007B Sample Midterm 1 - Revision history

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Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 7B for the midterm. An actual test may or may not be similar.''' '''Click on the''' '''<s..."

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'''This is a sample, and is meant to represent the material usually covered in Math 7B for the midterm. An actual test may or may not be similar.'''

'''Click on the''' '''<span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''
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== [[007B_Sample Midterm 1,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] ==
<span class="exam"> Let  <math style="vertical-align: -5px">f(x)=1-x^2</math>.

<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.

== [[007B_Sample Midterm 1,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] ==
<span class="exam"> A population grows at a rate

::<math>P'(t)=500e^{-t}</math>

<span class="exam">where  <math style="vertical-align: -5px">P(t)</math>  is the population after  <math style="vertical-align: 0px">t</math>  months.

<span class="exam">(a)   Find a formula for the population size after  <math style="vertical-align: 0px">t</math>  months, given that the population is  <math style="vertical-align: 0px">2000</math>  at  <math style="vertical-align: 0px">t=0.</math>

<span class="exam">(b)   Use your answer to part (a) to find the size of the population after one month.

== [[007B_Sample Midterm 1,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] ==
<span class="exam">Evaluate the following integrals.

<span class="exam">(a)   <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>

== [[007B_Sample Midterm 1,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] ==
<span class="exam"> Evaluate the following integrals.

<span class="exam">(a)   <math>\int x^2 e^x~dx</math>

<span class="exam">(b)   <math>\int \frac{5x-7}{x^2-3x+2}~dx</math>

== [[007B_Sample Midterm 1,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] ==
<span class="exam"> Find the area bounded by  <math style="vertical-align: -5px">y=\sin(x)</math>  and  <math style="vertical-align: -5px">y=\cos(x)</math>  from  <math style="vertical-align: -1px">x=0</math>  to  <math style="vertical-align: -14px">x=\frac{\pi}{4}.</math>

'''Contributions to this page were made by [[Contributors|Kayla Murray]]'''