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% Worksheets are Copyright 2008 by Eric Hsu, erichsu@math.sfsu.edu
% Usage Notes at http://math.sfsu.edu/hsu
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\begin{center}
\textbf{Calculus I With Group Work} \\
by Eric Hsu\\[.25in]
Raw Materials Piloted in Fall 2008\\ at San Francisco State University\\[.25in]
Usage Notes are at \texttt{http://math.sfsu.edu/hsu}\\[2in]
Copyright 2008 by Eric Hsu, \texttt{erichsu@math.sfsu.edu}\\
(except anything that I clearly stole from someone else)
\end{center}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 1}
\end{center}
\begin{enumerate}
\item \label{flag} At summer camp, a child comes out every morning to raise a flag. Consider the height of the flag as a function of time. Sketch what such a graph might look like.
\item Consider these candidates for the graph in (\ref{flag}). Explain what each graph would mean. What would reasonable units be for each axis? Which one seems the most realistic to you? The least realistic?\\[1cm]
(A)
\includegraphics[width=3in]{plot/flag1.pdf} $\;$ (B)
\includegraphics[width=3in]{plot/flag2.pdf} \\[.2in]
(C)
\includegraphics[width=3in]{plot/flag3.pdf} $\;$ (D)
\includegraphics[width=3in]{plot/flag4.pdf} \\[.2in]
(E)
\includegraphics[width=3in]{plot/flag5.pdf} $\;$ (F)
\includegraphics[width=3in]{plot/flag6.pdf} \\[.2in]
\end{enumerate}
% Extension:
% \item For each graph, try to give a formula for a function with a matching graph.
\newpage
\begin{center}
\textbf{Math 226 Worksheet 2}
\end{center}
\begin{enumerate}
\item Without using a graphing calculator, try to match the following six equations to the six graphs in Worksheet 1:
\[ y=-\frac{1}{5}\cos(5x)+\frac{3}{10}+x; \;\;\;\;\;\; y=\frac{1}{2}(x-2)^3+4; \;\;\;\;\;\; x=3;\]
\[ y=\frac{3}{2}x;\;\;\;\;\;\; y=-\cos(4x)+1; \;\;\;\;\;\; y=(\frac{x}{2})^2; \]
\item There is a speed trap in my hometown. It is a straight stretch of road and at the start, the speed limit suddenly drops to 60 miles per hour. There is a machine that makes a graph for each car that drives on this road, graphing the position of the car (in miles) as a function of time (in \underline{minutes}).
Suppose the six graphs from Worksheet 1 are graphs that this machine recorded.
\begin{enumerate}
\item Did Car (A) break the speed limit? What was its velocity, and at which times?
\item Which of the other cars broke the speed limit? Tell me when the violations happened.
\item For each car, pick a specific time and estimate as well as you can its velocity at that moment. You can use a calculator for this part.
\item Somewhere near the 4 mile mark, there is a stop sign. Did any cars actually stop for it? \\[1in]
\end{enumerate}
\end{enumerate}
\begin{center}
\textbf{Math 226 Worksheet 2}
\end{center}
\begin{enumerate}
\item Without using a graphing calculator, try to match the following six equations to the six graphs in Worksheet 1:
\[ y=-\frac{1}{5}\cos(5x)+\frac{3}{10}+x; \;\;\;\;\;\; y=\frac{1}{2}(x-2)^3+4; \;\;\;\;\;\; x=3;\]
\[ y=\frac{3}{2}x;\;\;\;\;\;\; y=-\cos(4x)+1; \;\;\;\;\;\; y=(\frac{x}{2})^2; \]
\item There is a speed trap in my hometown. It is a straight stretch of road and at the start, the speed limit suddenly drops to 60 miles per hour. There is a machine that makes a graph for each car that drives on this road, graphing the position of the car (in miles) as a function of time (in \underline{minutes}).
Suppose the six graphs from Worksheet 1 are graphs that this machine recorded.
\begin{enumerate}
\item Did Car (A) break the speed limit? What was its velocity, and at which times?
\item Which of the other cars broke the speed limit? Tell me when the violations happened.
\item For each car, pick a specific time and estimate as well as you can its velocity at that moment. You can use a calculator for this part.
\item Somewhere near the 4 mile mark, there is a stop sign. Did any cars actually stop for it?
\end{enumerate}
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 3}
\end{center}
\begin{enumerate}
\item For Cars (A), (B) and (D) in Worksheet 2, calculate the average velocity on the interval $[2,3]$, i.e. between 2 minutes and 3 minutes. Then calculate the average velocity over $[2,2.5]$. Which calculation seems closer to the instantaneous velocity at 2 minutes?
\item \label{shrinkintervals} Your textbook suggests estimating the instantaneous velocity at 2 minutes by calculating average velocities over $[2,2+h]$, and choosing smaller and smaller values for the interval length $h$. Carry out this strategy for Cars (A), (B) and (D).
\item Create a graph of a position function where the average velocity over $[2,3]$ is a \underline{better} estimate for the instantaneous velocity at 2 minutes than the average velocity over $[2,2.5]$.
\item \label{powerrule} For Cars (A), (B) and (D), use the strategy in (\ref{shrinkintervals}) to estimate the instantaneous velocities at $x=0,1,2,3$. Try to come up with a formula for the velocity function of each of the Cars. This function should take as input \underline{any} time (not just $x= 0,1,2, 3$) and should output the instantaneous velocity at that time. Make sure your formula fits your data and try it out on different input times.
\item Suppose a car has a position $x^n$ miles for any time $x$ minutes. Use your work in (\ref{powerrule}) to hypothesize what its velocity function should be.
\item For each of Cars (A), (B) and (D), write the successive approximations in (\ref{shrinkintervals}) as a formal limit of a function. Try to evaluate the limit and prove your hypothesis in (\ref{powerrule}).\\[.5in]
\end{enumerate}
\begin{center}
\textbf{Math 226 Worksheet 3}
\end{center}
\begin{enumerate}
\item For Cars (A), (B) and (D) in Worksheet 2, calculate the average velocity on the interval $[2,3]$, i.e. between 2 minutes and 3 minutes. Then calculate the average velocity over $[2,2.5]$. Which calculation seems closer to the instantaneous velocity at 2 minutes?
\item \label{shrinkintervals2} Your textbook suggests estimating the instantaneous velocity at 2 minutes by calculating average velocities over $[2,2+h]$, and choosing smaller and smaller values for the interval length $h$. Carry out this strategy for Cars (A), (B) and (D).
\item Create a graph of a position function where the average velocity over $[2,3]$ is a \underline{better} estimate for the instantaneous velocity at 2 minutes than the average velocity over $[2,2.5]$.
\item \label{powerrule2} For Cars (A), (B) and (D), use the strategy in (\ref{shrinkintervals2}) to estimate the instantaneous velocities at $x=0,1,2,3$. Try to come up with a formula for the velocity function of each of the Cars. This function should take as input \underline{any} time (not just $x= 0,1,2, 3$) and should output the instantaneous velocity at that time. Make sure your formula fits your data and try it out on different input times.
\item Suppose a car has a position $x^n$ miles for any time $x$ minutes. Use your work in (\ref{powerrule2}) to hypothesize what its velocity function should be.
\item For each of Cars (A), (B) and (D), write the successive approximations in (\ref{shrinkintervals2}) as a formal limit of a function. Try to evaluate the limit and prove your hypothesis in (\ref{powerrule2}).
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 4}
\end{center}
\begin{enumerate}
\item \label{calcisland} The population of Calculus Island is modeled by the function $c(x) = x^2$, where the inputs are years after 2000 and the outputs are thousands of people.
\begin{enumerate}
\item Calculate the average rate of change of the population between 2002 and 2003.
\item Calculate the instantaneous rate of change of the population at the start of 2002.
\end{enumerate}
\item Suppose Precalculus Island has a population modeled by the function $p(x) = x^2 + 3$, inputs and outputs as above.
\begin{enumerate}
\item Calculate the analogous answers to the two questions in (\ref{calcisland}).
\item \label{comparecalcprecalc} Compare your answers for Calculus and Precalculus Islands. Explain why your answers are related in that way.
\item Do you think the relationship in (\ref{comparecalcprecalc}) holds for the instantaneous rate of change at ANY time $x$?
\end{enumerate}
\item We find it tiresome to write `the instantaneous rate of change of' a function $f$, so we will often use the equivalent term `\textbf{derivative of}' a function $f$. We will often write this as $f'$. Notice the accent. (Sometimes it is also written as $\frac{df}{dx}$.)
\begin{enumerate}
\item What is the derivative of the population function of Calculus Island at the start of 2002?
\item What is the derivative of $c(x)$ at $x=2$?
\item What is $p'(2)$ ?
\item Write an equation relating $c'(2)$ and $p'(2)$.
\item Go back to that good old position function for Car B. Let's name the position function $b$. What are $b'(0), b'(1), b'(2)$ and $b'(3)$?
\item Write a formula for $b'$ and prove your formula is true.
\end{enumerate}
\item Recall that we've defined the instantaneous rate of change of a function $f$ at input $a$ as a limit of average rates of change over intervals $[a,a+h]$ where $h$ tends to 0. Use this definition to write the derivative of $f$ at input $a$ as a formal limit.
\item (bonus) Suppose $d(x) = b(x) + k$, where $k$ is a constant that does not depend on $x$. Prove $d'(x) = b'(x)$.
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 5}
\end{center}
Each of the graphs (a) through (d) is the derivative of one of the graphs numbered 27-30. Match them up and explain why they match. Be able to discuss the significance of the derivative's graph, zeroes and sign.
\begin{center} \includegraphics[width=5in]{plot/Haas22.pdf}
\end{center}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 6}
\end{center}
Haas, Section 2.3, is full of great shortcuts for calculating derivative functions. Use it as a reference for this worksheet.
\begin{enumerate}
\item Figure out: \[ \frac{d}{dx}(4+5), \;\;\;\;\;\; (5x^7)', \;\;\;\;\;\; \frac{d}{dt}(6 + 7t + 8t^2), \;\;\;\;\;\; \frac{d}{dx}(6 + 7t + 8t^2), \;\;\;\;\;\; (-\frac{1}{x^2})' \;\; \]
\item Figure out the derivative of $f(x)=x(3x-17)$ in two ways: one time using the product rule and one time by multiplying it out and finding other useful rules.
\item Consider the graph of a function $f$. Recall that the slope of the \emph{tangent line} at a point $x=a$ is equal to the instantaneous rate of change at $x=a$.
The equation of the tangent line to the graph of a function $f$ at $x=4$ is $y=3x-17$.
\begin{enumerate}
\item What is $f(4)$? What is $f'(4)$?
\item Given that $f(x)=ax^3+b$, find the constants $a$ and $b$.
\end{enumerate}
\item Find the derivative of each of the following functions by using the product and quotient rule and then doing it with only the constant multiple, sum, difference and power rules. Do you get the same answer?
\[ x \sqrt{x} \;\;\;\;\;\; \frac{1}{\sqrt[3]{x}} \;\;\;\;\;\; \frac{x^2 + 4x + 3}{\sqrt{x}} \]
\\[.25in]
\end{enumerate}
\begin{center}
\textbf{Math 226 Worksheet 6}
\end{center}
Haas, Section 2.3, is full of great shortcuts for calculating derivative functions. Use it as a reference for this worksheet.
\begin{enumerate}
\item Figure out: \[ \frac{d}{dx}(4+5), \;\;\;\;\;\; (5x^7)', \;\;\;\;\;\; \frac{d}{dt}(6 + 7t + 8t^2), \;\;\;\;\;\; \frac{d}{dx}(6 + 7t + 8t^2), \;\;\;\;\;\; (-\frac{1}{x^2})' \;\; \]
\item Figure out the derivative of $f(x)=x(3x-17)$ in two ways: one time using the product rule and one time by multiplying it out and finding other useful rules.
\item Consider the graph of a function $f$. Recall that the slope of the \emph{tangent line} at a point $x=a$ is equal to the instantaneous rate of change at $x=a$.
The equation of the tangent line to the graph of a function $f$ at $x=4$ is $y=3x-17$.
\begin{enumerate}
\item What is $f(4)$? What is $f'(4)$?
\item Given that $f(x)=ax^3+b$, find the constants $a$ and $b$.
\end{enumerate}
\item Find the derivative of each of the following functions by using the product and quotient rule and then doing it with only the constant multiple, sum, difference and power rules. Do you get the same answer?
\[ x \sqrt{x} \;\;\;\;\;\; \frac{1}{\sqrt[3]{x}} \;\;\;\;\;\; \frac{x^2 + 4x + 3}{\sqrt{x}} \]
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 7}
\end{center}
\begin{enumerate}
\item \label{avgrateofchange} Let's think about figuring out the instantaneous rate of change of the sine and cosine functions with respect to angle changes. In the left figure below, we drew a unit circle and the point $\alpha$ radians counter-clockwise. Remind yourself that $(\cos \alpha, \sin \alpha)$ are the coordinates of that point. Starting there, we increased the angle by some amount $\Delta \alpha$ radians. This results in a horizontal change by $\Delta x$ and a vertical change by $\Delta y$ as labeled in the first diagram.
\begin{enumerate}
\item \label{horizontalanswer} Write an expression for the average rate of change in the horizontal direction between angle $\alpha$ and angle $\alpha + \Delta \alpha$.
\item Relate your answer to (\ref{horizontalanswer}) to the average rate of change of cosine between angle $\alpha$ and angle $\alpha + \Delta \alpha$
\end{enumerate}
\begin{center}
\includegraphics[width=2.2in]{plot/trig1.pdf}
\includegraphics[width=2.2in]{plot/trig2.pdf}
\includegraphics[width=2.2in]{plot/trig3.pdf}
\end{center}
\item We could now figure out the derivative with the usual move of taking a limit of the average rate of change in (\ref{avgrateofchange}) as $\Delta \alpha$ shrinks to 0. (In fact, the book does this... it's a messy limit.) However, we want to explore a different approach. Instead, suppose that instead of moving along the curve of the circle, the point moved along a tangent line for the \underline{same} distance. The middle figure illustrates this rough approximation to the original movement. Amazingly, we now have enough information to figure out formulas for $\Delta x$ and $\Delta y$, as follows.
\begin{enumerate}
\item In the third figure, we extended the vertical line down to the $x$-axis and we marked three angles as right angles. Convince yourself that the three angles indeed have to be right angles. You may cite correct theorems from geometry without proof.
\item I assured you that, by design, the length of the hypotenuse of the upper triangle equals the length of the arc that the original point travels when the angle changes by $\Delta \alpha$. What is this length?
\item Figure out the angle marked `?'.
\item Figure out $\Delta x$ and $\Delta y$ in terms of the mystery angle `?' and the hypotenuse length.
\item Figure out the average rate of change of the point's horizontal position and vertical position between angle $\alpha$ and angle $\alpha + \Delta \alpha$.
\item Figure out the instantaneous rate of change of the horizonal position and vertical position at angle $\alpha$.
\item This was a rough approximation to the original movement of the point. Convince yourself that the approximation gets better and better as $\Delta \alpha$ shrinks to 0.
\item Conclude something about the derivative of sine and cosine, assuming this rough approximation becomes perfectly precise as $\Delta \alpha$ shrinks to 0.
\end{enumerate}
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 8}
\end{center}
\begin{enumerate}
\item The following graphs are of the functions $\sin x, -\sin x$ and $\cos x$. Which is which?
\begin{center}
\includegraphics[width=5in]{plot/sin.pdf}\\[.5in]
\includegraphics[width=5in]{plot/cos.pdf}\\[.5in]
\includegraphics[width=5in]{plot/negsin.pdf}
\end{center}
\item Consider the graphs of \( \sin x \) and \( \cos x \). Convince yourself that one is the derivative of the other. Pay attention to zeroes and sign.
\item Consider the graphs of \( -\sin x \) and \( \cos x \). Convince yourself that one is the derivative of the other.
\item Using the quotient rule and the derivatives of \( \sin x \) and \( \cos x \), find the derivative of \( \tan x \). Find the derivative of \( \sec x \) in the same way.
\item Differentiate \( (x^3)^5 \) using the chain rule and check it using the power rule. Do the answers match?
\item Differentiate \( f(x)/g(x) = f(x)(g(x))^{-1}\) using the product rule and the chain rule. Do you get the Quotient Rule?
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 9}
\end{center}
\begin{enumerate}
\item Differentiate the following with respect to $t$.
\[
e^{5t}, \;\;\;\; \sin(t^2), \;\;\;\; (\sin(t))^2 + (\cos (t))^2,\;\;\;\; \sin(e^{5t})
\]
\item There is a colony of rabbits on Calculus Island. Their population for time $t$ (years after 2000) is described by $p(t)$. The population changes in a very predictable way: if there are $P$ rabbits at time $t$, then the instantaneous rate of change of the population is $P$ rabbits added per year.
\begin{enumerate}
\item Suppose there were 25,000 rabbits at the start of 2003, what is $p'(3)$? Get the units right.
\item \label{pequalspprime} Write an equation relating $p$ and its derivative at time $t$.
\item Try to guess a population function that fits the relationship in (\ref{pequalspprime}). How many other functions can you think of that fit?
\item On Precalculus Island, the rabbits there breed much faster. There if there are $P$ rabbits at time $t$, the instantaneous population growth rate is $10P$. Write an equation relating the population and its derivative with respect to time. Find a population function that fits this new situation.
\end{enumerate}
\item Weeble Knieval is a stuntman.
\begin{enumerate}
\item Let his height above the ground at time $t$ seconds be $h(t)$. Give meaningful interpretations to $h'(t)$ and $h''(t)$.
\item A simple model of gravity is that it accelerates an object towards the ground at (about) $9.8$ meters per second per second, or $9.8$ $m/s^2$. Convince yourself the units make sense.
\item Suppose Weeble is only experiencing acceleration due to gravity. Write an formula for $h''$. What is $h'(1) - h'(0)$? Get the units and sign right.
\item Using your knowledge of $h''$, guess a formula for $h'$. There should be an unknown constant (that doesn't change with time) in your formula. Consider time $t=0$ to figure out your constant in terms of $h$.
\item Using your knowledge of $h'$, guess a formula for $h$.
\item Weeble is shot out of a cannon at ground level at an initial velocity of $4.9$ $m/s$, straight up in the air. Where is he after 1 second? Where is he at time $t$?
\end{enumerate}
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 10}
\end{center}
\begin{enumerate}
\item Recall that the \underline{logarithm base $b$ of x}, written $\log_b x$, is defined as the number $y$ for which $b^y = x$.
\begin{enumerate}
\item Figure out $\log_{10} 10$, $\log_{10} 1000$, $\log_{10} 1000000$, $\log_{10} 1$, $\log_{10} \frac{1}{100}$.
\item Without using a calculator decide the two closest integers to $\log_{10} 45$.
\item Using a calculator, but NOT using the $\log_{10}x$ (or $\ln x$) button, use trial and error to figure out $\log_{10} 45$ to three decimal places.
\item Without using a calculator decide the two closest integers to $\log_{e} 45$.
\item Using a calculator, but only using the $10^x$ button and NOT the $\log_{10}x$ (or $\ln x$) button, use trial and error to figure out $\log_{e} 45$ to three decimal places.
\end{enumerate}
\item $\log_e x$ is so important that we write it with the special symbol $\ln x$ and call it the natural logarithm.
\begin{enumerate}
\item \label{etotheln} Simplify $e^{(\ln x)}$.
\item \label{invderivtwoways} Suppose the derivative of the natural logarithm exists at $x$, i.e. $\frac{d}{dx}(\ln x)$ exists. Take the derivative of $e^{(\ln x)}$ with respect to $x$ in two ways.
\begin{enumerate}
\item Take the derivative of your answer to (\ref{etotheln}).
\item Use the chain rule on $e^{(\ln x)}$. You should leave $\frac{d}{dx}(\ln x)$ in that form when it appears in your answer.
\end{enumerate}
\item Convince yourself your answers to (\ref{invderivtwoways}) are equal. Then solve for $\frac{d}{dx}(\ln x)$.
\end{enumerate}
\item Suppose a function $f$ has an inverse $f^{-1}$ whose derivative exists at $x$.
\begin{enumerate}
\item Simplify $f(f^{-1}(x))$.
\item Follow the general strategy in (\ref{invderivtwoways}) etc. to find a formula for the derivative of $f^{-1}$, $\frac{d}{dx}(f^{-1}(x))$
\end{enumerate}
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 11}
\end{center}
\begin{enumerate}
\item Look up in your textbook the definitions of \emph{local minimum/maximum/extremum}, \emph{global minimum/maximum/extremum}, and \emph{critical point}. For each of the following letters, draw a graph of a function \( f \) defined on all of \( [0,5] \) that
\begin{enumerate}
\item is continuous and has a local maximum that is not a global maximum.
\item is continuous and has a local maximum that is also a global maximum.
\item is continuous and has a local maximum where \( f' \) is zero.
\item is continuous and has a local minimum where \( f' \) is zero.
\item is continuous and has a point where \( f' \) is zero which is not an extremum.
\item is continuous and has a local maximum where \( f' \) does not exist.
\item is continuous and has a local minimum where \( f' \) does not exist.
\item is continuous and has a critical point that is not a local extremum.
\item is continuous and has no local extrema.
\item has a global maximum that is also a global minimum.
\item has a local maximum at $x=3$ but no absolute maximum.
\item has a local maximum and min, but no absolute extrema.
\item has three local minima, but no local maxima.
\end{enumerate}
\item The height of a ball is expressed by $h(t) = 6t - t^2$ feet between time $t=0$ and $t=5$. Check the critical points and endpoints to see what the maximum and minimum height of the ball is in that time interval.
\item The population of owls on Calculus Island is modeled by the function $p(t)= t^3-6t^2+9t+1$, where the input is in years and the output is in thousands of owls. Find the greatest population of owls between time $t=0$ and $t=3.5$, and when it occurs. What is the smallest population in that time interval?
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 12}
\end{center}
\begin{enumerate}
\item
The graph of the $\underline{\text{derivative}}$ of a function
$ f( x)$ is given below.
\centerline{\includegraphics[width=4truein]{plot/wsf17-2.pdf}}
\begin{enumerate}
\item What are the critical points of $ f( x)$?
\item Make a rough sketch of $y= f(x)$. Be sure to get right the intervals where the function is increasing and decreasing, and where it levels off.
\item Which critical point(s) correspond to relative extrema?
\end{enumerate}
\item Draw a graph of a function $f$ that is concave down on $[0,5]$.
\begin{enumerate}
\item \label{derivconcavedown} What can you conclude about the derivative of $f$?
\item What can you conclude (from \ref{derivconcavedown}) about the derivative of $f'$ ?
\item Make an analogous argument about functions whose graphs are concave up on an interval.
\end{enumerate}
\item Weeble Knieval is a stuntman.
\begin{enumerate}
\item Let his height above the ground at time $t$ seconds be $h(t)$. Give meaningful interpretations to $h'(t)$ and $h''(t)$.
\item A simple model of gravity is that it accelerates an object towards the ground at (about) $9.8$ meters per second per second, or $9.8$ $m/s^2$. Convince yourself (and me) the units make sense.
\item Suppose Weeble is only experiencing acceleration due to gravity. Write an formula for $h''$. What is $h'(1) - h'(0)$? Get the units and sign right.
\item \label{derivguess} Using your knowledge of $h''$, guess a formula for $h'$.
\item Your guessed formula in (\ref{derivguess}) is not the only possible guess, because if we add any constant to it, it will not change the derivative. Now let me give you more information. Suppose Weeble's vertical velocity at time $t=0$ is $4.9$ $m/s$. Find the unique formula for $h'$ that fits the known information.
\item Using your knowledge of $h'$, guess a formula for $h$.
\item Suppose Weeble began on the ground. How high is Weeble after 1 second? Where is he at time $t$?
\item Write and prove a general formula for the height of a ``projectile'' that experiences vertical acceleration only due to gravity. Your formula should be in terms of the initial height (at time $t=0$) and initial velocity.
\end{enumerate}
\item Suppose that $g$ and $h$ are increasing functions on an
interval $\it I$. For the following functions, either show that they
must be increasing on $\it I$ or give a counter-example.
$$a)\; g+h \qquad\qquad b)\;g\cdot h \qquad\qquad c)\;g\circ h$$
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 13b}
\end{center}
\begin{enumerate}
\item State a handy test for determining whether critical points are local extrema, merely by checking the sign of the first derivative near each critical point. Why does it work? (Hint: WS12.1)
\item \label{secondderivsamples} Look at WS12.1 and estimate $f''(a)$ for the critical points of $f$. What is the difference between $f''(a)$ for the local extrema and the other critical points?
\item Suppose there is an $a$ where $f'(a)=0$. Tell me a condition on $f''(a)$ that will ensure that there is a local extremum of $f$ at $a$. (Hint: \ref{secondderivsamples}.)
\item Suppose $g''(a)$ is positive. What can you say about the concavity of $g$ at $a$? Give a good explanation why this has to be so. What if $g''(x)$ is negative?
\item Continue 12.3 and 12.4. \\[.5in]
\end{enumerate}
\begin{center}
\textbf{Math 226 Worksheet 13b}
\end{center}
\begin{enumerate}
\item State a handy test for determining whether critical points are local extrema, merely by checking the sign of the first derivative near each critical point. Why does it work? (Hint: WS12.1)
\item \label{secondderivsamples1} Look at WS12.1 and estimate $f''(a)$ for the critical points of $f$. What is the difference between $f''(a)$ for the local extrema and the other critical points?
\item Suppose there is an $a$ where $f'(a)=0$. Tell me a condition on $f''(a)$ that will ensure that there is a local extremum of $f$ at $a$. (Hint: \ref{secondderivsamples1}.)
\item Suppose $g''(a)$ is positive. What can you say about the concavity of $g$ at $a$? Give a good explanation why this has to be so. What if $g''(x)$ is negative?
\item Continue 12.3 and 12.4. \\[.3in]
\end{enumerate}
\begin{center}
\textbf{Math 226 Worksheet 13b}
\end{center}
\begin{enumerate}
\item State a handy test for determining whether critical points are local extrema, merely by checking the sign of the first derivative near each critical point. Why does it work? (Hint: WS12.1)
\item \label{secondderivsamples2} Look at WS12.1 and estimate $f''(a)$ for the critical points of $f$. What is the difference between $f''(a)$ for the local extrema and the other critical points?
\item Suppose there is an $a$ where $f'(a)=0$. Tell me a condition on $f''(a)$ that will ensure that there is a local extremum of $f$ at $a$. (Hint: \ref{secondderivsamples2}.)
\item Suppose $g''(a)$ is positive. What can you say about the concavity of $g$ at $a$? Give a good explanation why this has to be so. What if $g''(x)$ is negative?
\item Continue 12.3 and 12.4. \\[.5in]
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 14}
\end{center}
\begin{enumerate}
\item A rectangle has an 8 meter perimeter.
\begin{enumerate}
\item Suppose one of its sides is $x$ meters long. Write a formula for $A(x)$, the area of the rectangle as a function of $x$.
\item What are the largest and smallest values of $x$ that make sense as inputs?
\item \label{optimalrect} Use critical points to find the maximum that $A(x)$ can be on its domain.
\item What is special about the dimensions about the optimal rectangle you found in (\ref{optimalrect})? Conclude something in general about maximizing the area of rectangles with fixed perimeter.
\end{enumerate}
\item An open-top box is to be made by cutting small congruent squares from the corners of a 12 inch by 12 inch sheet of tin and bending up the sides.
\begin{enumerate}
\item Suppose the square cut from each corner has a side of $x$ inches. Write a formula for $V(x)$, the volume of the resulting box after cutting away a $x$ inch square from each corner and folding up.
\item What is the most and least that $x$ can be and still have the construction make sense?
\item Use critical points to find the maximum and minimum volume that can be constructed as described.
\item How large should the squares cut from the corners be to make the box hold as \textbf{much} as possible?\\[.25in]
\end{enumerate}
\end{enumerate}
\begin{center}
\textbf{Math 226 Worksheet 14}
\end{center}
\begin{enumerate}
\item A rectangle has an 8 meter perimeter.
\begin{enumerate}
\item Suppose one of its sides is $x$ meters long. Write a formula for $A(x)$, the area of the rectangle as a function of $x$.
\item What are the largest and smallest values of $x$ that make sense as inputs?
\item \label{optimalrect2} Use critical points to find the maximum that $A(x)$ can be on its domain.
\item What is special about the dimensions about the optimal rectangle you found in (\ref{optimalrect2})? Conclude something in general about maximizing the area of rectangles with fixed perimeter.
\end{enumerate}
\item An open-top box is to be made by cutting small congruent squares from the corners of a 12 inch by 12 inch sheet of tin and bending up the sides.
\begin{enumerate}
\item Suppose the square cut from each corner has a side of $x$ inches. Write a formula for $V(x)$, the volume of the resulting box after cutting away a $x$ inch square from each corner and folding up.
\item What is the most and least that $x$ can be and still have the construction make sense?
\item Use critical points to find the maximum and minimum volume that can be constructed as described.
\item How large should the squares cut from the corners be to make the box hold as \textbf{much} as possible?
\end{enumerate}
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 15}
\end{center}
\begin{enumerate}
\item Suppose $f'(x)= 3x^2$. What could $f(x)$ be? Check your answer. Are there other possible answers?
\item Suppose $g(x)=x^5$ and $f'(x)=g(x)$. What could $f(x)$ be?
\item When $f'(x) = g(x)$ for all $x$ in an interval $I$, we say $f$ is the \emph{antiderivative} of $g$ on $I$. Find antiderivatives for the following functions.
\[ x^5 - 3x^2 + 1,\;\;\;\;\;\; 2-\frac{5}{x^2},\;\;\;\;\;\;\frac{1}{2\sqrt[3]{x}},\;\;\;\;\;\;\frac{2}{x}
\]
\item Find antiderivatives for the following functions.
\[ \sin x,\;\;\;\;\;\; \cos 5x,\;\;\;\;\;\; 6e^x,\;\;\;\;\;\;6e^{3x},
\]
\item (Hard) Find antiderivatives for the following functions.
\[ 2x \sin x^2,\;\;\;\;\;\; \frac{\cos (\ln x)}{x},\;\;\;\;\;\; 6x^2e^{x^3}
\]
\item Suppose at time $t$ hours, an owl on Calculus Island has a position of $p(t)$ miles down a straight road. Its velocity $v(t)$ is the derivative of $p(t)$. Suppose $v(t) = 1 + \sin t$.
\begin{enumerate}
\item Write all possible formulas for $p(t)$.
\item Suppose you know the owl was 5 miles down the road at time $t=0$. Now how many possible formulas for $p(t)$ can you find?
\item Suppose instead you know that the owl was 7 miles down the road at $t=\frac{9\pi}{2}$. Now what possible formulas for $p(t)$ are there?
\end{enumerate}
\item (Bonus) Suppose that $g$ and $h$ are increasing functions on an
interval $\it I$. For the following functions, either show that they
must be increasing on $\it I$ or give a counter-example.
$$a)\; g+h \qquad\qquad b)\;g\cdot h \qquad\qquad c)\;g\circ h$$
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 16a}
\end{center}
Consider a function $f$ whose graph is above the x-axis between $x=a$ and $x=b$. It turns out that the area of the region between the graph of $y=f(x)$, the x-axis and between $x=a$ and $x=b$ is very important to study.
This area is so important that we often just call it ``the area under the graph of $f$ between $a$ and $b$''. In fact, there is a strange standard notation for this area which we will practice using today: $\displaystyle \int^{b}_{a} f(x) dx$.
\begin{enumerate}
\item Draw the regions associated with the following areas. Then figure out what the areas actually are. \label{rawregions}
\begin{enumerate}
\item \label{car1}$\displaystyle \int_{0}^{2} f(x) \; dx$ where $f(x)=60$.
\item $\displaystyle \int_{0}^{2} f(x) \; dx$ where $f(x)=60x$.
\item \label{car2}$\displaystyle \int_{2}^{4} f(x) \; dx$ where $f(x)=60x$.
\item $\displaystyle \int_{0}^{5} f(x) \; dx$ where $f(x) =
\begin{cases}
3x & \text{if $x < 1$,}\\
3 & \text{if $1 \leq x \leq 2$,}\\
5-x &\text{if $x > 2$.}
\end{cases}$.
\end{enumerate}
\item Suppose for each case in (\ref{rawregions}), the graphed function $f(x)$ represents the \textbf{velocity} of a car along a straight track in miles per hour at time $x$ hours.
\begin{enumerate}
\item For each case, describe in plain English what is happening to the car over the graphed time intervals.
\item For (\ref{car1}) through (\ref{car2}), make an intuitive argument calculating how far the car has moved over the time interval (i.e. its displacement). \textbf{Don't} refer to the area under the curve.
\item \label{ftc-intuit} Conjecture a relationship between the area under the graph of $f$ and the displacement of the car.
\item \label{parabolacar} Suppose $f(x)=30x^2$ is the velocity function. Find the displacement of the car from time 0 to 2. (Hint: antiderivative.) What should $\displaystyle \int_{0}^{2} f(x) \; dx$ be according to (\ref{ftc-intuit})?
\item (bonus) Suppose $F$ is an antiderivative of $f$. What should $\displaystyle \int_{a}^{b} f(x) \; dx$ be according to (\ref{ftc-intuit})?
\end{enumerate}
\item There is a particular method people use to estimate areas under curves, called \textit{Riemann Sums}. Let's construct a Riemann Sum with 4 subintervals for the function \( f(x) = 30x^2 \) on \( [0,2] \) :
\begin{enumerate}
\item Divide the interval \( [0,2] \) into 4 equal subintervals.
\item Approximate the area above each subinterval with a rectangle whose height is equal to the value of \( f \) at the midpoint. What estimate do you get? Be sure you draw a picture of these rectangles with the graph of $f$.
\item This estimate is called the Midpoint Rule. How close is that answer to your guess from (\ref{parabolacar})?
\item Divide up the interval into 8 subintervals, and get another estimate. How do you think this estimate compares with the 4 subinterval estimate? (And how close is it to your guess?)
\end{enumerate}
\end{enumerate}
\newpage
\begin{center}
\textbf{Math 226 Worksheet 17a}
\end{center}
\begin{enumerate}
\item Write the following sums in sigma notation.
\begin{enumerate}
\item \( 1 + 2 + \dots + N \)
\item \( 3 + 3 + 3 + 3 + 3 \)
\item \( 1 + 4 + 9 + 16 + 25 + ... \)
\end{enumerate}
\item Write out the following sigma-notated sums as full explicit sums.
\begin{enumerate}
\item \( \displaystyle \sum_{i=1}^{4} \frac{1}{i+3} \)
\item \( \displaystyle \sum_{i=2}^{5} \sin x_{i} \cdot \Delta x \)
\item \( \displaystyle \sum_{i=-2}^{2} \frac{x^i}{i^2}\)
\item (bonus) $ \displaystyle \lim_{N \to \infty} \sum_{i=0}^N \frac{1}{i!}$. Use a calculator to guess the limit.
\end{enumerate}
\item Write the Riemann Sums from Worksheet 16.3 in sigma notation. You should define the width of each subinterval as $\Delta x$ to make your sum tidier.
\item For each equation in the list below, draw a corresponding picture involving areas under curves. Explain why the equation is true for all functions $f$ and $g$, or please provide a counterexample. Here \( a, b, c \mbox{ and } k \) are constants with respect to \( x \), and \( c \neq 0 \). You may assume $a** 0$ if $a,b > 0$
\item $\int_{a}^{b} f(x)\; dx + \int^{c}_{b} f(x)\; dx = \int^{c}_{a} f(x)\; dx$
\item $\int_{a}^{b} k \cdot f(x)\; dx = k \cdot \int^{b}_{a} f(x)\; dx $
\item \( \int_{a}^{b} c + f(x) \; dx = c + \int_{a}^{b} f(x) \; dx \)
\item If $M < f(x) **