Difference between revisions of "009C Sample Final 1"
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| − | '''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar. | + | '''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. 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.''' | ||
<div class="noautonum">__TOC__</div> | <div class="noautonum">__TOC__</div> | ||
| Line 5: | Line 7: | ||
<span class="exam">Compute | <span class="exam">Compute | ||
| − | <span class="exam">a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math> | + | ::<span class="exam">a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math> |
| − | <span class="exam">b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math> | + | ::<span class="exam">b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math> |
== [[009C_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] == | == [[009C_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] == | ||
<span class="exam"> Find the sum of the following series: | <span class="exam"> Find the sum of the following series: | ||
| − | <span class="exam">a) <math>\sum_{n=0}^{\infty} (-2)^ne^{-n}</math> | + | ::<span class="exam">a) <math>\sum_{n=0}^{\infty} (-2)^ne^{-n}</math> |
| − | <span class="exam">b) <math>\sum_{n=1}^{\infty} \bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg)</math> | + | ::<span class="exam">b) <math>\sum_{n=1}^{\infty} \bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg)</math> |
== [[009C_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] == | == [[009C_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] == | ||
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::::::<math>f(x)=\sum_{n=1}^{\infty} nx^n</math> | ::::::<math>f(x)=\sum_{n=1}^{\infty} nx^n</math> | ||
| − | <span class="exam">a) Find the radius of convergence of the power series. | + | ::<span class="exam">a) Find the radius of convergence of the power series. |
| − | <span class="exam">b) Determine the interval of convergence of the power series. | + | ::<span class="exam">b) Determine the interval of convergence of the power series. |
| − | <span class="exam">c) Obtain an explicit formula for the function <math style="vertical-align: -5px">f(x)</math>. | + | ::<span class="exam">c) Obtain an explicit formula for the function <math style="vertical-align: -5px">f(x)</math>. |
== [[009C_Sample Final 1,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] == | == [[009C_Sample Final 1,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] == | ||
| Line 45: | Line 47: | ||
::::::<math>r=1+\sin\theta</math> | ::::::<math>r=1+\sin\theta</math> | ||
| − | <span class="exam">a) Sketch the curve. | + | ::<span class="exam">a) Sketch the curve. |
| − | <span class="exam">b) Compute <math style="vertical-align: -12px">y'=\frac{dy}{dx}</math>. | + | ::<span class="exam">b) Compute <math style="vertical-align: -12px">y'=\frac{dy}{dx}</math>. |
| − | <span class="exam">c) Compute <math style="vertical-align: -12px">y''=\frac{d^2y}{dx^2}</math>. | + | ::<span class="exam">c) Compute <math style="vertical-align: -12px">y''=\frac{d^2y}{dx^2}</math>. |
== [[009C_Sample Final 1,_Problem_8|<span class="biglink"><span style="font-size:80%"> Problem 8 </span>]] == | == [[009C_Sample Final 1,_Problem_8|<span class="biglink"><span style="font-size:80%"> Problem 8 </span>]] == | ||
| Line 57: | Line 59: | ||
::::::<math>0\leq \theta \leq 2\pi</math> | ::::::<math>0\leq \theta \leq 2\pi</math> | ||
| − | <span class="exam">a) Sketch the curve. | + | ::<span class="exam">a) Sketch the curve. |
| − | <span class="exam">b) Find the area enclosed by the curve. | + | ::<span class="exam">b) Find the area enclosed by the curve. |
== [[009C_Sample Final 1,_Problem_9|<span class="biglink"><span style="font-size:80%"> Problem 9 </span>]] == | == [[009C_Sample Final 1,_Problem_9|<span class="biglink"><span style="font-size:80%"> Problem 9 </span>]] == | ||
| Line 76: | Line 78: | ||
::::::<span class="exam"><math>0\leq t \leq 2\pi</math> | ::::::<span class="exam"><math>0\leq t \leq 2\pi</math> | ||
| − | <span class="exam">a) Sketch the curve. | + | ::<span class="exam">a) Sketch the curve. |
| − | <span class="exam">b) Compute the equation of the tangent line at <math>t_0=\frac{\pi}{4}</math>. | + | ::<span class="exam">b) Compute the equation of the tangent line at <math>t_0=\frac{\pi}{4}</math>. |
Revision as of 17:15, 18 April 2016
This is a sample, and is meant to represent the material usually covered in Math 9C 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
Compute
- 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 \lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}}
- 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 \lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}}
Problem 2
Find the sum of the following series:
- 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 \sum_{n=0}^{\infty} (-2)^ne^{-n}}
- 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 \sum_{n=1}^{\infty} \bigg(\frac{1}{2^n}-\frac{1}{2^{n+1}}\bigg)}
Problem 3
Determine whether the following series converges or diverges.
- 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 \sum_{n=0}^{\infty} (-1)^n \frac{n!}{n^n}}
Problem 4
Find the interval of convergence of the following series.
- 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 \sum_{n=0}^{\infty} (-1)^n \frac{(x+2)^n}{n^2}}
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)=\sum_{n=1}^{\infty} nx^n}
- a) Find the radius of convergence of the power series.
- b) Determine the interval of convergence of the power series.
- c) Obtain an explicit formula for 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 f(x)} .
Problem 6
Find the Taylor polynomial of degree 4 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)=\cos^2x} 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 a=\frac{\pi}{4}} .
Problem 7
A curve is given in polar coordinates 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=1+\sin\theta}
- a) Sketch the curve.
- b) Compute 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'=\frac{dy}{dx}} .
- c) Compute 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''=\frac{d^2y}{dx^2}} .
Problem 8
A curve is given in polar coordinates 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=1+\sin 2\theta}
- 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\leq \theta \leq 2\pi}
- a) Sketch the curve.
- b) Find the area enclosed by the curve.
Problem 9
A curve is given in polar coordinates 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=\theta}
Find the length of the curve.
Problem 10
A curve is given in polar parametrically 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 x(t)=3\sin t}
- 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(t)=4\cos t}
- 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\leq t \leq 2\pi}
- a) Sketch the curve.
- b) Compute the equation of the tangent line 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=\frac{\pi}{4}} .