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\centerline{MATH 136 -- Calculus 2}
\centerline{Retest Problems for the \emph{Core Learning Outcomes}}
\vskip 10pt
\noindent
{\it BP 1} -- I understand the definition of the definite integral $\int_a^b f(x)\ dx$ as a limit of Riemann sums
\vskip 10pt
\begin{enumerate}
\item The following limit yields the value of a definite integral $\int_a^b f(x)\ dx$:
$$\lim_{n\to\infty} \sum_{k=1}^n \left(\frac{3k}{n} + 1\right)^3 \frac{3}{n}$$
Give a possible $f(x)$ and interval $[a,b]$.  (There is more than one correct answer here; any one correct answer is OK.)
\item The following limit yields the value of a definite integral $\int_a^b f(x)\ dx$:
$$\lim_{n\to\infty} \sum_{k=1}^n \left(\frac{k-1}{n} + 6\right)^2 \frac{1}{n}$$
Give a possible $f(x)$ and interval $[a,b]$.  (There is more than one correct answer here; any one correct answer is OK.)
\item The following limit yields the value of a definite integral $\int_a^b f(x)\ dx$:
$$\lim_{n\to\infty} \frac{4}{n} \sum_{k=1}^n \sqrt{\frac{4k}{n} + 7} $$
Give a possible $f(x)$ and interval $[a,b]$.  (There is more than one correct answer here; any one correct answer is OK.)
\item The following limit yields the value of a definite integral $\int_a^b f(x)\ dx$:
$$\lim_{n\to\infty} \frac{\pi}{n} \sum_{k=1}^n \sin\left(\frac{\pi (k-1)}{n}\right) $$
Give a possible $f(x)$ and interval $[a,b]$.  (There is more than one correct answer here; any one correct answer is OK.)
\item The following limit yields the value of a definite integral $\int_a^b f(x)\ dx$:
$$\lim_{n\to\infty}  \sum_{k=1}^n \frac{k^6}{n^7} $$
Give a possible $f(x)$ and interval $[a,b]$.  (There is more than one correct answer here; any one correct answer is OK.)
\end{enumerate}
\vfill\eject
\noindent
{\it BP 2} -- I understand the interpretation of the value of  $\int_a^b f(x)\ dx$ as a signed area.
\vskip 10pt
\begin{enumerate}
\item Sketch the graph $y = -3x + 2$ on the interval $[0,2]$ and find the value of $\int_0^2 -3x + 2\ dx$ by using the geometry of your graph.
\item Sketch the graph 
$$f(x) = \begin{cases} 2x - 2 & \text{ if } 0 \le x \le 2\\
                                4-x & \text{ if } 2 \le x \le 6
         \end{cases}$$
on the interval $[0,5]$ and find the value of $\int_0^5 f(x)\ dx$ by using the geometry of your graph.
\item Sketch the graph 
$$f(x) = \begin{cases} 2x- 2 & \text{ if } 0 \le x \le 2\\
                                4-x & \text{ if } 2 \le x \le 6
         \end{cases}$$
on the interval $[1,6]$ and find the value of $\int_1^6 f(x)\ dx$ by using the geometry of your graph.
\item Sketch the graph 
$$f(x) = \begin{cases} \sqrt{1 - x^2} & \text{ if } 0 \le x \le 1\\
                                   -\sqrt{4 - (x - 3)^2} & \text{ if } 2 \le x \le 5
         \end{cases}$$
on the interval $[0,5]$ and use it to determine $\int_0^3 f(x)\ dx$.
\item Sketch the graph 
$$f(x) = \begin{cases} \sqrt{1 - x^2} & \text{ if } 0 \le x \le 1\\
                                   -\sqrt{4 - (x - 3)^2} & \text{ if } 2 \le x \le 5
         \end{cases}$$
on the interval $[0,5]$ and use it to determine $\int_0^5 f(x)\ dx$.
\end{enumerate}
\vfill\eject
{\it BP 3} -- I understand the Fundamental Theorem of Calculus (Parts I and/or II).
\vskip 10pt
\begin{enumerate}
\item Let 
$$F(x) = \int_3^x \frac{t}{t^4 + 4t^2 + 1}\ dt.$$  What is the slope of the tangent line to $y = F(x)$ at $x = 1$?
\item Let 
$$F(x) = \int_3^x \frac{t}{t^4 + 4t^2 + 1}\ dt.$$ 
Does $F$ have a local maximum, a local minimum, or neither at $x = 0$?
\item Let 
$$G(x) = \int_{x}^{10} \frac{t}{t^4 + 4t^2 + 1}\ dt.$$  
What is $F'(x)$?
\item Let 
$$G(x) = \int_0^{e^x} \frac{t}{t^4 + 4t^2 + 1}\ dt.$$  
What is $G'(x)$?
\item Let ${\rm Si}(x)$ be the function defined by $\displaystyle {\rm Si}(x) = \int_0^x \frac{\sin(t)}{t}\ dt$.  Find an antiderivative 
$G(x)$ of $\displaystyle f(x) = \frac{\sin(x)}{x}$ satisfying $G(2) = 0$ and give a formula for $G(x)$ using ${\rm Si}(x)$. 
\item Let ${\rm erf}(x)$ be the function defined by $\displaystyle {\rm erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2/2}\ dt$.  Find an antiderivative 
$H(x)$ of $\displaystyle f(x) = \frac{2}{\sqrt{\pi}} e^{-x^2/2}$ satisfying $H(1) = 0$ and give a formula for $H(x)$ using ${\rm erf}(x)$. 
\item Compute the definite integral 
$$\int_0^1 x^4 - 3\sqrt{x} + \sin(\pi x)\ dx.$$
\item Compute the definite integral 
$$\int_1^4 x^{3/2} - \frac{2}{x^2} + \cos(2\pi x)\ dx.$$
\item Compute the definite integral 
$$\int_0^3 12 x - 2e^x - \frac{1}{1 + x^2}\ dx.$$
\end{enumerate}
\vfill\eject
\noindent
{\it BP 4} -- I understand how the integral of a rate of change gives the total change and I can apply that idea.
\vskip 10pt
\begin{enumerate}
\item The velocity of a car moving along a straight line road at time $t$ is give by $v(t) = 15 t^2 - 30 t$ ($t$ in hours, $v$ in miles per hour).  What is the difference between the 
car's positions at $t = 0$ and $t = 3$?
\item The velocity of a car moving along a straight line road at time $t$ is give by $v(t) = 15 t^2 - 30 t$ ($t$ in hours, $v$ in miles per hour).   What is the total distance
traveled by the car between $t = 0$ and $t = 3$?  
\item Water is being poured into large holding tank at a rate of $5 + t$ gallons per minute ($t$ in minutes).  What is the difference between the amount of water in the 
tank at $t = 1$ and $t = 2$?  Include the appropriate units in your answer.
\item At time $t$ (in hours), the rate of change of the population of a colony of bacteria is $r(t) = e^{0.1 t}$ (in units of $1000$ bacteria per hour).  How 
much does the population change between $t = 1$ and $t = 2$ hours?
\item The velocity of a car moving along a straight line road at time $t$ is give by $v(t) = 20 \sin(t) + 30$ ($t$ in hours, $v$ in miles per hour).  What is the difference between the 
car's positions at $t = 0$ and $t = \pi$?
\end{enumerate}
\vfill\eject
\noindent
{\it BP 5} -- I understand what a differential equation is, what the order means, what a slope field shows, and what a solution is.
\vskip 10pt
\begin{enumerate}
\item Given:  The rate of change of the temperature $T$ of a frozen dinner in a $350^\circ$ oven at time $t$ is proportional to the difference between $350$ and the 
dinner's temperature.  Express this statement as a differential equation, including a proportionality constant $k$.  What is the order of your equation?  
Is the function $T(t) = 350 - 340 e^{kt}$ a solution of your equation?
\item What is the order of the differential equation $y'' - 4y' + 4y = 0$ with unknown $y = y(x)$?  Is $y = x e^{3x}$ a solution of this equation?
\item What is the order of the differential equation $y'' + y = e^t$ with unknown $y = y(t)$?  Is $y = \sin(t) + \frac{e^t}{2}$ a solution of this equation?
\item The plot (see figure above) shows the slope field of the differential equation $y' = 10 - y$ for $(x,y)$ in the rectangle $[0,10] \times [0,20]$.  On the slope
field plot, sketch the graphs of the solutions with $y(0) = 10$ and $y(0) = 5$.  
\begin{figure}
\begin{center}
\includegraphics[width=200pt]{SlopeField1.png}
\caption{Slope field for $y' = 10 - y$}
\end{center}
\label{slopefield}
\end{figure}
\item The plot (see figure below) shows the slope field of the differential equation $y' = y\left(1 - \frac{y}{10}\right)$ for $(x,y)$ in the rectangle $[0,10] \times [0,12]$.  On the slope
field plot, sketch the graphs of the solutions with $y(0) = 12$ and $y(0) = 5$.  
\begin{figure}
\begin{center}
\includegraphics[width=200pt]{SlopeField2.png}
\caption{Slope field for $y' = y\left(1 - \frac{y}{10}\right)$}
\end{center}
\label{slopefield}
\end{figure}

\end{enumerate}
\vfill\eject
\noindent
{\it BP 6} -- I understand what sequences and series are and what it means for each of them to converge or diverge.
\vskip 10pt
\begin{enumerate}
\item Let $\displaystyle a_n = \frac{n + 1}{n^2 + 7}$.  Is this a sequence or a series?  Does it converge?  How can you tell?
\item Consider $\displaystyle \sum_{n=1}^\infty \frac{n + 1}{n^2 + 7}$.  Is this a sequence or a series?  Does it converge?  How can you tell?
\item Let $b_n = e^{-n}$ Is this a sequence or a series?  Does it converge?  How can you tell?
\item Consider $\displaystyle \sum_{n=0}^\infty e^{-n}$.  Is this a sequence or a series?  Does it converge?  How can you tell?
\item Let $\displaystyle s_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$.  Is this a sequence or a series?  Does it converge?  How can you tell?
\item Consider $\displaystyle \sum_{n=1}^\infty \left(1 + \frac{1}{2} + \cdots + \frac{1}{n}\right)$.  Is this a sequence or a series?  Does it converge?  How can you tell?
\end{enumerate}

\vfill\eject
\noindent
{\it BP 7} --  I understand what the Taylor series of a function is, and I know the Taylor series for $e^x, \sin(x), \cos(x)$  expanding at 0.
\vskip 10pt
\begin{enumerate}
\item Find the Taylor series for $f(x) = x^4$, expanding at $a = 1$.  Use the definition.  
\item Find the Taylor series for $f(x) = e^{9x}$, expanding at $a = 0$.  How does your answer relate to the Taylor series for $e^x$?
\item Find the Taylor series for $\displaystyle f(x) = \frac{e^x + e^{-x}}{2}$, expanding at $a = 0$.  There are a number of different methods for this.  Any correct one is OK.
How does your answer relate to the Taylor series for $g(x) = \cos(x)$?
\item Find the Taylor series for $\displaystyle f(x) = \frac{e^x - e^{-x}}{2}$, expanding at $a = 0$.  There are a number of different methods for this.  Any correct one is OK.
How does your answer relate to the Taylor series for $g(x) = \sin(x)$?
\item Find the Taylor series for $f(x) = \ln(1 - x)$, expanding at $a = 0$.  There are a number of different methods for this.  Any correct one is OK.  How
does your answer relate to the geometric series with ratio $r = x$ and first term $c = 1$?
\end{enumerate}

\vfill\eject
\noindent
{\it S 1} --  I understand how to set up and evaluate Riemann sums.
\vskip 10pt
\begin{enumerate}
\item Compute the left- and right-hand Riemann sums for $f(x) = x^3 - 1$ on $[a,b] = [1,3]$ using the regular subdivision with $n = 4$.   Express your answers
in exact form as fractions.  One of these is an overestimate for $\int_1^3 x^3 - 1\ dx$ and one is an underestimate.  Which is which?
\item Compute the midpoint Riemann sum for $f(x) = x^2 + 1$ on $[a,b] = [1,3]$ using the regular subdivision with $n = 4$.  Express your answer
in exact form as a fraction.  Is your answer an overestimate or an underestimate for $\int_1^3 x^2 + 1\ dx$?  How can you tell?
\item What is the difference between the left- and right-hand Riemann sums for $f(x) = e^x$ with $n = 1000$ on the interval $[0,1]$.  (Hint: You do not need
to evaluate the left- and right-hand sums separately to find the difference!)
\item  Compute the left- and right-hand Riemann sums for $f(x) = e^{-x}$ on $[a,b] = [0,5]$ using the regular subdivision with $n = 5$.   Express your answers
in decimal form.  One of these is an overestimate for $\int_0^5 e^{-x}\ dx$ and one is an underestimate.  Which is which?
\item  Compute the midpoint Riemann sum for $f(x) = \sqrt{x}$ on $[a,b] = [2,3]$ using the regular subdivision with $n = 5$.   Express your answer
in decimal form.  Is your answer an overestimate or an underestimate for $\int_2^3 \sqrt{x}\ dx$?  How can you tell?
\end{enumerate}
\vfill\eject
\noindent
{\it S 2} -- I can find antiderivatives by the basic rules with power functions, exponential and logarithm functions, trigonometric and inverse trigonometric functions.
\vskip 10pt
\begin{enumerate}
\item Find the most general antiderivative for 
$$f(x) = \sqrt{x} + \frac{1}{x^{3/4}} - e^{5x} + \ln(x) + \cos(3x) - \frac{3}{\sqrt{1 - x^2}}.$$
\item Find the most general antiderivative for 
$$f(x) = \sqrt[3]{x} + \frac{1}{x^7} - e^{2x} + \ln(x) + \sin(4x) - \frac{2}{1 + x^2}.$$
\item Find the most general antiderivative for 
$$f(x) = x^{5/7} - \frac{1}{x^6} - e^{-x} + \ln(x) + \sec^2(5x) - \frac{8}{\sqrt{1 - x^2}}.$$
\item Find the most general antiderivative for 
$$f(x) = \sqrt[5]{x} + \frac{1}{x^4} - e^{-2x} + \ln(x) + \sec(5x)\tan(5x) - \frac{1}{|x| \sqrt{x^2 - 1}}.$$
\item Find the most general antiderivative for 
$$f(x) = x^{1/9} + \frac{1}{\sqrt{x}} + e^{3x} + \ln(x) + \csc(2x)\cot(2x) - \frac{16}{1 + x^2}.$$
\end{enumerate}
\vfill\eject
\noindent
{\it S 3} -- I can find indefinite integrals by the u-substitution method.
\vskip 10pt
\begin{enumerate}
\item Integrate:
$$\int (x + 3) \sqrt{5 x^2 + 30 x + 8}\ dx$$
\item Integrate:
$$\int \frac{e^{6x}}{\sqrt[3]{e^{6x} + 7}}\ dx$$
\item Integrate:
$$\int \frac{e^{\tan(3x)}}{\cos^2(3x)}\ dx$$
\item Integrate:
$$\int \frac{x + 4}{x^2 + 8x + 23}\ dx$$
\item Integrate:
$$\int \frac{1}{x\ln(x)}\ dx$$
\end{enumerate}
\vfill\eject
\noindent
{\it S 4} -- I can recognize when integration by parts applies and apply the parts formula.
\vskip 10pt
\begin{enumerate}
\item Consider integrals of the form $\int x^n \sin(a x)\ dx$ where $n$ is a positive integer and $a$ is a constant.  Choose a $u$ and a $dv$ for an integration by parts and apply the parts formula to rewrite this in an equivalent form.  You \emph{do not} need to evaluate the resulting integral on the right-hand side of your equation.  
\item Consider integrals of the form $\int x^n \cos(a x)\ dx$ where $n$ is a positive integer and $a$ is a constant.  Choose a $u$ and a $dv$ for an integration by parts and apply the parts formula to rewrite this in an equivalent form.  You \emph{do not} need to evaluate the resulting integral on the right-hand side of your equation.  
\item Consider integrals of the form $\int x^n \ln(a x)\ dx$ where $n$ is a positive integer and $a > 0$ is a constant.  Choose a $u$ and a $dv$ for an integration by parts and apply the parts formula to rewrite this in an equivalent form.  Evaluate the resulting integral on the right-hand side of your equation.
\item  Consider integrals of the form $\int e^{ax} \sin(b x)\ dx$ where $a,b$ are constants.  Choose a $u$ and a $dv$ for an integration by parts and apply the parts formula to rewrite this in an equivalent form.  You \emph{do not} need to evaluate the resulting integral on the right-hand side of your equation.  
\item  Consider integrals of the form $\int e^{ax} \cos(b x)\ dx$ where $a,b$ are constants.  Choose a $u$ and a $dv$ for an integration by parts and apply the parts formula to rewrite this in an equivalent form.  You \emph{do not} need to evaluate the resulting integral on the right-hand side of your equation.  
\end{enumerate}
\vfill\eject
\noindent
{\it S 5} -- I can recognize when integration by trigonometric substitution applies, select the appropriate substitution and convert an integral to an equivalent trigonometric form.
\vskip 10pt
\begin{enumerate}
\item Does the trigonometric substitution method apply to the following integral:
$$\int \frac{x^2}{\sqrt{16 - x^2}}\ dx?$$
Why or why not?  If so, 
what trigonometric substitution would you apply to evaluate it?
Make the substitution and convert to a trigonometric integral in $\theta$.  Simplify as much as possible.
\item Does the trigonometric substitution method apply to the following integral:
$$\int \frac{x^2}{\sqrt{x^3 - 36}}\ dx?$$
If so, what trigonometric substitution would you apply to evaluate it?
Make the substitution and convert to a trigonometric integral in $\theta$.  Simplify as much as possible.
\item Does the trigonometric substitution method apply to the following integral:  
$$\int \frac{x^2}{\sqrt{x^2 + 49}}\ dx?$$
Why or why not?  
Make the substitution and convert to a trigonometric integral in $\theta$.  Simplify as much as possible.
\item  Does the trigonometric substitution method apply to the following integral:
$$\int \frac{1}{x^2 \sqrt{x^2 - 16}}\ dx?$$
Why or why not?  If so, what trigonometric substitution would you apply to evaluate it?
Make the substitution and convert to a trigonometric integral in $\theta$.  Simplify as much as possible.
\item Does the trigonometric substitution method apply to the following integral:  
$$\int \frac{1}{x \sqrt{9 - x^5}}\ dx?$$
Why or why not?  What trigonometric substitution would you apply to evaluate it?
Make the substitution and convert to a trigonometric integral in $\theta$.  Simplify as much as possible.
\end{enumerate}
\vfill\eject
\noindent
{\it S 6} --  I can recognize when integration by partial fractions applies, and set up and solve for the coefficients in the partial fractions.

\vskip 10pt
\begin{enumerate}
\item Does the partial fractions method apply to the following integral:
$$\int \frac{x^3}{\sqrt{16 - x^2}}\ dx?$$
Why or why not?  If so, divide if necessary, then set up and solve for the coefficients in the partial fractions.
\item Does the partial fractions method apply to the following integral:
$$\int \frac{x^2+1}{x^3 + 3x}\ dx?$$
Why or why not?  If so, divide if necessary, then set up and solve for the coefficients in the partial fractions.
\item Does the partial fractions method apply to the following integral:
$$\int \frac{x^2+x+2}{x^3 - 9x}\ dx?$$
Why or why not?  If so, divide if necessary, then set up and solve for the coefficients in the partial fractions.
\item Does the partial fractions method apply to the following integral:
$$\int \frac{e^x}{x^2 - 9x - 90}\ dx?$$
Why or why not?  If so, divide if necessary, then set up and solve for the coefficients in the partial fractions.
\item Does the partial fractions method apply to the following integral:
$$\int \frac{x^2}{x^2 - 9x - 90}\ dx?$$
Why or why not?  If so, divide if necessary, then set up and solve for the coefficients in the partial fractions.
\end{enumerate}

\vfill\eject
\noindent
{\it S 7} -- I can use combinations of the methods in S2 through S6 as appropriate to completely work out indefinite integrals.  
\vskip 10pt
\begin{enumerate}
\item Use combinations of the methods in S2 through S6 as appropriate to compute the indefinite integral 
$$\int \frac{x^2}{\sqrt{9 - x^2}}\ dx.$$
If you used a trigonometric substitution, convert back to a function of $x$.
\item Use combinations of the methods in S2 through S6 as appropriate to compute the indefinite integral 
$$\int \frac{x^2 + 1}{x^2 - 5x + 6}.$$
If you used a trigonometric substitution, convert back to a function of $x$.
\item Use combinations of the methods in S2 through S6 as appropriate to compute the indefinite integral 
$$\int \frac{x}{\sqrt{1 - x^4}}\ dx$$
If you used a trigonometric substitution, convert back to a function of $x$.
\item Use combinations of the methods in S2 through S6 as appropriate to compute the indefinite integral 
$$\int x^5 \cos(5x^2)\ dx$$
If you used a trigonometric substitution, convert back to a function of $x$.
\item Use combinations of the methods in S2 through S6 as appropriate to compute the indefinite integral 
$$\int \sin^{-1}(x)\ dx$$
If you used a trigonometric substitution, convert back to a function of $x$.
\item Use combinations of the methods in S2 through S6 as appropriate to compute the indefinite integral 
$$\int \frac{1}{e^x + 1}\ dx$$
Hint:  Multiply numerator and denominator by $e^x$ to start off.
If you used a trigonometric substitution, convert back to a function of $x$.
\item Use combinations of the methods in S2 through S6 as appropriate to compute the indefinite integral 
$$\int x\tan^{-1}(x)\ dx$$
If you used a trigonometric substitution, convert back to a function of $x$.
\item Use combinations of the methods in S2 through S6 as appropriate to compute the indefinite integral 
$$\int x^2\tan^{-1}(x)\ dx$$
If you used a trigonometric substitution, convert back to a function of $x$.
\end{enumerate}

\vfill\eject
\noindent
{\it S 8} --  I understand what makes an integral improper, what it means for an improper integral to converge or diverge, and how to 
tell by setting and evaluating appropriate limits.
\vskip 10pt
\begin{enumerate}
\item Is the integral 
$$\int_0^\infty e^{-x}\ dx$$
improper?  Why or why not?  If it is improper, set up and evaluate the limit(s) 
to determine whether the integral converges or diverges.  If it is not improper, evaluate the integral.
\item Is the integral 
$$\int_1^3 \frac{1}{x^2 - 3x + 2}\ dx$$
improper?  Why or why not?  If it is improper, set up and evaluate the limit(s) 
to determine whether the integral converges or diverges.  If it is not improper, evaluate the integral.
\item Is the integral 
$$\int_5^{10} \frac{1}{x^2 - 3x + 2}\ dx$$
improper?  Why or why not?  If it is improper, set up and evaluate the limit(s) 
to determine whether the integral converges or diverges.  If it is not improper, evaluate the integral.
\item Is the integral 
$$\int_{-\infty}^\infty \frac{1}{x^2 + 9}\ dx$$
improper?  Why or why not?  If it is improper, set up and evaluate the limit(s) 
to determine whether the integral converges or diverges.  If it is not improper, evaluate the integral.
\item Is the integral 
$$\int_0^3 x\ln(x)\ dx$$
improper?  Why or why not? If it is improper, set up and evaluate the limit(s) 
to determine whether the integral converges or diverges.  If it is not improper, evaluate the integral.
\end{enumerate}

\vfill\eject
\noindent
{\it S 9} -- I can recognize separable differential equations and solve them by integration.
\vskip 10pt
\begin{enumerate}
\item Is the first-order differential equation 
$$y' = xy - 3y + 5x -15$$
separable?  Why or why not?  If it is separable, find the general solution and express in explicit form $y = \text{function of }  x$.
\item Is the first-order differential equation 
$$y' = \frac{x^2}{\cos(y)}$$
separable?  Why or why not?  If it is separable, find the general solution and express in explicit form $y = \text{function of } x$.
\item Is the first-order differential equation 
$$y' = x^3 (y^2 + 1)$$
separable?  Why or why not?  If it is separable, find the general solution and express in explicit form $y = \text{function of } x$.
\item Is the first-order differential equation 
$$y' = \frac{3}{x} y + \frac{4}{x}$$
separable?  Why or why not?  If it is separable, find the general solution and express in explicit form $y = \text{function of } x$.
\item Is the first-order differential equation 
$$y' = 4y(1 - y)$$
separable?  Why or why not?  If it is separable, find the general solution and express in explicit form $y = \text{function of } x$.
\end{enumerate}
\vfill\eject
\noindent
{\it S 10} --  I can recognize harmonic, geometric, and p-series and determine whether they converge or diverge.
\vskip 10pt
\begin{enumerate}
\item  Of the following three infinite series, only one is either harmonic, geometric, or a $p$-series.  Indicate the one that falls into at least
one of those categories, say which categories it belongs to, and say whether it converges or not by applying what we know about those types of series.
$$\text{(a)}\quad \sum_{n=1}^\infty \frac{1}{n^5} \qquad \text{(b)} \quad  \sum_{n=0}^\infty \frac{1}{2^n + 3^n} \qquad \text{(c)} \quad \sum_{n=1}^\infty \frac{n + 3}{n^2 - n + 7}$$
\item  Of the following three infinite series, only one is either harmonic, geometric, or a $p$-series.  Indicate the one that falls into at least 
one of those categories, say which categories it belongs to, and say whether it converges or not by applying what we know about those types of series.
$$\text{(a)}\quad \sum_{n=1}^\infty \frac{1}{n^3 + 8} \qquad \text{(b)} \quad  \sum_{n=0}^\infty \frac{8^n}{n!} \qquad \text{(c)} \quad \sum_{n=1}^\infty \frac{1}{n}$$
\item  Of the following three infinite series, only one is either harmonic, geometric, or a $p$-series.  Indicate the one that falls into at least 
one of those categories, say which categories it belongs to, and say whether it converges or not by applying what we know about those types of series.
$$\text{(a)}\quad \sum_{n=1}^\infty \frac{9^n}{8^n} \qquad \text{(b)} \quad  \sum_{n=0}^\infty \frac{n!}{e^n} \qquad \text{(c)} \quad \sum_{n=1}^\infty \frac{1}{n^2 - 5}$$
\item  Of the following three infinite series, only one is either harmonic, geometric, or a $p$-series.  Indicate the one that falls into at least
one of those categories, say which categories it belongs to, and say whether it converges or not by applying what we know about those types of series.
$$\text{(a)}\quad \sum_{n=1}^\infty \frac{n^4}{n^5 + 1} \qquad \text{(b)} \quad  \sum_{n=0}^\infty \frac{(-1)^n}{3^n} \qquad \text{(c)} \quad \sum_{n=1}^\infty \frac{\sin(n)}{n}$$
\item  Of the following three infinite series, only one is either harmonic, geometric, or a $p$-series.  Indicate the one that falls into at least
one of those categories, say which categories it belongs to, and say whether it converges or not by applying what we know about those types of series.
$$\text{(a)}\quad \sum_{n=1}^\infty \frac{n^2}{n^2 + 3n - 1} \qquad \text{(b)} \quad  \sum_{n=0}^\infty \frac{\cos(n)}{e^n} \qquad \text{(c)} \quad \sum_{n=1}^\infty \frac{1}{n^{5/8}} $$
\end{enumerate}
\vfill\eject
\noindent
{\it S 11} -- I understand when the nth term, integral, alternating series, or ratio tests apply and I can use them to determine whether a series converges or diverges.
\begin{enumerate}
\item Why does the integral test apply to the infinite series
$$\sum_{n=1}^\infty \frac{1}{n^2 + 5}?$$
Apply it to determine whether the series converges or diverges.  State your conclusion clearly.  
\item Why does the integral test apply to the infinite series
$$\sum_{n=2}^\infty \frac{1}{n\ln(n)}?$$
Apply it to determine whether the series converges or diverges.  State your conclusion clearly.   
\item According to the $n$th term test, what can we say about the infinite series
$$\sum_{n=1}^\infty \frac{n^2}{n^2 + 3n - 1}?$$
State your conclusion clearly.
\item  According to the $n$th term test, what can we say about the infinite series
$$\sum_{n=1}^\infty \frac{n}{n^2 + 3n - 1}?$$
State your conclusion clearly.
\item Let $p(n)$ be the $n$th prime number, so $p(1) = 2$, $p(2) = 3$, $p(3) = 5$, and so on.  Why does the 
alternating series test apply to the infinite series
$$\sum_{n=1}^\infty \frac{(-1)^n}{p(n)}?$$
Apply the test to determine whether the series converges or diverges.  State your conclusion clearly.
\item Why does the alternating series test apply to the infinite series
$$\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt[5]{n}}?$$
Apply the test to determine whether the series converges or diverges.  State your conclusion clearly.
\item What does the ratio test say about the infinite series
$$\sum_{n=0}^\infty \frac{(-1)^n 8^n}{(2n)!}?$$
Explain what you are doing to apply the test and state your conclusion clearly.
\item What does the ratio test say about the infinite series
$$\sum_{n=0}^\infty \frac{(-1)^n n^4}{e^n}?$$
Explain what you are doing to apply the test and state your conclusion clearly.
\end{enumerate}

\vfill\eject
\noindent
{\it S 12} -- I can use the ratio test to determine the interval of absolute convergence for a power series, and test convergence at the endpoints of the interval (if there are any).
\vskip 10pt
\begin{enumerate}
\item Use the ratio test to determine the interval of absolute convergence for the power series
$$\sum_{n=1}^\infty \frac{x^n}{n}$$
and test convergence at the endpoints of the interval (if there are any).
\item Use the ratio test to determine the interval of absolute convergence for the power series
$$\sum_{n=1}^\infty \frac{(-1)^n (x-1)^n}{n}$$
and test convergence at the endpoints of the interval (if there are any).
\item Use the ratio test to determine the interval of absolute convergence for the power series
$$\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}$$
and test convergence at the endpoints of the interval (if there are any).
\item Use the ratio test to determine the interval of absolute convergence for the power series
$$\sum_{n=1}^\infty n^2 (x - 3)^n$$
and test convergence at the endpoints of the interval (if there are any).
\item Use the ratio test to determine the interval of absolute convergence for the power series
$$\sum_{n=1}^\infty e^n (x - 2)^n$$
and test convergence at the endpoints of the interval (if there are any).
\end{enumerate}

\vfill\eject
\noindent
{\it A 1} -- I understand how approximations to areas, average values, volumes, moments, etc. lead to Riemann sums and where the integral formulas come from. 

\begin{enumerate}
\item The area between two graphs $y = f(x)$ and $y = g(x)$ and $a \le x \le b$ can be approximated by subdividing the $x$-interval into $n$ equal 
subintervals, picking one point $x_k^*$ in the $k$th subinterval, and forming the sum
$$A \doteq \sum_{k=1}^n (f(x_k^*) - g(x_k^*)) \Delta x$$
if $f(x) \ge g(x)$ on the whole interval.  What do we want to do to this sum to get the exact value of the area?  What is the corresponding integral
formula for the area.  What would need to be changed if $f(x) < g(x)$ on part or all of the interval?
\item The average value $\overline{y}$ of $f(x)$ on the interval $a \le x \le b$ can be approximated by subdividing the $x$-interval into $n$ equal 
subintervals, picking one point $x_k^*$ in the $k$th subinterval, and forming the sum
$$\overline{y} \doteq \frac{1}{n} \sum_{k = 1}^n f(x_k^*)$$
 How does this need to be rearranged before
 we can recognize a Riemann sum in this?  What do we want to do to this sum to get the exact average value? 
 What is the corresponding integral formula for the average value?
 \item Suppose a solid is lined up along the $x$-axis and occupies the interval $a \le x \le b$.  If the area of the cross-section at $x$ is 
 $A(x)$, then the volume of the solid can be approximated by by subdividing the $x$-interval into $n$ equal 
subintervals, picking one point $x_k^*$ in the $k$th subinterval, and forming the sum
$$V \doteq \sum_{k=1}^n A(x_k^*)\ \Delta x.$$
What do we want to do to this sum to get the exact value of the volume?  What is the corresponding integral
formula for the volume. 
\item Suppose a solid is generated by rotating the region between $y = f(x) \ge 0$ and the $x$-axis for $a \le x \le b$ \emph{about the $x$-axis}.  
What is the area $A(x)$ of the cross-section at $x$ in this case?
The volume of the solid can be approximated by subdividing the $x$-interval into $n$ equal 
subintervals, picking one point $x_k^*$ in the $k$th subinterval, and forming the sum
$$V \doteq \sum_{k=1}^n A(x_k^*)\ \Delta x.$$
What do we want to do to this sum to get the exact value of the volume?  What is the corresponding integral
formula for the volume, expressed using the function $f(x)$? 
\item Suppose a solid is generated by rotating the region between $y = f(x) \ge 0$ and the $x$-axis for $a \le x \le b$ \emph{about the $y$-axis}.  
The volume of the solid can be approximated by subdividing the $x$-interval into $n$ equal 
subintervals, picking one point $x_k^*$ in the $k$th subinterval, and forming the sum
$$V \doteq \sum_{k=1}^n 2\pi\, x_k^*\, f(x_k^*)\ \Delta x.$$
Where does the $2\pi\, x_k^*\, f(x_k^*)$ come from here?  (What is the shape of the part of the solid we get by rotating 
one strip of the region under $y = f(x)$ about the $y$-axis?)
What do we want to do to this sum to get the exact value of the volume?  What is the corresponding integral
formula for the volume, expressed using the function $f(x)$?
\end{enumerate}
\vfill\eject
\noindent
{\it A 2} -- I can find the area between two graphs for some specified range of $x$-values or by finding where the graphs cross.
\vskip 10pt
\begin{enumerate}
\item Find the area below $y = e^{-x}$ and above $y = 1/2$ for $x > 0$.
\item Find the area below $y = \frac{1}{1 + x^2}$ and above $y = \frac{1}{5}$.  
\item Find the area between $y = x^2$ and $y = 2x$ for $-1 \le x \le 3$.  Note: These graphs cross at least once in the given 
interval.
\item Find the area between $y = \sin(x)$ and $y = \cos(x)$ for $0 \le x \le \pi$.  Note:  These graphs cross at least once
in the given interval.
\item Find the area between $y = x^3$ and $y = x/3$ for $-1 \le x \le 1$.  Note: These graphs cross at least once
in the given interval.
\item Set up an integral or integrals to find the area inside the unit circle centered at $(0,0)$ and above the parabola $y = x^2$.  
Do not evaluate.  
\item Set up an integral or integrals to find the total area between upper branch of the hyperbola $y^2 - x^2 = 1$ and its asymptote line
$y = x$ for $x > 0$.  Do not evaluate.  What kind of integral is this?  
\end{enumerate}

\vfill\eject
\noindent
{\it A 3} -- I can find volumes by slicing and by cylindrical shells.
\vskip 10pt
\begin{enumerate}
\item A solid has a base equal to the unit circle in the $xy$-plane.  The cross-sections by planes perpendicular to the $x$-axis are squares extending the 
full width of the base at that $x$.  What is the volume of the solid?
\item A solid has a base equal to the unit circle in the $xy$-plane.  The cross-sections by planes perpendicular to the $x$-axis are equilateral triangles extending the 
full width of the base at that $x$.  What is the volume of the solid?
\item  A solid is generated by rotating the region between $y = \sqrt{x^2 + 1}$ and $y = x$ for $0 \le x \le 2$ about the $x$-axis.  What is its volume?  (Any correct method is OK.)
\item  A solid is generated by rotating the region between $y = \sqrt{x^2 + 1}$ and $y = x$ for $0 \le x \le 2$ about the $y$-axis.  What is its volume?  (Any correct method is OK.)
\item  A solid is generated by rotating the region below $y = e^{-x}$ and above $y = 1/2$ for $x > 0$ about the $x$-axis.  What is its volume?  (Any correct method is OK.)
\item  A solid is generated by rotating the region below $y = e^{-x}$ and above $y = 1/2$ for $x > 0$ about the $y$-axis.  What is its volume?  (Any correct method is OK.)
\item  A solid is generated by rotating the region bounded above by $y = \sqrt{1 - x^2}$ and below by $y = 1/3$ about the $x$-axis.  What is its volume? (Any correct method is OK.)
\item Is it true that 
$$\int_{-2\sqrt{2}/3}^{2\sqrt{2}/3} \pi\cdot (\sqrt{1 - x^2})^2 - \pi \left(\frac{1}{3}\right)^2\ dx = 2\int_{1/3}^1 2\pi x \sqrt{1 - x^2}\ dx?$$
Explain, either by interpreting the value of each integral as a certain volume or by computing both sides.
\end{enumerate}
\vfill\eject
\noindent
{\it A 4} --  I can solve modeling problems about exponential growth/decay, heating and cooling, and logistic population growth using solutions of differential equations.

\vskip 10pt
\noindent
\begin{enumerate}
\item In radioactive decay, the rate of change of the amount of an isotope present is proportional to the amount present at all times.  A sample of radioactive
carbon-14 contained $100$ grams at time $t = 0$ years.  After $t = 5730$ years, the amount had decayed to $50$ grams.  How long after $t = 0$ will 
it take for the amount to decay to $15$ grams?
\item The population of a colony of bacteria is growing at a rate proportional to the population size.  The population was $1000$ bacteria at $t = 0$ and 
by $t = 1$ hour it had grown to $1500$ bacteria.  How long will it take for the population to reach $5000$ bacteria?
\item A frozen pizza is taken out of $0^\circ$F freezer at $t = 0$ and placed into an oven held at a constant $400^\circ$F.  After $5$ minutes, the temperature of the pizza
has increased to $50^\circ$F.  How long will it take for the pizza to reach $300^\circ$F?  Assume that the situation is governed by Newton's Law
of Heating and Cooling: the rate of change of the temperature of the pizza is proportional to the difference between its temperature and
the oven temperature.
\item A pizza with temperature $400^\circ$F is taken out of the oven at $t = 0$ and placed on a counter a kitchen held at a constant $70^\circ$F to cool before it is served. 
After $5$ minutes, the temperature of the pizza
has decreased to $250^\circ$F.  How much longer will it take for the pizza to reach $200^\circ$F?  Assume that the situation is governed by Newton's Law
of Heating and Cooling: the rate of change of the temperature of the pizza is proportional to the difference between its temperature and
the room temperature.
\item A population of rabbits is living in a habitat that can support at most $100$ of them in the long run.  At time $t = 0$ months there are $30$ rabbits and the 
population has grown to $40$ rabbits by time $t = 3$ months.  How long will it take for the population to reach $95$ rabbits?  Assume 
logistic growth -- that is, if $P = P(t)$ is the population, then $P' = rP(1 - P/K)$, where $K$ is the carrying capacity, and $r$ is a proportionality constant.
\item At $t = 0$, a population of $120$ rabbits is introduced into a habitat that can only support at most $100$ of them in the long run.  The population has
fallen to $110$ by time $t = 3$ months.  How long will it take for the population to reach $105$ rabbits?  Assume 
logistic growth -- that is, if $P = P(t)$ is the population, then $P' = rP(1 - P/K)$, where $K$ is the carrying capacity, and $r$ is a proportionality constant.
\end{enumerate}
\end{document}