Difference between revisions of "009B Sample Midterm 1"

Revision as of 10:58, 9 April 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

Evaluate the indefinite and definite integrals.

(a) $\int x^{2}{\sqrt {1+x^{3}}}~dx$

(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}{2}} \frac{\cos(x)}{\sin^2(x)}~dx}

Problem 2

Otis Taylor plots the price per share of a stock that he owns as a function of time

and finds that it can be approximated by the function

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 s(t)=t(25-5t)+18}

where 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 t} is the time (in years) since the stock was purchased.

Find the average price of the stock over the first five years.

Problem 3

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^2 e^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_{1}^{e} x^3\ln x~dx}

Problem 4

Evaluate the integral:

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 \sin^3x \cos^2x~dx}

Problem 5

Let 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)=1-x^2} .

(a) Compute the left-hand Riemann sum approximation 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 \int_0^3 f(x)~dx} with 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 n=3} boxes.

(b) Compute the right-hand Riemann sum approximation 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 \int_0^3 f(x)~dx} with 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 n=3} boxes.

(c) Express 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_0^3 f(x)~dx} as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.

Contributions to this page were made by Kayla Murray

@@ Line 1: / Line 1: @@
 '''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''' '''<span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
@@ Line 7: / Line 7: @@
 <span class="exam">Evaluate the indefinite and definite integrals.
-::<span class="exam">a) <math>\int x^2\sqrt{1+x^3}~dx</math>
+<span class="exam">(a) &nbsp; <math>\int x^2\sqrt{1+x^3}~dx</math>
-::<span class="exam">b) <math>\int _{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos(x)}{\sin^2(x)}~dx</math>
+<span class="exam">(b) &nbsp; <math>\int _{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos(x)}{\sin^2(x)}~dx</math>
 == [[009B_Sample Midterm 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
-<span class="exam"> Find the average value of the function on the given interval.
+<span class="exam"> Otis Taylor plots the price per share of a stock that he owns as a function of time
+<span class="exam">and finds that it can be approximated by the function
+::<math>s(t)=t(25-5t)+18</math>
-::<math>f(x)=2x^3(1+x^2)^4,~~~[0,2]</math>
+<span class="exam">where &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; is the time (in years) since the stock was purchased.
+<span class="exam">Find the average price of the stock over the first five years.
 == [[009B_Sample Midterm 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
 <span class="exam"> Evaluate the indefinite and definite integrals.
-::<span class="exam">a) <math>\int x^2 e^x~dx</math>
+<span class="exam">(a) &nbsp; <math>\int x^2 e^x~dx</math>
-::<span class="exam">b) <math>\int_{1}^{e} x^3\ln x~dx</math>
+<span class="exam">(b) &nbsp; <math>\int_{1}^{e} x^3\ln x~dx</math>
 == [[009B_Sample Midterm 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
@@ Line 28: / Line 35: @@
 == [[009B_Sample Midterm 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
-<span class="exam"> Let <math style="vertical-align: -5px">f(x)=1-x^2</math>.
+<span class="exam"> Let &nbsp;<math style="vertical-align: -5px">f(x)=1-x^2</math>.
+<span class="exam">(a) Compute the left-hand Riemann sum approximation of &nbsp;<math style="vertical-align: -14px">\int_0^3 f(x)~dx</math>&nbsp; with &nbsp;<math style="vertical-align: 0px">n=3</math>&nbsp; boxes.
+<span class="exam">(b) Compute the right-hand Riemann sum approximation of &nbsp;<math style="vertical-align: -14px">\int_0^3 f(x)~dx</math>&nbsp; with &nbsp;<math style="vertical-align: 0px">n=3</math>&nbsp; boxes.
+<span class="exam">(c) Express &nbsp;<math style="vertical-align: -14px">\int_0^3 f(x)~dx</math>&nbsp; as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.
-::<span class="exam">a) Compute the left-hand Riemann sum approximation of <math style="vertical-align: -14px">\int_0^3 f(x)~dx</math> with <math style="vertical-align: 0px">n=3</math> boxes.
-::<span class="exam">b) Compute the right-hand Riemann sum approximation of <math style="vertical-align: -14px">\int_0^3 f(x)~dx</math> with <math style="vertical-align: 0px">n=3</math> boxes.
-::<span class="exam">c) Express <math style="vertical-align: -14px">\int_0^3 f(x)~dx</math> as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.
 '''Contributions to this page were made by [[Contributors|Kayla Murray]]'''