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

calculus:resources:calculus_flipped_resources:limits:1.4-2.1_rates_of_change_tex

2014-08-31T19:52:32-04:00

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

\begin{document} \begin{frame} The cost (in dollars) of producing $x$ units of a certain commodity is $C(x) = 5000 + 6x + 0.05x^2$. \vskip 40pt Find the average rate of change of $C$ with respect to $x$ when the production level is changed from $x = 100$ to the given value: (Round your answers to the nearest cent.) \begin{enumerate} \item $x = 103$ \item $x = 101$ \end{enumerate} \end{frame} \begin{frame} \Large Each limit below represents the derivative of some function $f$ at some number $a$, find them. \vskip 15pt \LARGE \begin{enumerate}[a)] \item $\dlim_{h\to 0}\dfrac{(16+h)^{1/4}-2}{h}$ \vskip 15pt \item $\dlim_{x\to\pi/4}\dfrac{\tan(x)-1}{x-\pi/4}$ \vskip 15pt \item $\dlim_{t\to 1}\dfrac{t^5+t-2}{t-1}$ \vskip 15pt \end{enumerate} \end{frame} \begin{frame} The number of gallons of water in a tank $t$ minutes after the tank has started to drain is $Q(t)=200(30-t)^2$. \begin{enumerate} \item \begin{enumerate} \item What is the average rate at which the water flows out during the first ten minutes? \item during the five minutes from $t=5$ to $t=10$? \item during the two minutes from $t=8$ to $t=10$? \item during the minute from $t=9$ to $t=10$? \end{enumerate} \item Estimate how fast the water is running out of the tank at the end of ten minutes. \item Draw a graph of the function $Q$ for $0\leq t\leq20$. Draw the secant lines for the four time intervals used in part a). What are their slopes? \end{enumerate} \end{frame} \begin{frame} \Large The cost (in dollars) of producing $x$ units of a certain commodity is $C(x) = x^2 -2x + 10$. \vskip 10pt \begin{enumerate}[a)] \item Find the average rate of change of $C$ with respect to $x$ when the production level is changed from $x = 5$ to $x=7$ and for the change from $x=5$ to $x=6$. \vskip 15pt \item Find the instantaneous rate of change of $C$ with respect to $x$ when $x = 5$. \end{enumerate} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:limits:1.5_limit_tex

2014-08-31T19:51:05-04:00

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

\begin{document} \begin{frame} \large The statement ”Whether or not $\displaystyle\lim_{x\rightarrow a} f(x)$ exists, depends on how $f(a)$ is defined,” is true \begin{itemize} \item[(a)] sometimes, \item[(b)] always, \item[(c)] never. \end{itemize} \end{frame} \begin{frame} \Large Find the following limits. \vskip 15pt \begin{columns} \begin{column}{0.5\textwidth} \begin{enumerate} \item[\bf a)] $\displaystyle\lim_{x\to 7^-}\displaystyle\frac{x+6}{x-7}$ \vskip 30pt \item[\bf b)] $\displaystyle\lim_{x\to 4}\displaystyle\frac{3-x}{(x-4)^2}$ \end{enumerate} \end{column} \begin{column}{0.5\textwidth} \begin{enumerate} \item[\bf c)] $\displaystyle\lim_{x\to 1^+}\displaystyle\frac{8}{x^3-1}$ \vskip 30pt \item[\bf d)] $\displaystyle\lim_{x\to 1^-}\displaystyle\frac{8}{x^3-1}$ \vskip 30pt \end{enumerate} \end{column} \end{columns} \end{frame} \begin{frame} \LARGE If a function $f$ is not defined at $x=a$, \begin{enumerate}[a)] \item $\displaystyle{\lim_{x\rightarrow a} f(x)}$ cannot exist \item $\displaystyle{\lim_{x\rightarrow a} f(x)}$ could be $0$ \item $\displaystyle{\lim_{x\rightarrow a} f(x)}$ must approach $\infty$ \item none of the above. \end{enumerate} \end{frame} \begin{frame} \Large Draw the graph of a function $f(x)$ such that $\displaystyle\lim_{x\to 4} f(x)=5$ and $f(4)=5$, or explain why this is impossible. \vskip 30pt Draw the graph of a function $g(x)$ such that $\displaystyle\lim_{x\to 4} g(x)=5$ and $g(4)=4$, or explain why this is impossible. \vskip 30pt Draw the graph of a function $h(x)$ such that $\displaystyle\lim_{x\to 4} h(x)=5$ and $h(4)$ is undefined, or explain why this is impossible. \end{frame} \begin{frame} \Large Draw the graph of a function $f(x)$ such that $\displaystyle\lim_{x\to 6^-} f(x)=5$ and $\displaystyle\lim_{x\to 6^+} f(x)=7$, or explain why this is impossible. \vskip 30pt Draw the graph of a function $g(x)$ such that $\displaystyle\lim_{x\to 6^-} g(x)=5$ and $\displaystyle\lim_{x\to 6^+} g(x)=7$ and $g(6)=10$, or explain why this is impossible. \vskip 30pt Draw the graph of a function $h(x)$ such that $\displaystyle\lim_{x\to 6^-} g(x)=5$ and $\displaystyle\lim_{x\to 6^+} g(x)=5$ and $\dlim_{x\to 6} g(x)$ is undefined, or explain why this is impossible. \end{frame} \begin{frame} If all that you know about a function $g(x)$ is that $g(5)=-3$ and $g'(5)=4$, what is your best estimate of $g(7)$? \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:limits:1.6_limit_laws_tex

2014-08-31T19:49:27-04:00

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

\begin{document} \begin{frame} If $f(x) = \dfrac{x^2 - 4}{x - 2}$ and $g(x) = x + 2$, then we can say the functions $f$ and $g$ are equal. \end{frame} \begin{frame} \Large $$\dlim_{x\to 2} f(x) = 4 \hskip 10pt \dlim_{x\to 2} g(x) = -2 \hskip 10pt \dlim_{x\to 2} h(x) = 0$$ Find the limits, if they exist: \begin{columns} \begin{column}{0.5\textwidth} \begin{enumerate} \item[\bf (a)] $\displaystyle\lim_{x\to 2 }[f(x) + 5g(x)]$ \vskip 30pt \item[\bf (b)] $\displaystyle\lim_{x\to 2 }[g(x)]^3$ \vskip 30pt \item[\bf (c)] $\displaystyle\lim_{x\to 2 }\dfrac{1}{f(x)}$ \end{enumerate} \end{column} \begin{column}{0.5\textwidth} \begin{enumerate} \item[\bf (d)] $\displaystyle\lim_{x\to 2 } 4f(x)g(x)$ \vskip 30pt \item[\bf (e)] $\displaystyle\lim_{x\to 2 } \dfrac{g(x)}{h(x)}$ \vskip 30pt \item[\bf (f)] $\displaystyle\lim_{x\to 2 } \dfrac{g(x)h(x)}{f(x)}$ \end{enumerate} \end{column} \end{columns} \end{frame} \begin{frame} \LARGE \begin{columns} \begin{column}{0.5\textwidth} $$\displaystyle\lim_{x \to 1} \frac{x^2 -4x + 3}{x -1}$$ \vskip 20pt $$\displaystyle\lim_{x \to -1} \frac{x^2 -4x }{x^2-3x-4}$$ \vskip 20pt $$\displaystyle\lim_{h \to 0} \dfrac{(-4+h)^2-16 }{h}$$ \end{column} \begin{column}{0.5\textwidth} $$\displaystyle\lim_{ h \to 0} \dfrac{\sqrt{9 + h}- 3}{h}$$ \vskip 20pt $$\displaystyle\lim_{x \to -4 } \dfrac{1/4+1/x }{4+x}$$ \vskip 20pt $$\displaystyle\lim_{x \to 0} \dfrac{9 }{t}- \frac{9 }{t^2+t}$$ \end{column} \end{columns} \end{frame} \begin{frame} \begin{block}{}\begin{center}\LARGE \textbf{True} or \textbf{False}. \end{center}\end{block} \vskip 30pt \Large Consider a function $f(x)$ with the property that $\displaystyle{\lim_{x\rightarrow a} f(x) =0}$. Now consider another function $g(x)$ also defined near $a$. Then $$\displaystyle{\lim_{x\rightarrow a} [f(x)g(x)] = 0}$$ \end{frame} \begin{frame} \begin{block}{}\begin{center}\LARGE \textbf{True} or \textbf{False}. \end{center}\end{block} \vskip 30pt \Large If $\displaystyle{\lim_{x\rightarrow a} f(x) =\infty}$ and $\displaystyle{\lim_{x\rightarrow a} g(x) =\infty}$, then $$\displaystyle{\lim_{x\rightarrow a} [f(x)-g(x)] =0}$$ \end{frame} \begin{frame} \Large Find the following limits. $$\displaystyle\lim_{x\to 3} 8x+|x-3|$$ \vskip 15pt $$\displaystyle\lim_{x\to -3} \frac{4x+12}{|x+3|}$$ \vskip 15pt If $$2x -2 \leq f(x) \leq x^2 -2x + 2$$ for $x \geq 0$, find $\displaystyle\lim_{ x\to 2} f(x)$. \end{frame} \begin{frame} Consider the function \[f(x)=\left\{\begin{array}{ll} x^2 & \mbox{$x$ is rational, $x\neq 0$} \\ -x^2 & \mbox{$x$ is irrational} \\ \mbox{undefined} & x=0 \end{array}\right.\] Then \begin{enumerate} \item there is no $a$ for which $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists \item there may be some $a$ for which $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists, but it is impossible to say without more information \item $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists only when $a=0$ \item $\displaystyle{\lim_{x\rightarrow a}f(x)}$ exists for infinitely many $a$ \end{enumerate} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:limits:1.8_continuity_tex

2014-08-31T12:24:30-04:00

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

\begin{document} \begin{frame} \Large Where are the following functions continuous? \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[] $$f(x)=\frac{\sqrt{x}}{1+\sin(x)}$$ \item[] $$g(x)=(\sec(x))^2+x$$ \item[] $$a(x)=\frac{x}{|x|}$$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[] $$b(x)=\frac{1}{|x-2|}$$ \item[] $$c(x)=\frac{1}{|x-2|+1}$$ \item[] $$e(x)=\frac{1}{1+\sqrt{x}}$$ \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} Let $P(t) =$ the cost of parking in New York City's parking garages for $t$ hours. So, $$P(t) = \mbox{\$20 per hour or fraction thereof}$$ For example, if you are in the garage for two hours and one minute, you pay $\$60$. Graph the function $P$ and discuss the continuity. \end{frame} \begin{frame} \begin{block}{} \begin{center}{\bf \huge True or False}\end{center} \end{block} If $t_0$ closely approximates some time, $T$, then $P(t_0)$ closely approximates $P(T)$. Be prepared to justify your answer. \end{frame} \begin{frame} You decide to estimate $\pi^2$ by squaring longer decimal approximations of $\pi = 3.14159\ldots$. Choose which of the following can be justified with what you've learned so far: \begin{itemize} \item[i)] This is a good idea because $\pi$ is a rational number. \item[ii)] This is a good idea because $f(x) = x^2$ is a continuous function. \item[iii)] This is a bad idea because $\pi$ is irrational. \item[iv)] This is a good idea because $f(x) = \pi^x$ is a continuous function. \end{itemize} \end{frame} \begin{frame} Define the function $$f(x)=\left\{\begin{array}{ll} 1+x^2&\mbox{ if $x\leq 0$}\\ 4-x&\mbox{ if $0<x\leq 4$}\\ (x-4)^2&\mbox{ if $x>4$} \end{array}\right.$$ Where is $f$ continuous? At the points where it's not continuous, state whether it's continuous from the left, from the right, or neither. AFTER you've done this, sketch the graph of f . \end{frame} \begin{frame} Find all values ${\bf a}$ such that the function $$g(x)=\left\{\begin{array}{ll} x^2&\mbox{ if $x\leq 1$}\\ x+a&\mbox{ if $x> 1$} \end{array}\right.$$ is continuous. \end{frame} \begin{frame} Use the Intermediate Value Theorem to show that the equation $$x^4 + x - 4 = 0$$ has a root in the interval $(1, 2)$. \end{frame} \begin{frame} Argue using the Intermediate Value Theorem that my hair was 6 inches long at some point in the past. If I boast that my beard was once over a foot long, would I be able to use the Intermediate Value Theorem and my present beard length as proof of my claim? \end{frame} \end{document}