Difference between revisions of "007B Sample Midterm 1"
Kayla Murray (talk | contribs) (→ Problem 5 ) |
Kayla Murray (talk | contribs) |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | '''This is a sample, and is meant to represent the material usually covered in Math | + | '''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.''' | '''Click on the''' '''<span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.''' | ||
<div class="noautonum">__TOC__</div> | <div class="noautonum">__TOC__</div> | ||
| − | == [[ | + | == [[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"> Let <math style="vertical-align: -5px">f(x)=1-x^2</math>. | ||
| Line 13: | Line 13: | ||
<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. | <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 | <span class="exam"> A population grows at a rate | ||
| Line 24: | Line 24: | ||
<span class="exam">(b) Use your answer to part (a) to find the size of the population after one month. | <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">Evaluate the following integrals. | ||
| Line 31: | Line 31: | ||
<span class="exam">(b) <math>\int _{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos(x)}{\sin^2(x)}~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"> Evaluate the following integrals. | ||
| Line 38: | Line 38: | ||
<span class="exam">(b) <math>\int \frac{5x-7}{x^2-3x+2}~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> | <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]]''' | '''Contributions to this page were made by [[Contributors|Kayla Murray]]''' | ||
Latest revision as of 15:01, 2 November 2017
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 boxed problem numbers to go to a solution.
Problem 1
Let .
(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 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.
Problem 2
A population grows at a rate
- 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 P'(t)=500e^{-t}}
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 P(t)} is the population after 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} months.
(a) Find a formula for the population size after 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} months, given that the population is 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 2000} at 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=0.}
(b) Use your answer to part (a) to find the size of the population after one month.
Problem 3
Evaluate 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\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 4
Evaluate 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 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 \frac{5x-7}{x^2-3x+2}~dx}
Problem 5
Find the area bounded by 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)} and 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=\cos(x)} from 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 x=0} to
Contributions to this page were made by Kayla Murray