Difference between revisions of "009C Sample Final 3"

Latest revision as of 16:35, 3 December 2017

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

Which of the following sequences 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_n)_{n\ge 1}} converges? Which diverges? Give reasons for your answers!

(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 a_n=\bigg(1+\frac{1}{2n}\bigg)^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 a_n=\cos(n\pi)\bigg(\frac{1+n}{n}\bigg)^n}

Problem 2

Consider the 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=2}^\infty \frac{(-1)^n}{\sqrt{n}}.}

(a) Test if the series converges absolutely. Give reasons for your answer.

(b) Test if the series converges conditionally. Give reasons for your answer.

Problem 3

Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test.

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} \frac{n^3+7n}{\sqrt{1+n^{10}}}}

Problem 4

Determine if the following series converges or diverges. Please give your reason(s).

(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=1}^{\infty} \frac{n!}{(2n)!}}

(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} (-1)^n\frac{1}{n+1}}

Problem 5

Consider 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)=e^{-\frac{1}{3}x}.}

(a) Find a formula for 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 n} th derivative 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^{(n)}(x)} 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} and then find 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'(3).}

(b) Find the Taylor series 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 f(x)} 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 x_0=3,} i.e. write 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)} in the form

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=0}^\infty a_n(x-3)^n.}

Problem 6

Consider the power 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^{n+1}}{n+1}.}

(a) Find the radius of convergence of the above power series.

(b) Find the interval of convergence of the above power series.

(c) Find the closed 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)} to which the power series converges.

(d) Does the 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 \frac{1}{(n+1)3^{n+1}}}

converge?

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+\cos^2(2\theta).}

(a) Show that the point with Cartesian coordinates 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,y)=\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg)} belongs to the curve.

(b) Sketch the curve.

(c) In Cartesian coordinates, find 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 \bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg).}

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=4+3\sin \theta.}

(a) Sketch the curve.

(b) Find the area enclosed by the curve.

Problem 9

A wheel of radius 1 rolls along a straight line, say 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. A point 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} is located halfway between the center of the wheel and the rim; assume 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} starts at the point 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 \bigg(0,\frac{1}{2}\bigg).} As the wheel rolls, 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} traces a curve. Find parametric equations for the curve.

Problem 10

A curve is described 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^2}

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^3-t}

(a) Sketch the curve 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 -2\le t \le 2.}

(b) Find the equation of the tangent line to the curve at the origin.

@@ Line 5: / Line 5: @@
 == [[009C_Sample Final 3,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
-<span class="exam">Compute
+<span class="exam"> Which of the following sequences &nbsp;<math style="vertical-align: -5px">(a_n)_{n\ge 1}</math>&nbsp; converges? Which diverges? Give reasons for your answers!
-::<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) &nbsp;<math style="vertical-align: -15px">a_n=\bigg(1+\frac{1}{2n}\bigg)^n</math>
-::<span class="exam">b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math>
+<span class="exam">(b) &nbsp;<math style="vertical-align: -15px">a_n=\cos(n\pi)\bigg(\frac{1+n}{n}\bigg)^n</math>
 == [[009C_Sample Final 3,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
-<span class="exam"> Find the sum of the following series:
+<span class="exam"> Consider the series
-::<span class="exam">a) <math>\sum_{n=0}^{\infty} (-2)^ne^{-n}</math>
+::<math>\sum_{n=2}^\infty \frac{(-1)^n}{\sqrt{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">(a) Test if the series converges absolutely. Give reasons for your answer.
+<span class="exam">(b) Test if the series converges conditionally. Give reasons for your answer.
 == [[009C_Sample Final 3,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
-<span class="exam">Determine whether the following series converges or diverges.
+<span class="exam">Test if the following series converges or diverges. Give reasons and clearly state if you are using any standard test.
-::::::<math>\sum_{n=0}^{\infty} (-1)^n \frac{n!}{n^n}</math>
+::<math>\sum_{n=1}^{\infty} \frac{n^3+7n}{\sqrt{1+n^{10}}}</math>
 == [[009C_Sample Final 3,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
-<span class="exam"> Find the interval of convergence of the following series.
+<span class="exam"> Determine if the following series converges or diverges. Please give your reason(s).
+<span class="exam">(a) &nbsp;<math>\sum_{n=1}^{\infty} \frac{n!}{(2n)!}</math>
-::::::<math>\sum_{n=0}^{\infty} (-1)^n \frac{(x+2)^n}{n^2}</math>
+<span class="exam">(b) &nbsp;<math>\sum_{n=1}^{\infty} (-1)^n\frac{1}{n+1}</math>
 == [[009C_Sample Final 3,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
-<span class="exam"> Let
+<span class="exam"> Consider the function
-::::::<math>f(x)=\sum_{n=1}^{\infty} nx^n</math>
+::<math>f(x)=e^{-\frac{1}{3}x}.</math>
-::<span class="exam">a) Find the radius of convergence of the power series.
+<span class="exam">(a) Find a formula for the &nbsp;<math>n</math>th derivative &nbsp;<math style="vertical-align: -5px">f^{(n)}(x)</math>&nbsp; of &nbsp;<math style="vertical-align: -5px">f</math>&nbsp; and then find &nbsp;<math style="vertical-align: -5px">f'(3).</math>
-::<span class="exam">b) Determine the interval of convergence of the power series.
+<span class="exam">(b) Find the Taylor series for &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; at &nbsp;<math style="vertical-align: -5px">x_0=3,</math>&nbsp; i.e. write &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; in the form
-::<span class="exam">c) Obtain an explicit formula for the function <math style="vertical-align: -5px">f(x)</math>.
+::<math>f(x)=\sum_{n=0}^\infty a_n(x-3)^n.</math>
 == [[009C_Sample Final 3,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
-<span class="exam"> Find the Taylor polynomial of degree 4 of <math style="vertical-align: -5px">f(x)=\cos^2x</math> at <math>a=\frac{\pi}{4}</math>.
+<span class="exam"> Consider the power series
+::<math>\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}.</math>
+<span class="exam">(a) Find the radius of convergence of the above power series.
+<span class="exam">(b) Find the interval of convergence of the above power series.
+<span class="exam">(c) Find the closed formula for the function &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; to which the power series converges.
+<span class="exam">(d) Does the series
+::<math>\sum_{n=0}^\infty \frac{1}{(n+1)3^{n+1}}</math>
+<span class="exam">converge?
 == [[009C_Sample Final 3,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
 <span class="exam">A curve is given in polar coordinates by
-::::::<math>r=1+\sin\theta</math>
-::<span class="exam">a) Sketch the curve.
+::<math>r=1+\cos^2(2\theta).</math>
+<span class="exam">(a) Show that the point with Cartesian coordinates &nbsp;<math style="vertical-align: -15px">(x,y)=\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg)</math>&nbsp; belongs to the curve.
-::<span class="exam">b) Compute <math style="vertical-align: -12px">y'=\frac{dy}{dx}</math>.
+<span class="exam">(b) Sketch the curve.
-::<span class="exam">c) Compute <math style="vertical-align: -12px">y''=\frac{d^2y}{dx^2}</math>.
+<span class="exam">(c) In Cartesian coordinates, find the equation of the tangent line at &nbsp;<math style="vertical-align: -15px">\bigg(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\bigg).</math>
 == [[009C_Sample Final 3,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
-<span class="exam">A curve is given in polar coordinates by
+<span class="exam">A curve is given in polar coordinates by &nbsp;<math style="vertical-align: -2px">r=4+3\sin \theta.</math>
-::::::<math>r=1+\sin 2\theta</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 3,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
-<span class="exam">A curve is given in polar coordinates by
+<span class="exam">A wheel of radius 1 rolls along a straight line, say the &nbsp;<math>x</math>-axis. A point &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; is located halfway between the center of the wheel and the rim; assume &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; starts at the point &nbsp;<math style="vertical-align: -15px">\bigg(0,\frac{1}{2}\bigg).</math>&nbsp; As the wheel rolls, &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; traces a curve. Find parametric equations for the curve.
-::::::<span class="exam"><math>r=\theta</math>
-::::::<span class="exam"><math>0\leq \theta \leq 2\pi</math>
-<span class="exam">Find the length of the curve.
 == [[009C_Sample Final 3,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
-<span class="exam">A curve is given in polar parametrically by
+<span class="exam">A curve is described parametrically by
-::::::<span class="exam"><math>x(t)=3\sin t</math>
+::<span class="exam"><math>x=t^2</math>
-::::::<span class="exam"><math>y(t)=4\cos t</math>
+::<span class="exam"><math>y=t^3-t</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 for &nbsp;<math style="vertical-align: -2px">-2\le t \le 2.</math>
-::<span class="exam">b) Compute the equation of the tangent line at <math>t_0=\frac{\pi}{4}</math>.
+<span class="exam">(b) Find the equation of the tangent line to the curve at the origin.