Difference between revisions of "007B Sample Final 1"
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<div class="noautonum">__TOC__</div> | <div class="noautonum">__TOC__</div> | ||
| − | == [[007B_Sample Final | + | == [[007B_Sample Final 1,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] == |
<span class="exam">Divide the interval <math style="vertical-align: -5px">[-1,1]</math> into four subintervals of equal length <math style="vertical-align: -14px">\frac{1}{2}</math> and compute the left-endpoint Riemann sum of <math style="vertical-align: -5px">y=1-x^2.</math> | <span class="exam">Divide the interval <math style="vertical-align: -5px">[-1,1]</math> into four subintervals of equal length <math style="vertical-align: -14px">\frac{1}{2}</math> and compute the left-endpoint Riemann sum of <math style="vertical-align: -5px">y=1-x^2.</math> | ||
| − | == [[007B_Sample Final | + | == [[007B_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] == |
<span class="exam"> Evaluate the following integrals. | <span class="exam"> Evaluate the following integrals. | ||
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<span class="exam">(c) <math>\int_1^e \frac{\cos(\ln(x))}{x}~dx</math> | <span class="exam">(c) <math>\int_1^e \frac{\cos(\ln(x))}{x}~dx</math> | ||
| − | == [[007B_Sample Final | + | == [[007B_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] == |
<span class="exam">The rate <math style="vertical-align: 0px">r</math> at which people get sick during an epidemic of the flu can be approximated by | <span class="exam">The rate <math style="vertical-align: 0px">r</math> at which people get sick during an epidemic of the flu can be approximated by | ||
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<span class="exam">(c) How many people get sick altogether? | <span class="exam">(c) How many people get sick altogether? | ||
| − | == [[007B_Sample Final | + | == [[007B_Sample Final 1,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] == |
<span class="exam"> Find the volume of the solid obtained by rotating about the <math>x</math>-axis the region bounded by <math style="vertical-align: -4px">y=\sqrt{1-x^2}</math> and <math>y=0.</math> | <span class="exam"> Find the volume of the solid obtained by rotating about the <math>x</math>-axis the region bounded by <math style="vertical-align: -4px">y=\sqrt{1-x^2}</math> and <math>y=0.</math> | ||
| − | == [[007B_Sample Final | + | == [[007B_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] == |
<span class="exam"> Find the following integrals. | <span class="exam"> Find the following integrals. | ||
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<span class="exam">(b) <math>\int \sin^3(x)\cos^2(x)~dx</math> | <span class="exam">(b) <math>\int \sin^3(x)\cos^2(x)~dx</math> | ||
| − | == [[007B_Sample Final | + | == [[007B_Sample Final 1,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] == |
<span class="exam">Does the following integral converge or diverge? Prove your answer! | <span class="exam">Does the following integral converge or diverge? Prove your answer! | ||
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::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math> | ::<math>\int_1^\infty \frac{\sin^2(x)}{x^3}~dx</math> | ||
| − | == [[007B_Sample Final | + | == [[007B_Sample Final 1,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] == |
<span class="exam">Solve the following differential equations: | <span class="exam">Solve the following differential equations: | ||
Latest revision as of 22:56, 2 December 2017
This is a sample, and is meant to represent the material usually covered in Math 7B for the final. 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 into four subintervals of equal length 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 \frac{1}{2}} and compute the left-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=1-x^2.}
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
Problem 2
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_0^{\frac{\sqrt{3}}{4}} \frac{1}{1+16x^2}~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{\sqrt{x+1}}{x}~dx}
(c) 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 \frac{\cos(\ln(x))}{x}~dx}
Problem 3
The 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 r} at which people get sick during an epidemic of the flu can be approximated 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 r=1600te^{-0.2t}}
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 r} is measured in people/day 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 t} is measured in days since the start of the epidemic.
(a) Sketch a graph 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 r} as a function 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 t.}
(b) When are people getting sick fastest?
(c) How many people get sick altogether?
Problem 4
Find the volume of the solid obtained by rotating about the 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} -axis the region 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=\sqrt{1-x^2}} 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=0.}
Problem 5
Find 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\cos(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 \sin^3(x)\cos^2(x)~dx}
Problem 6
Does the following integral converge or diverge? Prove your answer!
- 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^\infty \frac{\sin^2(x)}{x^3}~dx}
Problem 7
Solve the following differential equations:
(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 \frac{dy}{dx}=3y,} 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 y_0=2} for 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=0}
(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 \frac{dy}{dx}=(y-1)(y-2)} 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 y_0=0} for 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=0}