Department of Mathematics and Statistics, Binghamton University calculus:resources:calculus_flipped_resources:applications

calculus:resources:calculus_flipped_resources:applications:3.4_horizontal_asymptotes_tex

2015-08-28T22:19:04-04:00

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\begin{document} \begin{frame} Find the following limits, if they exist. \vskip 5pt \begin{itemize} \item[\bf a)] $\dlim_{x\to\infty}\dfrac{7x^2 - x + 1}{3x^2 + 5x - 5} $ and $ \dlim_{x\to-\infty}\dfrac{7x^2 - x + 1}{3x^2 + 5x - 5}$. \vskip 30pt \item[\bf b)] $\dlim_{x\to\infty}\dfrac{8x - 9}{2x + 4}$ and $\dlim_{x\to-\infty}\dfrac{8x - 9}{2x + 4}$. \vskip 30pt \item[\bf c)] $\dlim_{x\to\infty}\dfrac{x - 8}{x^2 + 7}$ and $\dlim_{x\to-\infty}\dfrac{x - 8}{x^2 + 7}$. \end{itemize} \end{frame} \begin{frame} Find the following limits, if they exist. \vskip 5pt \begin{itemize} \item[\bf d)] $\dlim_{x\to\infty}\dfrac{\sqrt{4x^6-x}}{x^3+3}$ and $\dlim_{x\to-\infty}\dfrac{\sqrt{4x^6-x}}{x^3+3}$. \vskip 20pt \item[\bf e)] $\dlim_{x\to\infty}(\sqrt{25x^2+x}-5x)$ and $\dlim_{x\to-\infty}(\sqrt{25x^2+x}-5x)$. \vskip 20pt \pause \item[\bf f)] $\dlim_{x\to-\infty}(x+\sqrt{x^2+2x})$ \vskip 10pt \item[\bf g)] $\dlim_{x\to\infty} 6\cos(x)$ \vskip 10pt \item[\bf h)] $\dlim_{x\to\infty}\frac{x^4 - 3x^2 + x}{x^3 - x + 3}$ \end{itemize} \end{frame} \begin{frame} Find the horizontal and vertical asymptotes of each curve. \begin{enumerate}[a)] \item $$y=\frac{8x + 3}{x - 4}$$ \item $$y=\frac{x^2 + 1}{9x^2 - 80x - 9}$$ \item $$y=\frac{x^2 - x}{x^2 - 8x + 7}$$ \end{enumerate} \end{frame} \begin{frame} Let $P$ and $Q$ be polynomials with positive coefficients. \begin{enumerate}[a)] \item If the degree of $P$ is less than the degree of $Q$, what is $$\lim_{x\to\infty}\frac{P(x)}{Q(x)}?$$ \item If the degree of $P$ is greater than the degree of $Q$, what is $$\lim_{x\to\infty}\frac{P(x)}{Q(x)}?$$ \item If the degree of $P$ equals the degree of $Q$, what is $$\lim_{x\to\infty}\frac{P(x)}{Q(x)}?$$ \end{enumerate} \end{frame} \begin{frame} A tank contains 120 L of pure water. Brine that contains 25 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. \vskip 15pt \begin{enumerate}[a)] \item Find the concentration of salt after $t$ minutes (in grams per liter). \vskip 15pt \item As $t$ approaches infinity, what does the concentration approach? \end{enumerate} \end{frame} \begin{frame} Find $$\lim_{x\to\infty}(\sqrt{x^2+cx}-\sqrt{x^2+dx}).$$ (Here $c$ and $d$ represent arbitrary real numbers.) \vskip 25pt Find $$\lim_{x\to -\infty}(x^2+x^3).$$ \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:3.7_optimization_tex

2015-08-28T22:19:11-04:00

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.

\begin{document} \begin{frame} \LARGE Find two numbers whose difference is $140$ and whose product is a minimum. \vspace{\stretch{1}} Find the dimensions of a rectangle with perimeter $60$ meters whose area is as large as possible. \end{frame} \begin{frame} Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, $3$ feet wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. \begin{enumerate}[a)] \item Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. \vskip 15pt \item Draw a diagram illustrating the general situation. Let $x$ denote the length of the side of the square being cut out. Let $y$ denote the length of the base. \vskip 15pt \item Write an expression for the volume $V$ in terms of $x$ and $y$. \vskip 15pt \item Use the given information to write an equation that relates the variables $x$ and $y$. \vskip 15pt \item Use part (d) to write the volume as a function of $x$. \vskip 15pt \item Finish solving the problem by finding the largest volume that such a box can have. \end{enumerate} \end{frame} \begin{frame} \LARGE A rectangular storage container with an open top is to have a volume of $10$ cubic meters. The length of this base is twice the width. Material for the base costs $\$5$ per square meter. Material for the sides costs $\$3$ per square meter. Find the cost of materials for the cheapest such container.\\ (Round your answer to the nearest cent.) \end{frame} \begin{frame} \Large A manufacturer has been selling $1000$ flat-screen TVs a week at $\$350$ each. A market survey indicates that for each $\$10$ rebate offered to the buyer, the number of TVs sold will increase by $100$ per week. \begin{enumerate}[a)] \item Find the demand function (price $p$ as a function of units sold $x$). \item How large a rebate should the company offer the buyer in order to maximize its revenue? \item If its weekly cost function is $$C(x) = 60,000 + 120x$$ how should the manufacturer set the size of the rebate in order to maximize its profit? \end{enumerate} \end{frame} \begin{frame} \LARGE A boat leaves a dock at $1$ PM and travels due south at a speed of $20$ km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at $2$ PM. How many minutes after $1$ PM were the two boats closest together? \end{frame} \begin{frame} \LARGE At which points on the curve $$y = 1 + 40x^3 -3x^5$$ does the tangent line have the largest slope? \end{frame} \begin{frame} \LARGE A piece of wire 30m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. \vskip 5pt \begin{enumerate}[a)] \item How much wire should be used for the square in order to maximize the total area? \vskip 15pt \item How much wire should be used for the square in order to minimize the total area? \end{enumerate} \end{frame} \begin{frame} \LARGE A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter $x$ of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 32 ft, find the value of $x$ so that the greatest possible amount of light is admitted. \vspace{\stretch{1}} \end{frame} \begin{frame} A designer wants to introduce a new line of bookcases. He wants to make at least 100 bookcases, but not more than $2000$ of them. He predicts the cost of producing $x$ bookcases is $C(x)$. Assume that $C(x)$ is a differentiable function. Which of the following must he do to find the minimum average cost, $$c(x)=\dfrac{C(x)}{x}?$$ \begin{enumerate} \item[\bf I] find the points where $c'(x)=0$ and evaluate $c(x)$ there \item[\bf II] compute $c''(x)$ to check which of the critical points in (I) are local maxima. \item[\bf III] check the values of $c$ at the endpoints of its domain. \end{enumerate} \begin{enumerate} \item I only \item I and II only \item I and III only \item I, II and III \end{enumerate} \end{frame} \begin{frame} The rate (in appropriate units) at which photosynthesis takes place for a species of phytoplankton is modeled by the function $$P = \dfrac{120I}{I^2 + I + 4}$$ where $I$ is the light intensity (measured in thousands of foot-candles). For what light intensity is $P$ a maximum? \end{frame} \begin{frame} What is the maximum vertical distance between the line $$y = x + 6$$ and the parabola $$y = x^2$$ for $-2 \leq x \leq 3$? \end{frame} \begin{frame} Find the area of the largest rectangle that can be inscribed in the ellipse $$ \dfrac{x^2}{a^2} +\dfrac{ y^2}{b^2}=1$$ \vspace{\stretch{1}} \end{frame} \begin{frame} An oil refinery is located $1$ km north of the north bank of a straight river that is $1$ km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $\$400,000$ km over land to a point P on the north bank and $\$800,000$ km under the river to the tanks. To minimize the cost of the pipeline, how far downriver from the refinery should the point P be located? \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:3.9_antiderivatives_tex

2015-08-28T22:35:21-04:00

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.
There is one png image needed to compile slides:

antiderivative.png

\begin{document} \begin{frame} \begin{center} Which function from $\{a,b,c\}$ is an antiderivative of $f$? \end{center} \begin{center} \includegraphics[height=190pt]{antiderivative.png} \end{center} \end{frame} \begin{frame} \begin{block}{} \begin{center} {\LARGE {\bf True} or \bf{False}} \end{center} \end{block} \vskip 15pt An antiderivative of a sum of functions, $f+g$, is an antiderivative of $f$ plus an antiderivative of $g$. \vskip 20pt An antiderivative of a product of functions, $fg$, is an antiderivative of $f$ times an antiderivative of $g$. \end{frame} \begin{frame} Suppose you are told that the acceleration function of an object is a continuous function $a(t)$. Let's say you are given that $v(0)=1$. \vskip 20pt \begin{block}{} \begin{center} {\LARGE {\bf True} or \bf{False}} \end{center} \end{block} \vskip 15pt You can find the position of the object at any time $t$. \end{frame} \begin{frame} Find the most general antiderivative of each function. \vskip 5pt \begin{itemize} \item[\bf (i)] $f(x)=\dfrac{1}{2}x^2-2x+6$ \vskip 15pt \item[\bf (ii)] $g(x)=(x+5)(2x-6)$ \vskip 15pt \item[\bf (iii)] $h(x)=\dfrac{3+t+t^2}{\sqrt{t}}$ \end{itemize} \end{frame} \begin{frame} Let $f$ be a function so that $f''(x)=12x+\sin(x)$. \vskip 5pt \begin{itemize} \item[\bf (i)] If you know nothing else about $f$, give the best formula you can for $f$. \vskip 15pt \item[\bf (ii)] If you know $f'(\pi)=1$, give the best formula you can for $f$. \vskip 15pt \item[\bf (iii)] If you know $f'(\pi)=1$, and $f(\pi)=0$, give the best formula you can for $f$. \end{itemize} \end{frame} \begin{frame} Find $f$ if $f''(\theta)=\sin(\theta)+\cos(\theta)$, $f(0)=3$, and $f'(0)=3$. \vskip 100pt Find $f$ if $f'''(x)=\cos(x)$, $f(0)=5$, $f'(0)=1$, and $f''(0)=8$. \end{frame} \begin{frame} $$f(x)=\dfrac{1}{x^2}$$ If $F(x)$ is an antiderivative of $f$ with the property $F(1)=1$. \vskip 15pt \begin{block}{} \begin{center} {\LARGE {\bf True} or \bf{False}} \end{center} \end{block} \vskip 10pt $$F(-1)=3$$ \end{frame} \begin{frame} Find a function $f$ such that $f'(x)=2x^3$ and the line $2x+y$ is tangent to the graph of $f$. \vskip 100pt In each of the following, a particle is moving with the given data. Find the position function of the particle. \begin{enumerate}[a)] \item $v(t)=1.5\sqrt{t}$, $s(16)=67$. \item $a(t)=2t+5$, $s(0)=2$, $v(0)=-5$. \end{enumerate} \end{frame} \begin{frame} A stone was dropped off a cliff and hit the ground with a speed of 112 ft/s. What is the height of the cliff? (Use 32 ft/$\mbox{s}^2$ for the acceleration due to gravity.) \vskip 100pt What constant acceleration is required to increase the speed of a car from 25 mi/h to 53 mi/h in 3 s? \end{frame} \begin{frame} If a diver of mass $m$ stands at the end of a diving board with length $L$ and linear density $\rho$, then the board takes on the shape of a curve $y = f(x)$, where $$EIy'' = mg(L - x) + \frac{1}{2}\rho g(L - x)^2.$$ $E$ and $I$ are positive constants that depend on the material of the board and $g$ ($< 0$) is the acceleration due to gravity. \begin{enumerate}[a)] \item Find an expression for the shape of the curve. \item Use $f(L)$ to estimate the distance below the horizontal at the end of the board. \end{enumerate} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:4.3_fundamental_theorem_tex

2015-08-28T22:35:28-04:00

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.
There is one png image needed to compile slides:

ftc_graph.png

\begin{document} \begin{frame} Calculate the following integrals using Part II of the Fundamental Theorem of Calculus. \begin{alignat*}{2} &a)\;\displaystyle\int_{-1}^2(x^3-4x)\,dx &\qquad\qquad\qquad\qquad &b)\;\displaystyle\int_4^9 \sqrt{x}\,dx \\ &c)\;\displaystyle\int_{\frac{\pi}{6}}^{\pi} \sin(\theta)\,d\theta &\qquad\qquad\qquad\qquad &d)\;\displaystyle\int_0^1 (x+3)(x-6)\,dx \\ &e)\;\displaystyle\int_1^16 \dfrac{x-3}{\sqrt{x}}\,dx &\qquad\qquad\qquad\qquad &f)\;\displaystyle\int_{-2}^1 x^{-4}\,dx \end{alignat*} \end{frame} \begin{frame} You are traveling with velocity $v(t)$ that varies continuously over the interval $[a, b]$ and your position at time $t$ is given by $s(t)$. Which of the following represent your average velocity for that time interval: \vskip 5pt \begin{itemize} \item[\bf (I)] $\dfrac{1}{b-a}\displaystyle\int_a^b v(t)\,dt$ \vskip 5pt \item[\bf (II)] $\dfrac{s(b)-s(a)}{b-a}$ \vskip 5pt \item[\bf (III)] $v(c)$ for at least one $c$ between $a$ and $b$. \end{itemize} \vskip 15pt \begin{enumerate}[a)] \item I, II, and III \item I only \item I and II only \end{enumerate} \end{frame} \begin{frame} Below is the graph of a function $f$. \begin{center} \includegraphics[height=1.6in]{FTC_Graph.png} \end{center} Let $g(x) = \displaystyle\int_0^x f(t)\,dt$. Then for $0 < x < 2$, $g(x)$ is \begin{itemize} \item[\bf (a)] increasing and concave up. \item[\bf (b)] increasing and concave down. \item[\bf (c)] decreasing and concave up. \item[\bf (d)] decreasing and concave down. \end{itemize} \end{frame} \begin{frame} \begin{block}{} \begin{center} {\LARGE \bf True or False} \end{center} \end{block} If $f$ is continuous on the interval $[a,b]$, then $$ \dfrac{d}{dx}\left(\displaystyle\int_a^b f(x)\,dx\right) = f(x) $$ \end{frame} \begin{frame} \begin{block}{} \begin{center} {\LARGE \bf True or False} \end{center} \end{block} Let $f$ be continuous on the interval $[a,b]$. There exist two constants $m$ and $M$, such that $$ m(b-a) \leq \int_a^b f(x)\,dx \leq M(b-a) $$ \end{frame} \begin{frame} \begin{block}{} \begin{center} {\LARGE \bf True or False} \end{center} \end{block} If $f'(x) = g'(x)$, then $f(x)=g(x)$. \end{frame} \begin{frame} Calculate the following derivatives using Part I of the Fundamental Theorem of Calculus. \begin{alignat*}{2} &a)\;\frac{d}{dx}{\displaystyle\int_0^x {\frac{\text{dt} }{ 1+t^2}}} & \qquad\qquad\qquad\qquad &b)\;\frac{d}{dx}{\displaystyle\int_0^{x^2}{\frac{\text{dt} }{ 1+t^2}}} \\ &c)\;\frac{d}{dx}{\displaystyle\int_{{-x^2}}^{x^2}{\frac{\text{dt} }{ 1+t^2}}} & \qquad\qquad\qquad\qquad &d)\;\frac{d^2 }{ dx^2}{\displaystyle\int_0^x {\frac{\text{dt} }{ 1+t^2}}} \\ &e)\;\frac{d}{dx}{\displaystyle\int_{1}^{{{\tan x}}}{{t^{10}\cos t}}\;dt} & \qquad\qquad\qquad\qquad &f)\;\frac{d}{dx}{\displaystyle\int_{{{x^3}}}^{{{x^5 +1}}}{\frac{1}{ t}}\;dt} \end{alignat*} \end{frame} \begin{frame} If $f$ is continuous and $f(x) < 0$ for all $x$ in the interval $[a,b]$, then $\displaystyle\int_a^b f(x)\,dx$ \begin{itemize} \item[\bf (a)] must be negative. \item[\bf (b)] might be zero. \item[\bf (c)] not enough information. \end{itemize} \end{frame} \begin{frame} If $f$ is a differentiable function, then $\displaystyle\int_0^x f'(t)\,dt = f(x)$ \begin{itemize} \item[\bf (a)] Always. \item[\bf (b)] Sometimes. \item[\bf (c)] Never. \end{itemize} \end{frame} \begin{frame} A sprinter practices by running various distances back and forth along a straight line. Her velocity at $t$ seconds is given by the function $v(t)$. What does $\displaystyle\int_0^{60} |v(t)|\,dt$ represent? \begin{itemize} \item[\bf (a)] The total distance the sprinter ran in one minute. \item[\bf (b)] The sprinter's average velocity in one minute. \item[\bf (c)] The sprinter's distance from the starting point after one minute. \item[\bf (d)] None of the above. \end{itemize} \end{frame} \begin{frame} Water is pouring out of a pipe at the rate of $f(t)$ gallons per minute. You collect the water that flows from the pipe between $t=2$ and $t=4$ minutes. The amount of water you collect can be represented by \begin{itemize} \item[\bf (a)] $\displaystyle\int_2^4 f(x)\,dx$ \item[\bf (b)] $f(4)-f(2)$ \item[\bf (c)] $(4-2)f(4)$ \item[\bf (d)] the average of $f(4)$ and $f(2)$ times the amount of time the elapsed. \end{itemize} \end{frame} \begin{frame} If $f$ is continuous on the interval $[a,b]$ then \begin{itemize} \item[\bf (i)] $\displaystyle\int_a^b f(x)\,dx$ is the area bounded by the graph of $f$, the $x$-axis, and the lines $x=a$ and $x=b$. \item[\bf (ii)] $\displaystyle\int_a^b f(x)\,dx$ is a number. \item[\bf (iii)] $\displaystyle\int_a^b f(x)\,dx$ is an antiderivative of $f(x)$. \item[\bf (iv)] $\displaystyle\int_a^b f(x)\,dx$ may not exist. \end{itemize} \end{frame} \begin{frame} If $ \displaystyle\int_{a}^{b} f(x) \; dx = b^{3} - a^{3} $ for all numbers $ a $ and $ b $ , what is $ \displaystyle\int_{a}^{b} f'(x) \; dx $ ? \vskip 30pt If $ \frac{d}{dx} \left( \displaystyle\int_{a}^{x} f(t) \; dt \right) = x^{3} - 1 $ , what is $ \displaystyle\int_{a}^{b } f'(x) \; dx $ ? \vskip 30pt If $ \displaystyle\int_{a}^{b} f(u(x)) u'(x) \; dx = (2/3)(b^{2}+1)^{3/2 }- (2/3)(a^{2}+1)^{3/2} $ for all numbers $ a $ and $ b $ , what might $ f(x) $ and $ u(x) $ be? Are they unique? \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:4.4_indefinite_integrals_tex

2014-09-07T07:53:44-04:00

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.

\begin{document} \begin{frame} \begin{block}{} \begin{center} {\LARGE {\bf True} or {\bf False}} \end{center} \end{block} \vskip 30pt If $f$ is continuous on the interval $[a,b]$, then $$\displaystyle\int_a^b f(x)\,dx$$ is a number. \end{frame} \begin{frame} Find each of the following derivatives, or specify that you don't have enough information to do so. \begin{enumerate}[a)] \item $$\frac{d}{dx}\int_3^8 f(x)dx$$ \item $$\frac{d}{dx}\int_3^x f(t)\,dt$$ \item $$\frac{d}{dx}\int_x^3 f(t)\,dt$$ \item $$\frac{d}{dx}\int f(x)\,dx$$ \end{enumerate} \end{frame} \begin{frame} If $w'(t)$ is the rate of growth of a child in pounds per year, what does $\displaystyle\int_5^{11}w'(t)\,dt$ represent? \vskip 10pt \begin{enumerate}[a)] \item The child's initial weight at birth. \item The decrease in the child's weight (in pounds) between the ages of 5 and 11. \item The child's weight at age 5. \item The increase in the child's weight (in pounds) between the ages of 5 and 11. \item The child's weight at age 11. \end{enumerate} \end{frame} \begin{frame} The current in a wire is defined as the derivative of the charge $$I(t) = Q'(t)$$ What does $\displaystyle\int_a^b I(t)\,dt$ represent? \vskip 10pt \begin{enumerate}[a)] \item $I$t represents the change in the current $I$ from time $t=a$ to $t=b$. \item It represents the charge $Q$ at time $t=b$. \item It represents the current $I$ at time $t=b$. \item It represents the charge $Q$ at time $t=a$. \item It represents the change in the charge $Q$ from time $t=a$ to $t=b$. \end{enumerate} \end{frame} \begin{frame} Find the general indefinite integral. $$\int (8\sqrt{x^3}+9\sqrt[3]{x^2})dx$$ \vskip 75pt Find the particular indefinite integral of $\displaystyle\int (8\sqrt{x^3}+9\sqrt[3]{x^2})dx$ whose value at $x=0$ is $4$. \end{frame} \begin{frame} Find the general indefinite integrals, and evaluate the definite integrals. \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (i)] $\displaystyle\int 7v(v^2 + 8)^2\,dv$ \vskip 20pt \item[\bf (ii)] $\displaystyle\int_0^2 (6x-3)(4x^2+9)\,dx$ \vskip 20pt \item[\bf (iii)] $\displaystyle\int_0^2 (6x-3)(4x^2+9)\,dx$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (iv)] $\displaystyle\int_9^{16}\frac{3x-3}{\sqrt{x}}\,dx$ \vskip 20pt \item[\bf (v)] $\displaystyle\int_1^4 \sqrt{t}(5+7t)\,dt$ \vskip 20pt \item[\bf (vi)] $\displaystyle\int_{-1}^2 (x-6|x|)\,dx$ \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} Find the indefinite integrals and evaluate the definite integrals. \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (i)] $\displaystyle\int 7(1+\tan^2(\alpha))\,d\alpha$ \vskip 20pt \item[\bf (ii)] $\displaystyle\int 5\frac{\sin(2x)}{\sin(x)}\,dx$ \vskip 20pt \item[\bf (iii)] $\displaystyle\int_0^\pi (4\sin(\theta)-17\cos(\theta))\,d\theta$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (iv)] $\displaystyle\int_0^{\frac{\pi}{4}}\frac{2+3\cos^2(\theta)} {cos^2(\theta)}\,d\theta$ \vskip 20pt \item[\bf (v)] $\displaystyle\int_0^{\frac{2\pi}{3}}\frac{7\sin(\theta)(1+\tan^2(\theta))} {\sec^2(\theta)}\,d\theta$ \vskip 20pt \item[\bf (vi)] $\displaystyle\int_0^{\frac{3\pi}{2}} 5|\sin(x)|\,dx$ \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} The velocity function (in meters per second) for a particle moving along a line is $$v(t) = 3t- 8$$ \begin{enumerate}[a)] \item Find the displacement. \item Find the distance traveled from time $t=0$ to time $t=4$. \end{enumerate} \vskip 60pt A particle is moving along a line so that its acceleration at time $t$ is $a(t) = 2t + 2$ and its initial velocity is $v(0)=-3$. \begin{enumerate}[a)] \item Find the velocity at time $t$. \item Find the distance traveled from time $t=0$ to time $t=4$. \end{enumerate} \end{frame} \begin{frame} Water flows from the bottom of a storage tank at a rate of $r(t) = 400 - 8t$ liters per minute. Find the amount of water that flows from the tank during the first 30 minutes. \vskip 80pt Sketch the region bounded by the $y$-axis, the line $y=4$, and the curve $y=4\sqrt[4]{x}$. Find the area of this region in two ways: \begin{enumerate}[a)] \item by integrating an appropriate function of x, and \item by writing $x$ as a function of $y$ and integrating with respect to $y$. \end{enumerate} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:4.5_substitution_tex

2014-09-07T07:55:53-04:00

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.

\begin{document} \begin{frame} The differentiation rule that helps us understand why the Substitution rule works is: \vskip 20pt \begin{enumerate}[a)] \item The product rule. \vskip 10pt \item The chain rule. \vskip 10pt \item The quotient rule. \vskip 10pt \item All of the above. \end{enumerate} \end{frame} \begin{frame} Find the indefinite integrals. \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (i)] $\displaystyle\int x^2\sqrt{x^3+21}\,dx$ \vskip 20pt \item[\bf (ii)] $\displaystyle\int \cos^4(\theta)\sin(\theta)\,d\theta$ \vskip 20pt \item[\bf (iii)] $\displaystyle\int (9t+7)^{2.5}\,dt$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (iv)] $\displaystyle\int (x+5)\sqrt{10x+x^2}\,dx$ \vskip 20pt \item[\bf (v)] $\displaystyle\int \frac{z^3}{\sqrt[3]{3+z^4}}\,dz$ \vskip 20pt \item[\bf (vi)] $\displaystyle\int x(8x+7)^8\,dx$ \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} Find the indefinite integrals and evaluate the definite integrals. \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (i)] $\displaystyle\int x^3\sqrt{x^2+4}\,dx$ \vskip 20pt \item[\bf (ii)] $\displaystyle\int x^5\sin(x^6)\,dx$ \vskip 20pt \item[\bf (iii)] $\displaystyle\int \sec^2(\theta)\tan^7(\theta)\,d\theta$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (iv)] $\displaystyle\int \sqrt{x^5}\sin(2+x^{7/2})\,dx$ \vskip 20pt \item[\bf (v)] $\displaystyle\int \frac{\cos(\pi/x^{29})}{x^{30}}\,dx$ \vskip 20pt \item[\bf (vi)] $\displaystyle\int \sin(45t)\sec^2(\cos(45t))\,dt$ \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} If $f$ is continuous and $\displaystyle\int_0^4 f(x)\,dx=2$, find $\displaystyle\int_0^2 f(2x)\,dx$. \end{frame} \begin{frame} Evaluate the definite integrals. \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (i)] $\displaystyle\int_0^1 \sqrt[3]{1+7x}\,dx$ \vskip 20pt \item[\bf (ii)] $\displaystyle\int_0^{\sqrt[14]{\pi}} x^{13}\cos(x^{14})\,dx$ \vskip 20pt \item[\bf (iii)] $\displaystyle\int_0^{\pi/10} \cos(5x)\sin(\sin(5x))\,dx$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (iv)] $\displaystyle\int_0^{31}\frac{dx}{\sqrt[3]{(1+4x)^2}}$ \vskip 20pt \item[\bf (v)] $\displaystyle\int_9^{10} x\sqrt{x-9}\,dx$ \end{itemize} \end{column} \end{columns} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:5.1_area_tex

2015-08-28T22:35:25-04:00

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There is one png image needed to compile slides:

51graph.png

\begin{document} \begin{frame} For each of the two regions described below, sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area. \vskip 10pt $$y = 2x + 3\qquad y = 13 - x^2\qquad x = -1\qquad x = 2$$ \vskip 35pt $$x = 45 - 5y^2\qquad x = 5y^2 - 45$$ \end{frame} \begin{frame} Sketch the region enclosed by the given curves. Then find the area. \begin{enumerate}[a)] \item $$x = 6y^2\qquad x = 4 + 5y^2$$ \item $$y = 6 \cos(\pi x)\qquad y = 12x^2 - 3$$ \pause \item $$y = 4 \cos(6x)\qquad y = 4 \sin(12x)\qquad x = 0\qquad x = \pi/12$$ \item $$y = \sqrt{x} \qquad y = \frac{1}{2}x\qquad x = 25$$ \pause \item $$y = |3x|\qquad y = x^2 - 4$$ \end{enumerate} \end{frame} \begin{frame} Two cars, A and B, start side by side and accelerate from rest. The graphs of their velocity functions are given below. \begin{figure}[h]\centering{ \includegraphics[height=1.7in]{51graph.png}} \end{figure} \begin{enumerate}[a)] \item Which car is ahead at time $a$? Explain. \item Interpret the area of the shaded region in physical terms. \item Which car is ahead after $1.5a$ minutes? Explain. \end{enumerate} \end{frame} \begin{frame} Find the number $b$ such that the line $y = b$ divides the region bounded by the curves $y = 4x^2$ and $y = 16$ into two regions with equal area. \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:5.3_cylindrical_shells_tex

2014-09-07T08:01:31-04:00

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\begin{document} \begin{frame} The region $R$ is bounded by the curves $$ y=x^3 \hskip 20pt y=8 \hskip 20pt x=0 $$ Sketch $R$. For the following rotational axes, {\bf set-up} two integrals for the volume of the solid generated by revolving $R$ about the indicated axis, one representing the washer method and one the cylindrical shells method. \begin{itemize} \item[\bf (a)] $x$-axis. \item[\bf (b)] $y$-axis. \item[\bf (c)] $y=5$. \item[\bf (d)] $x=-2$. \end{itemize} \end{frame} \begin{frame} The region $R$ is bounded by the curves $$ y=1+\sin(x) \hskip 20pt y=1 \hskip 20pt x=0 \hskip 20pt x=2 $$ Sketch $R$. For the following rotational axes, {\bf set-up} two integrals for the volume of the solid generated by revolving $R$ about the indicated axis, one representing the washer method and one the cylindrical shells method. \begin{itemize} \item[\bf (a)] $x$-axis. \item[\bf (b)] $y$-axis. \item[\bf (c)] $y=-1$. \end{itemize} \end{frame} \begin{frame} The triangular region with vertices $(0,2)$, $(1,0)$, and $(0,1)$ is rotated about the line $x=4$. Find the volume of the solid generated by this rotation. \end{frame} \begin{frame} Let $B$ be the region bounded by the graphs of $x=y^2$ and $x=9$. Sketch $B$. For each part below, find the volume of the solid that has $B$ as its base if every cross section by a plane perpendicular to the $x$-axis is \begin{itemize} \item[\bf (a)] a square. \item[\bf (b)] a semicircle with diameter lying on $B$. \item[\bf (c)] an equilateral triangle. \end{itemize} \end{frame} \begin{frame} Find the volume of a wedge cut out of a cylinder of radius $r$ if the angle between the top and bottom of the wedge is $\frac{\pi}{6}$. \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:5.5_average_tex

2014-09-07T08:00:22-04:00

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\begin{document} \begin{frame} Find the average value of each function on the given interval. \vskip 10pt \begin{enumerate}[a)] \item $ f(x) = 10x - x^2$ on the interval $ [0, 2] $ \vskip 15pt \item $f(\theta) = 11 \sec^2(\theta/4)$ on the interval $ [0,\pi] $ \vskip 15pt \item $ h(x) = 7 \cos^4(x)\sin(x) $ on the interval $ [0,\pi] $ \end{enumerate} \end{frame} \begin{frame} Consider the function $$f(x) = 3\sqrt{x}$$ \begin{enumerate}[a)] \item Find the average value $f_{\mbox{ave}}$ of $f$ on the interval $[0, 16]$. \item Find all values $c$ such that $f_{\mbox{avg}}= f(c)$. \item Sketch the graph of $f$ and, in the same picture, a rectangle whose area is the same as the area under the graph of $f$. \end{enumerate} \end{frame} \begin{frame} Consider the function $$f(x) = (x-5)^2$$ \begin{enumerate}[a)] \item Find the average value $f_{\mbox{ave}}$ of $f$ on the interval $[4,7]$. \item Find all values $c$ such that $f_{\mbox{avg}}= f(c)$. \item Sketch the graph of $f$ and, in the same picture, a rectangle whose area is the same as the area under the graph of $f$. \end{enumerate} \end{frame} \begin{frame} Consider the function $$f(x) = 9 sin(4x)$$ \begin{enumerate}[a)] \item Find the average value $f_{\mbox{ave}}$ of $f$ on the interval $[-\pi, \pi]$. \item Find all values $c$ such that $f_{\mbox{avg}}= f(c)$. \item Sketch the graph of $f$ and, in the same picture, a rectangle whose area is the same as the area under the graph of $f$. \end{enumerate} \end{frame} \begin{frame} Find all numbers $b$ such that the average value of $$f(x) = 7 + 10x - 9x^2$$ on the interval $[0, b]$ is equal to 8. \vskip 65pt The velocity $v$ of blood that flows in a blood vessel with radius $R$ and length $L$ at a distance $r$ from the central axis is $$v(r) =\frac{ P}{4\eta L}(R^2 - r^2)$$ where $P$ is the pressure difference between the ends of the vessel and $\eta$ is the viscosity of the blood. Find the average velocity (with respect to $r$) over the interval $0 \leq r \leq R$. \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:closed_interval_method_tex

2014-08-29T08:32:53-04:00

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\begin{document} \begin{frame} \large Let $f(x)$ be a differentiable function on a closed interval with $x=a$ being one of the endpoints of the interval. If $f'(x)>0$ for all $x$, then \vskip 15pt \begin{itemize} \item[\bf (a)] $f$ could have either an absolute maximum or minimum at $x=a$. \vskip 15pt \item[\bf (b)] $f$ cannot have an absolute maximum at $x=a$. \vskip 15pt \item[\bf (c)] $f$ must have an absolute minimum at $x=a$. \vskip 15pt \item[\bf (d)] $x=a$ must be a critical number for $f$. \end{itemize} \end{frame} \begin{frame} \large If $f$ is continuous on $[a,b]$, then \vskip 15pt \begin{itemize} \item[\bf (a)] there must be local extreme values, but there may or may not be an absolute maximum or minimum value for the function. \vskip 15pt \item[\bf (b)] there must be numbers $m$ and $M$ such that $m\leq f(x) \leq M$, for all $x$ in $[a,b]$. \vskip 15pt \item[\bf (c)] any absolute maximum or minimum would be at either the endpoints of the interval, or at places in the domain where $f'(x)=0$. \end{itemize} \end{frame} \begin{frame} \large Find the absolute extrema of: \vskip 15pt \begin{itemize} \item[\bf (a)] $f(x)=x^3-3x+1$ on the interval $[0,3]$. \vskip 15pt \item[\bf (b)] $g(x)=\dfrac{x^2-4}{x^2+4}$ on the interval $[-4,4]$. \vskip 15pt \item[\bf (c)] $h(t)=t\sqrt{4-t^2}$ on the interval $[-1,2]$. \vskip 15pt \item[\bf (d)] $i(x)=x+\cot\left(\frac{x}{2}\right)$ on the interval $\left[\frac{\pi}{4},\frac{7\pi}{4}\right]$. \end{itemize} \end{frame} \begin{frame} \large Find the highest and lowest points on the graph of $ f(x) = x^{3}-3x+6 $ on the following intervals: \vskip 15pt \begin{itemize} \item[\bf (i)] $ [-2,2] $. \vskip 15pt \item[\bf (ii)] $ [-2,3] $. \vskip 15pt \item[\bf (iii)] $ (-2,3)$. \end{itemize} \end{frame} \begin{frame} \large Show that the maximum and minimum values of the function $$f(x) = x^{3}+ax^{2}+bx+c$$ on the interval $[p,q]$ occur at the endpoints if $a^{2} <3b$. \vskip 50pt If $a$ and $b$ are positive numbers, find the maximum value of $$f(x)=x^a(1-x)^b$$ on the interval $[0,1]$. \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:critical_points_tex

2014-08-29T08:33:45-04:00

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\begin{document} \begin{frame} \large Sketch the graph of $y=(x-1)^2+2$ on the closed interval $[-4,4]$. \vskip 15pt \begin{itemize} \item[\bf (a)] What are the local maximum and minimum values? points? \vskip 15pt \item[\bf (b)] What are the absolute maximum and minimum values? points? \end{itemize} \end{frame} \begin{frame} \large Find the critical number of the following functions \vskip 15pt \begin{itemize} \item[\bf (a)] $f(x) = 8x^3-12x^2-48x$ \vskip 15pt \item[\bf (b)] $g(x) = x^{\frac{3}{4}} - 9x^{\frac{1}{4}}$ \vskip 15pt \item[\bf (c)] $h(\theta) = 18\cos(\theta) + 9\sin^2(\theta)$ \end{itemize} \end{frame} \begin{frame} \large Show that $5$ is a critical number of the function $$g(x)=2+(x-5)^2$$ but $g$ does not have a local extreme value of $5$. \vskip 60pt If $f$ has a minimum value of $c$, does the function $g(x)=-f(x)$ have a maximum value of $c$? \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:differentials_tex.html

2014-08-28T19:05:03-04:00

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\begin{document} \begin{frame} \large Peeling an orange changes its volume V. What does $\Delta V$ represent? \vskip 10pt \begin{enumerate}[a)] \item the volume of the rind. \vskip 10pt \item the surface area of the orange. \vskip 10pt \item the volume of the "edible part" of the orange. \vskip 10pt \item $-1\times $(the volume of the rind). \end{enumerate} \end{frame} \begin{frame} \large Imagine that you increase the dimensions of a square with side $x_1$ to a square with side length $x_2$. The change in the area of the square, $\Delta A$, is approximated by the differential $dA$. Find $dA$: \vskip 10pt \begin{enumerate}[a)] \item $2x_1(x_2-x_1)$ \vskip 10pt \item $2x_2(x_2-x_1)$ \vskip 10pt \item $x_1^2-x_2^2$ \vskip 10pt \item $(x_2-x_1)^2$ \end{enumerate} \end{frame} \begin{frame} \large Imagine that you increase the dimensions of a square with side $x_1$ to a square with side length $x_2$. The change in the area of the square, $\Delta A$, is approximated by the differential $$dA=2x_1(x_2-x_1)$$ This approximation will result in an \vskip 5pt \begin{enumerate}[a)] \item overestimate \vskip 10pt \item underestimate \vskip 10pt \item exactly equal \end{enumerate} \end{frame} \begin{frame} Find the differential of each function: \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf a)] $y=\sqrt{1+x^2}$ \vskip 20pt \item[\bf b)] $y=x^2\sin(x)$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf c)] $y=\sec\left(\sqrt{7x}\right)$ \vskip 20pt \item[\bf d)] $y=\dfrac{3-t^2}{3+t^2}$ \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} \large The radius of a sphere is measured to be $84$ inches with a possible error of $0.5$ inches. \begin{itemize} \item[\bf a)] Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? \vskip 20pt \item[\bf b)] Use differentials to estimate the maximum error in the calculated volume. What is the relative error? \end{itemize} \end{frame} \begin{frame} \large Use differentials to estimate the amount of paint needed to apply a coat of paint $0.1$ cm thick to hemispherical dome with diameter $50$ meters. \end{frame} \begin{frame} \large A window has the shape of a square surmounted by a semicircle. \vskip 15pt The base of the window is measured as having width $50$ inches with a possible error in measurement of $0.1$ inches. \vskip 15pt Use differentials to estimate the maximum error possible in computing the area of the window. What is the maximum relative error? \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:linearization_tex.html

2014-08-28T18:51:18-04:00

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\begin{document} \begin{frame} Find the linearization of each function: \vskip 5pt \begin{itemize} \item[\bf a)] $h(x) = x^4-3x^2-1$ at $a=-1$. \vskip 20pt \item[\bf b)] $f(x)=\sin^2(x)$ at $a=\frac{\pi}{2}$. \vskip 20pt \item[\bf c)] $g(x) = \dfrac{1}{(1+3x)^4}$ at $a=0$. \vskip 20pt \item[\bf d)] $r(t) = t^{\frac{3}{4}}$ at $a=16$. \end{itemize} \end{frame} \begin{frame} \large Use a linear approximation to estimate the value of $\sqrt[3]{9}$. \vskip 30pt Use a linear approximation to estimate the value of $\tan(44^o)$. \end{frame} \begin{frame} \large The line tangent to the graph of $f(x)=\sin(x)$ at the point $(0,0)$ is $y=x$. This implies that \vskip 10pt \begin{enumerate}[a)] \item $\sin(0.0005) \approx 0.0005$ \vskip 10pt \item The line $y=x$ touches the graph of $f(x)=\sin(x)$ at exactly one point, $(0,0)$. \vskip 10pt \item $y=x$ is the best straight line approximation to the graph of $f$ for all $x$. \end{enumerate} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:applications:mean_value_theorem_tex

2014-08-29T08:23:59-04:00

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\begin{document} \begin{frame} Verify that the function satisfies the Mean Value Theorem on the given interval. Then find all numbers $c$ which satisfy the conclusion of the Mean Value Theorem. \vskip 10pt \begin{itemize} \item[\bf a)] $f(x) = 3x^2+2x+5$ on $[-1,1]$. \vskip 10pt \item[\bf b)] $g(x) = x^3+x-1$ on $[0,2]$. \vskip 10pt \item[\bf c)] $h(x) = \dfrac{x}{x+2}$ on $[1,4]$. \vskip 10pt \item[\bf d)] $i(x) = (x-2)^{-2}$ on $[1,4]$. \end{itemize} \end{frame} \begin{frame} On a toll road a driver takes a time stamped toll-card from the starting booth and drives directly to the end of the toll section. After paying the required toll, the driver is surprised to receive a speeding ticket along with the toll receipt. Which of the following describes the situation? \vskip 5pt \begin{itemize} \item[\bf a)] The booth attendant does not have enough information to prove that the driver was speeding. \vskip 5pt \item[\bf b)] The booth attendant can prove that the driver was speeding during their trip. \vskip 5pt \item[\bf c)] The driver will get a ticker for a lower speed than their actual maximum speed. \end{itemize} \end{frame} \begin{frame} {\bf True or False} \vskip 5pt An athlete is running back and forth along a straight path. She finishes her run at the place where she began. There must be at least one moment, other than the end of the race, where she was at a complete stop. \end{frame} \begin{frame} Two runners start a race at the same moment and finish in a tie. What must be true? \vskip 10pt \begin{itemize} \item[\bf a)] At some point during the race the two runners were not tied. \vskip 5pt \item[\bf b)] The runners' speeds at the end of the race must have been exactly the same. \vskip 5pt \item[\bf c)] The runners must have had the same speed at exactly the same time at some point in the race. \vskip 5pt \item[\bf d)] The runners had to have the same speed at some moment, but not necessarily at exactly the same time. \end{itemize} \end{frame} \begin{frame} Show that for all values $a$ and $b$ $$|\sin(a)-\sin(b)| \leq |a-b|$$ \vskip 35pt Suppose that $3\leq f'(x) \leq 5$ for all values of $x$. Show that $$18 \leq f(8)-f(2) \leq 30$$ \vskip 35pt Show that the polynomial $$f(x)=1+2x+x^3+4x^5$$ has exactly one real root. \end{frame} \end{document}