\int x^2\sin (x^3) ~dx</math>
+<span class="exam">(ii) &nbsp; Its graph passes through the point &nbsp;<math style="vertical-align: -5px">(0,1)</math>&nbsp; and has a horizontal tangent there.
-::<span class="exam">b) <math>\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2(x)\sin (x)~dx</math>
 == [[009B_Sample Midterm 3,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-<span class="exam"> Evaluate the integral:
+<span class="exam"> Compute the following integrals:
+<span class="exam">(a) &nbsp; <math>\int x^2\sin (x^3) ~dx</math>
-::<math>\int \sin (\ln x)~dx.</math>
+<span class="exam">(b) &nbsp; <math>\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos^2(x)\sin (x)~dx</math>
 == [[009B_Sample Midterm 3,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
 <span class="exam"> Evaluate the indefinite and definite integrals.
-::<span class="exam">a) <math>\int \tan^3x ~dx</math>
+<span class="exam">(a) &nbsp; <math>\int x\ln x ~dx</math>
-::<span class="exam">b) <math>\int_0^\pi \sin^2x~dx</math>
+<span class="exam">(b) &nbsp; <math>\int_0^\pi \sin^2x~dx</math>
 '''Contributions to this page were made by [[Contributors|Kayla Murray]]'''