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

calculus:resources:calculus_flipped_resources:derivatives:2.2_derivative_function_tex

2014-08-31T19:33:55-04:00

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

\begin{document} \begin{frame} Is the function $$f(x)=\left\{\begin{array}{ll} 2-x&\mbox{ if $x\leq 2$}\\ x^2-4x+4&\mbox{ if $x> 2$} \end{array}\right.$$ differentiable at 2? \end{frame} \begin{frame} Find all $a$ and $b$ such that the function $$g(x)=\left\{\begin{array}{ll} 2-x&\mbox{ if $x\leq 2$}\\ x^2+ax+b&\mbox{ if $x> 2$} \end{array}\right.$$ is differentiable for all $x$. \end{frame} \begin{frame} You are designing the first ascent and drop for a roller coaster. You want the slope of the ascent to be $.8$ and the slope of the drop to be $-1.6$. You will connect these two straight stretches by part of a parabola $$y=ax^2+bx+c$$ of width $100$ units. \begin{enumerate}[a)] \item Certainly you don't want a sharp corner in your tracks at the points where the linear parts meet the parabola. This puts a condition on the tangent lines of the parabola -- what's the condition? \item Find a formula for the parabola. \end{enumerate} \end{frame} \begin{frame} If $f + g$ is differentiable at $a$, are $f$ and $g$ necessarily differentiable at $a$? \end{frame} \begin{frame} If $f'(a)$ exists, $\displaystyle\lim_{x\rightarrow a} f(x)$ \begin{itemize} \item[i)] must exist, but there is not enough information to determine it exactly. \item[ii)] equals $f(a)$. \item[iii)] equals $f'(a)$. \item[iv)] may not exist. \end{itemize} \end{frame} \begin{frame} A slow freight train chugs along a straight track. The distance it has traveled after ${\bf x}$ hours is given by a function $f(x)$. An engineer is walking along the top of the box cars at the rate of $3$ miles per hour in the same direction as the train is moving. The speed of the man relative to the ground is \begin{itemize} \item[i)] $f(x) + 3$ \item[ii)] $f'(x) + 3$ \item[iii)] $f(x) - 3$ \item[iv)] $f'(x) - 3$ \end{itemize} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:derivatives:2.3_differentiation_formulas_tex

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

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.
There are two png images needed to compile slides:

product_linesegments.png

circ.png

\begin{document} \begin{frame} \large The functions $f(x)$ and $h(x)$ are graphed below: \begin{center} \includegraphics[height=4cm]{product_linesegments.png} \end{center} \begin{enumerate} \item Graph the function $g(x) = (2f - h)(x)$. \item Label the slopes along the line segments of $f$, $g$, and $h$. \item Plot $f'$, $g'$, and $h'$. Do these graphs agree with the differentiation rules? How are these derivatives related? \end{enumerate} \end{frame} \begin{frame} Assume the functions $f$ and $g$ are such that: $$ f(5) = 1 \hskip 30pt f '(5) = 9$$ $$ g(5) = -4 \hskip 30pt g'(5) = 5$$ Evaluate the following expressions: \begin{itemize} \item[\bf (a)] $(f+g)'(5)$ \item[\bf (b)] $(fg)'(5)$ \item[\bf (c)] $(f/g)'(5)$ \item[\bf (d)] $(g/f)'(5)$ \item[\bf (e)] $\dfrac{d}{dx}\left(\dfrac{g(x)}{x}\right)$ at $x=5$. \end{itemize} \end{frame} \begin{frame} \LARGE \begin{columns} \begin{column}{0.5\textwidth} Easier problems: $$f(x)=x^4-2x^2+6$$ \vskip 15pt $$g(x)=7x + 4x^{-1/8}$$ \vskip 15pt $$h(x)=\dfrac{x^4}{5-x^3}$$ \end{column} \begin{column}{0.5\textwidth} Harder problems: $$F(x)=\sqrt{x}(x-4)$$ \vskip 15pt $$G(x)=\dfrac{8x^2 + 2x + 4}{\sqrt{x}}$$ \vskip 15pt $$H(u)=\sqrt{6}u+\sqrt{5u}$$ \end{column} \end{columns} \end{frame} \begin{frame} \LARGE The Constant Multiple Rule tells us $$\dfrac{d}{dx}\left(cf(x)\right)=c \dfrac{d}{dx}\left(f(x)\right)$$ and the Product Rule says $$\dfrac{d}{dx}\left(cf(x)\right)=c\dfrac{d}{dx}\left(f(x)\right)+f(x)\dfrac{d}{dx}\left(c\right).$$ Why do these agree? \end{frame} \begin{frame} \LARGE Find the first and second derivatives. \begin{itemize} \item[\bf (a)] $f(x) = 2x^4 - 2x^3 + 4x$ \vskip 25pt \item[\bf (b)] $g(r)=\sqrt{r}+\sqrt[3]{r}$ \vskip 25pt \item[\bf (c)] $h(x)=\dfrac{x^2}{1+8x}$ \end{itemize} \end{frame} \begin{frame} \large Suppose you cut a slice of pizza from a circular pizza of radius $r$, \begin{center} \includegraphics[width=3.5cm]{circ.png} \end{center} As you change the size of the angle $\theta$, you change the area of the slice, $A=\frac{1}{2}r^2\theta$. Then $A'$ is \begin{center} (a) $r\theta$ \hf \hf \hf (b) $\frac{1}{2}r^2$ \hf \hf \hf (c) $r$ \hf \hf \hf (d) Unknowable \end{center} \end{frame} \begin{frame} Find an equation of the tangent line to the curve $$y=\frac{5x}{x+3}$$ at the point $(2,2)$. \vskip 40pt Find the points on the curve $$y=2x^3+3x^2-12x+3$$ where the tangent line is horizontal. \end{frame} \begin{frame} The equation of motion of a particle is $$s = t^3 -27t$$ where $s$ is in meters and $t$ is in seconds. (Assume $t \geq 0$.) \vskip 12pt \begin{itemize} \item[\bf (a)] Find the velocity and acceleration as functions of $t$. \vskip 12pt \item[\bf (b)] Find the acceleration when the velocity is zero. \end{itemize} \end{frame} \begin{frame} \large Use the product rule to show $$ \frac{d}{dx}(fgh)(x) = (fgh' + fhg' + ghf')(x) $$ Can you generalize this argument? ie What is the derivative of four, five, six, ... functions multiplied together? \vskip 35pt Find the derivative of $$ y=(3x+1)(2x-1)(x-2) \hskip 35pt y=(3x-2)^2(2x+3) $$ \end{frame} \begin{frame} \large Consider the function $f(x)=|x^2-25|$. \begin{enumerate} \item Sketch the graph of $f$. \item Find a formula for $f'$. \item For what values of $x$ is the function not differentiable? \end{enumerate} \end{frame} \begin{frame} \large What is the derivative of the function $f\cdot f$? What is the derivative of the function $f\cdot f\cdot f$? Can you generalize this for any positive integer $n$? What is the derivative of $(x^2+1)^6$? \end{frame} \begin{frame} \large The functions $$ y=x^2+ax+b \hskip 25pt y=cx-x^2 $$ share a tangent line at the point $(1,0)$. Find $a, b,$ and $c$. \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:derivatives:2.4_trig_derivatives_tex

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

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

arc_chord_limit.png

\begin{document} \begin{frame} \large \begin{block}{} \begin{center}$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sin(x)}{x} = 1 \hskip 15pt \displaystyle\lim_{x\rightarrow 0}\dfrac{\cos(x)-1}{x} = 0$\end{center} \end{block} Evaluate the limits: \vskip 5pt \begin{columns} \begin{column}{0.5\textwidth} $$\lim_{x\rightarrow 0}\dfrac{\sin(6x)}{\sin(9x)}$$ \vskip 15pt $$\lim_{\theta\to 0}\dfrac{\cos(7\theta)-1}{sin(9\theta)}$$ \vskip 15pt $$\lim_{t\to 0}\dfrac{\tan(16t)}{\sin(4t)}$$ \end{column} \begin{column}{0.5\textwidth} $$ \lim_{x\to 0}\dfrac{\sin(x^7)}{x}$$ \vskip 15pt $$ \lim_{x\to 2}\dfrac{\sin(x-2)}{x^2+6x-16}$$ \vskip 15pt $$ \lim_{x\rightarrow 2}\dfrac{3-3\tan(x)}{\sin(x)-\cos(x)} $$ \end{column} \end{columns} \end{frame} \begin{frame} \LARGE Use both the derivatives of $\sin(x)$ and $\cos(x)$ and the quotient rule to show: $$\frac{d}{dx}\left( \tan(x)\right)=\sec^2(x)$$ and $$\frac{d}{dx}\left( \sec(x)\right)=\sec(x)\tan(x)$$ \end{frame} \begin{frame} \LARGE \begin{columns} \begin{column}{0.5\textwidth} Find $f'(x)$: $$f(x)=5x^2+7\sin(x)$$ \vskip 20pt Find $g'(\theta)$: $$g(\theta)=\sec(\theta)\tan(\theta)$$ \end{column} \begin{column}{0.5\textwidth} Find $F'(x)$: $$F(x)=\dfrac{3-\sec(x)}{\tan(x)}$$ \vskip 20pt Evaluate: $$\frac{d^2}{d\theta^2}\left(\theta\sin(\theta)\right)$$ \end{column} \end{columns} \end{frame} \begin{frame} \LARGE Find the equation of the tangent line to the curve $y=14x\sin(x)$ at $x=\pi/2$. \vskip 55pt Find $$\frac{d^{103}}{dx^{103}}\left(\sin(x)\right) \mbox{ and } \frac{d^{201}}{dx^{201}}\left(\cos(x)\right)$$ \end{frame} \begin{frame} \emph{Step into your teacher's shoes. What is wrong (if anything) with the following calculations? Explain any errors and correct for them.} \vskip 5pt \begin{block}{} Find all values of x in the interval $[0,4\pi)$ that satisfy the equation $$\sin(2x) = \cos(x).$$ {\bf Solution:} Since $\sin(2x) = 2\sin(x)\cos(x)$, $$ \sin(2x) = \cos(x) \hskip 20pt \Rightarrow \hskip 20pt 2\sin(x)\begin{cancel}\cos(x)\end{cancel} = \begin{cancel}\cos(x)\end{cancel} $$ $$\Rightarrow \hskip 20pt 2\sin(x) = 1 \hskip 20pt \Rightarrow \hskip 20pt \sin(x) = \frac{1}{2}$$ Therefore, $x = \frac{\pi}{3}$ or $x=\frac{5\pi}{3}$. \end{block} \end{frame} \begin{frame} What does $\displaystyle\lim_{x\rightarrow 0} \frac{\sin(x)}{x} = 1$ mean? Explain why three of the options are false and one is true. \begin{itemize} \item[a)] $\frac{0}{0} = 1$. \item[b)] The tangent to the graph of $y =\sin(x)$ at $(0,0)$ is the line $y=x$. \item[c)] You can cancel the x's. \item[d)] $\sin(x) = x$ for x near 0. \end{itemize} \end{frame} \begin{frame} The figure shows a circular arc of length $s$ and a chord of length $d$, both subtended by a central angle $\theta$. Find $\displaystyle\lim_{\theta\rightarrow 0^+} \frac{s}{d}$. \begin{center} \includegraphics[height=2in]{arc_chord_limit.png} \end{center} It may be helpful to review the formulas associated to arcs and isosceles triangles. Further, why would the limit $\displaystyle\lim_{\theta\rightarrow 0} \frac{s}{d}$ not exist? \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:derivatives:2.5_chain_rule_tex

2014-09-01T09:51:28-04:00

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

\begin{document} \begin{frame} \large You own a company producing iSquids, (the latest portable electronic craze). Your big production limitation is a scarcity of Chip 187, produced by outside manufacturers. \vskip 5pt If $f(x)$ is the profit your company will make if it gets $x$ Chip 187's and $g(x)$ is a function giving the number of Chip 187's you can obtain for $x$ dollars, which of the following is of interest to you? \vskip 8pt \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (a)] $f\circ g$ \item[\bf (b)] $g\circ f$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (c)] both \item[\bf (d)] neither \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} \Large $$ f(x)=x+\frac{1}{x} \hskip 30pt g(x)=\dfrac{x+8}{x+2} \hskip 30pt h(x)=\sqrt{x} $$ Express each function as an equation. \\ What is the domain of each function? \vskip 10pt \begin{columns} \begin{column}{0.5\textwidth} $(f\circ g)(x)$ \vskip 30pt $g(f(x))$ \vskip 30pt $(g\circ g)(x)$ \end{column} \begin{column}{0.5\textwidth} $(h\circ f)(x)$ \vskip 30pt $(g\circ h)(x)$ \vskip 30pt $h(h(x))$ \end{column} \end{columns} \vskip 20pt \end{frame} \begin{frame} \large For each of the following functions, first express it as a composition of 2 functions. Then find the derivatives. \vskip 15pt \begin{columns} \begin{column}{0.5\textwidth} \begin{enumerate} \item[\bf a)] $F(x)=\sqrt[3]{1+5x}$ \vskip 30pt \item[\bf b)] $G(x)=(x^4+9x^2+3)^8$ \vskip 30pt \item[\bf c)] $F(t)=\sqrt[9]{1+\tan(t)}$ \end{enumerate} \end{column} \begin{column}{0.5\textwidth} \begin{enumerate} \item[\bf d)] $H(x)=\cos(3^7+x^7)$ \vskip 30pt \item[\bf e)] $G(x)=\left(\dfrac{x^2+8}{x^2-8}\right)^3$ \vskip 30pt \item[\bf f)] $S(z)=\sqrt{\dfrac{z-7}{z+7}}$ \end{enumerate} \end{column} \end{columns} \end{frame} \begin{frame} \large Find the derivatives. \vskip 15pt \begin{columns} \begin{column}{0.45\textwidth} \begin{enumerate} \item[\bf a)] $y=\dfrac{r}{\sqrt{r^2+3}}$ \vskip 20pt \item[\bf b)] $y=x\sin\left(\dfrac{7}{x}\right)$ \vskip 20pt \item[\bf c)] $f(t)=\sqrt{\dfrac{t}{t^2+1}}$ \vskip 20pt \item[\bf d)] $g(y)=\dfrac{(y-2)^6}{(y^2+4y)^9}$ \end{enumerate} \end{column} \begin{column}{0.55\textwidth} \begin{enumerate} \item[\bf e)] $y=\sin(\tan(8x))$ \vskip 20pt \item[\bf f)] $y=\cos(\cos(\cos(x)))$ \vskip 20pt \item[\bf g)] $y=(1+\sec(3\pi x+4\pi))^5$ \vskip 20pt \item[\bf h)] $y=\sqrt{11x+ \sqrt{11x+ \sqrt{11x}}}$ \vskip 20pt \item[\bf i)] $y = [x + (x + \sin(2 x))^6]^7$ \end{enumerate} \end{column} \end{columns} \end{frame} \begin{frame} \large If $h(x) = \sqrt{7 + 6f(x)}$, where \begin{center} $f(4) = 7$ and $f '(4) = 2$, \end{center} find $h'(4)$. \vskip 70pt Find the first and second derivatives of $y=\sin\left(x^2\right)$. \end{frame} \begin{frame} \large If $f$ and $g$ are both differentiable and $h=f\circ g$, $h^{\prime}(2)$ equals \vskip 20pt \begin{enumerate} \item $f^{\prime}(2)\circ g^{\prime}(2)$ \item $f^{\prime}(2)g^{\prime}(2)$ \item $f^{\prime}(g(2)) g^{\prime}(2)$ \item $f^{\prime}(g(x)) g^{\prime}(2)$ \end{enumerate} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:derivatives:2.6_implicit_differentiation_tex

2015-08-28T22:34:56-04:00

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.
There are four png images needed to compile slides:

\begin{document} \begin{frame} \large Draw a graph of $x=\sin y$ and find the slope of the line tangent to the graph at the point $(0,\pi)$. \vskip 45pt Find $dx/dy$ and $dy/dx$ if $y\sec(x) = 6x\tan(y)$. \end{frame} \begin{frame} \large Find $dy/dx$ by implicit differentiation. \vskip 15pt \begin{columns} \begin{column}{0.5\textwidth} \begin{enumerate} \item[\bf a)] $x^4+y^3=1$ \vskip 30pt \item[\bf b)] $7x^2 + 5xy - y^2 = 6$ \vskip 30pt \item[\bf c)] $x^7(x + y) = y^2(4x ? y)$ \end{enumerate} \end{column} \begin{column}{0.5\textwidth} \begin{enumerate} \item[\bf d)] $4 \cos(x) \sin(y) = 2$ \vskip 30pt \item[\bf e)] $5y\sin(x^2) = 9x\sin(y^2)$ \vskip 30pt \item[\bf f)] $\sqrt{7x+y}=6+x^2y^2$ \end{enumerate} \end{column} \end{columns} \end{frame} \begin{frame} \large Explain (without calculating) why the two following equations will yield the same formula for $dy/dx$. Does this mean that the two graphs will have exactly the same tangent lines? $$ x^3y+y^2+y=1$$ $$ x^3y+y^2+y=-1$$ \begin{columns} \begin{column}{0.5\textwidth} \includegraphics[height=5cm]{contour1.png} \end{column} \begin{column}{0.5\textwidth} \includegraphics[height=5cm]{contour2.png} \end{column} \end{columns} \end{frame} \begin{frame} \large Find an equation of the tangent line to the ellipse $$9x^2 + xy + 9y^2 = 19$$ at the point $(1, 1)$. \vskip 20pt Find an equation of the tangent line to the {\bf astroid} $x^{2/3}+y^{2/3} = 4$ at $(-3\sqrt{3},1)$. \begin{center} \includegraphics[height=4cm]{astroid.png} \end{center} \end{frame} \begin{frame} \large Find the points on the {\bf lemniscate} $8(x^2+y^2)^2=25(x^2-y^2)$ where the tangent is horizontal. \begin{figure}[htp] \centering{ \includegraphics[height=4cm]{lemniscate.png}} \end{figure} \end{frame} \begin{frame} \large If $f(x) + x^2[f(x)]^3 = 10$ and $f(1) = 2$, find $f '(1)$. \vskip 50pt Find $dx/dy$ and $dy/dx$ and $dz/dx$ if $$y\sec(z) = 6x\tan(y).$$ \end{frame} \begin{frame} \large Find $y''$ by implicit differentiation. $$4x^2+y^2=9$$ \end{frame} \begin{frame} \large When we introduced the Power Rule, we explained it for $y=x^n$ when $n$ is a nonnegative integer, and we promised that later we'd explain it when $n$ is a rational and/or negative number. The moment has come. In the following, you should use the Power Rule {\bf only for $n$ a nonnegative integer} to prove it the Power Rule for all rational numbers. \begin{enumerate}[a)] \item Warm-up: write $y=x^\frac{2}{3}$ as $y^3=x^2$. Then use Implicit Differentiation to show $y'=\frac{2}{3}x^{-\frac{1}{3}}$. \item Let $y=x^\frac{p}{q}$, where $p$ and $q$ are positive integers. Use the same method as the previous problem to show $y'=\frac{p}{q}x^{\frac{p}{q}-1}$. \item Warm-up: write $y=x^{-1}$ as $xy=1$. Then use Implicit Differentiation to show $y'=-x^{-2}$. \item Let $y=x^{-a}$, where $a$ is a positive rational number. Use the same method as the previous problem to show $y'=-ax^{-a-1}$. \end{enumerate} \end{frame} \begin{frame} \large When you solve for $y'$ in an implicit differentiation problem, you have to solve a quadratic equation \begin{enumerate} \item always \item sometimes \item never \end{enumerate} \end{frame} \begin{frame} \large Find equations of both the tangent lines to the ellipse $$x^2 + 9y^2 = 81$$ that pass through the point $(27, 3)$. \end{frame} \begin{frame} \large The Thin Lens Equation in optics relates the focal length $f$ of a lens, the distance $a$ from an object to the lens, and the distance $b$ from the object's image to the lens. The equation is $$\frac{1}{a}+\frac{1}{b}=\frac{1}{f}$$ Let's say you have a lens with focal length $10$ cm. \begin{itemize} \item[\bf (a)] Which of the following derivatives describes the rate at which the position of the image changes as you move the object? \centerline{ $\dfrac{da}{db}$\qquad $\dfrac{db}{da}$\qquad $\dfrac{da}{df}$ \qquad $\dfrac{df}{da}$} \item[\bf (b)] If the object is 20 centimeters from the lens and moving away from the lens, where is the object's image and in what direction is it moving? \end{itemize} \end{frame} \end{document}

calculus:resources:calculus_flipped_resources:derivatives:2.7_sciences_tex

2014-09-05T22:02:45-04:00

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

<nowiki> \begin{document}

\begin{frame}

  \large Beginning at time $t=0$, a particle moves along the number 
  line so that its position after $t$ seconds is
  $$f(t)=t^3-15t^2+72t$$
  \vskip 15pt

  \begin{enumerate}[a)]
      \item Find the velocity and acceleration at time $t$.
      \item At what time(s) is the particle moving 3 units/sec in the 
      negative direction?
      \item  At what time(s) is the particle at rest?
      \item When is the particle moving in the positive direction?
      \item At what times is the particle speeding up?
  \end{enumerate}

\end{frame}

\begin{frame}

  \large Given on the board are the graphs of the {\bf velocity} functions 
  of two particles. For each particle, answer the following questions.
  \vskip 15pt

  \begin{enumerate}[a)]
      \item When is it speeding up? When is it slowing down?
      \item When is it moving in the positive direction?
      \item When is it at rest?
  \end{enumerate}

\end{frame}

\begin{frame}

  \large The cost, in dollars, of producing x yards of a certain fabric is 
  $C(x)=1300+14x-0.1x^2+.0005x^3$.
  \vskip 15pt

  \begin{enumerate}[a)]
      \item Find the marginal cost function.
      \item Find $C'(300)$. This is the rate at which costs are increasing 
      with respect to the production level. Use $C(300)$ and $C'(300)$ to 
      estimate $C(301)$.
      \item Find the actual value of $C(301)$ and compare.
  \end{enumerate}

\end{frame}

\begin{frame}

  \large If a ball is thrown vertically upward with a velocity of 128 ft/s, 
  then its height after $t$ seconds is $s = 128t - 16t^2$ ft.
  \vskip 15pt

  \begin{enumerate}[a)]
      \item What is the velocity and acceleration after $t$ seconds?
      \item What is the maximum height reached by the ball?
  \vskip 15pt

      \item What is the velocity of the ball when it is 240 ft above the 
      ground on its way up? (Consider up to be the positive direction.)
      \item What is the velocity of the ball when it is 240 ft above the 
      ground on its way down?
  \end{enumerate}

\end{frame}

\begin{frame}

  \large Sodium chlorate crystals are easy to grow in the shape of cubes 
  by allowing a solution of water and sodium chlorate to evaporate slowly.
   \vskip 15pt

  If $V$ is the volume of such a cube with side length $x$, calculate 
  the derivative when $x = 4$ mm. What's the physical interpretation of 
  $V'(4)$, in plain English?

\end{frame}

\begin{frame}

  \large A stone is dropped into a lake, creating a circular ripple that 
  travels outward at a speed of 60 cm/s.
  \vskip 15pt

  \begin{enumerate}[a)]
      \item Find the rate at which the area within the circle is increasing 
      after $t$ seconds.
      \item Compare this rate at time $t$ versus time $2t$. That is, 
      after twice as much time has passed, how much faster is the area increasing?
      \item When the radius of the circle has doubled, how much has 
      the rate $dA/dt$ increased?
  \end{enumerate}

\end{frame}

\begin{frame}

  \large A spherical balloon is being inflated. Find the rate of 
  increase of the surface area ($S = 4\pi r^2$) with respect to the 
  radius $r$ when $r$ is each of the following.
  \vskip 15pt

  \begin{enumerate}[a)]
      \item 1 ft
      \item 5 ft
      \item 8 ft
  \end{enumerate}

\end{frame}

\begin{frame}

  \large Newton's Law of Gravitation says that the magnitude $F$ of 
  the force exerted by a body of mass $m$ on a body of mass $M$ is
  $$F = \frac{GmM}{r^2}$$
  where $G$ is the gravitational constant and $r$ is the distance 
  between the bodies.
  \vskip 15pt

  \begin{enumerate}[a)]
      \item Find $dF/dr$.
      \item What's the physical interpretation of $dF/dr$, in plain 
      English?
      \item What does the minus sign indicate?
  \end{enumerate}
  \vskip 15pt

  (The value of $G$ depends on the units you're using. In case 
  you're interested, in metric it's $6.67\times 10^{-11} Nm^2/{kg}^2$. 
  Note that $10^{-11}$ is really, really small.)

\end{frame}

\end{document} <\nowiki>

calculus:resources:calculus_flipped_resources:derivatives:2.8_related_rates_tex

2015-08-28T22:35:01-04:00

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

problem2.png

\begin{document} \begin{frame} \begin{block}{Steps to Solve a Related Rate} \begin{itemize} \item[\bf (i)] What is variable?\\ What is constant? \vskip 2pt \item[\bf (ii)] Which rates are known?\\ Which rates need to be found? \vskip 2pt \item[\bf (iii)] What equation relates the variables in (ii)? \vskip 2pt \item[\bf (iv)] Use Implicit Differentiation on the equation in (iii) to relate the rates. \end{itemize} \end{block} If $V$ is the volume of a cube with edge lengths $x$ and the cube expands as time passes, find $\frac{dV}{dt}$ in terms of $\frac{dx}{dt}$. \begin{itemize} \item[\bf a)] What is $\frac{dV}{dx}$ when $x=4$ inches and is growing at a rate of $2$ inches per minute? \item[\bf b)] What is $x$ if the volume is shrinking at $3$ cubic inches per minute and the side length is shrinking at $4$ inches per minute? \item[\bf c)] Can a cube have a shrinking volume and a growing sides? \end{itemize} \end{frame} \begin{frame} \large A spherical weather balloon is being inflated at a rate of $ 0.5 m^{3}/sec $. \vskip 5pt \begin{enumerate}[a)] \item How fast is the diameter increasing at the instant the diameter is 2 meters? \vskip 15pt \item How fast is the volume changing at that same instant? \vskip 15pt \item How fast is the surface area changing at that same instant? \end{enumerate} \end{frame} \begin{frame} \vskip 10pt As gravel is being poured into a conical pile, its volume $V$ changes with time. As a result, the height $h$ and radius $r$ also change with time. Knowing that at any moment $V=\frac{1}{3}\pi r^2 h$, the relationship between the changes with respect to time in the volume, radius and height is \vskip 10pt \begin{enumerate} \item $\displaystyle{\frac{dV}{dt}=\frac{1}{3}\pi \left( 2r\frac{dr}{dt} h+r^2\frac{dh}{dt}\right)}$ \item $\displaystyle{\frac{dV}{dt}=\frac{1}{3}\pi \left( 2r\frac{dr}{dt} \cdot \frac{dh}{dt}\right)}$ \item $\displaystyle{\frac{dV}{dr}=\frac{1}{3}\pi \left( 2rh+r^2\frac{dh}{dt}\right)}$ \item $\displaystyle{\frac{dV}{dh}=\frac{1}{3}\pi \left( (r^2)(1)+2r\frac{dr}{dh}h\right)}$ \end{enumerate} \end{frame} \begin{frame} {\large Imagine the following magic triangle. Its base is on a horizontal surface and no matter what you do to its height, the triangle always has area $10$ cm$^2$.} \vskip 20pt {\large If you push down on the top of the triangle so that it becomes shorter at a rate of $3$ cm/sec, how fast will the length of the base be changing when the triangle is $5$ cm tall?} \end{frame} \begin{frame} \large My neighbors have a very loud stereo. The volume knob turns half a circle (angles $\theta$ between $0^\circ$ and $180^\circ$) and the volume of the music is given by the function $V(\theta)=110\sin(\theta/2)$ decibels (dB). \vskip 20pt One night at $3:30$ in the morning I notice an increase from a volume of $88$ dB at a rate of $1$ decibel per second! At what rate can I deduce that my neighbor is turning his volume knob? \end{frame} \begin{frame} %%%%%%% Students should discover errors in this "solution" {\small Water is leaking out of a tank shaped like a right circular cone with height $5$ m and top radius $3$ m. When the water level in the cone is $2$ m, the water level is decreasing at a rate of $0.1 \frac{m}{s}$. How fast is the water leaking out of the cone?} \pause \begin{columns} \begin{column}{0.45\textwidth} \begin{center} \includegraphics[width=5cm]{problem2.png} \end{center} \end{column} \pause \begin{column}{0.55\textwidth} {\small The volume of the water in the cone is $V = \frac{1}{3}\pi r^2h$ and using the figure above and similar triangles $\frac{r}{h} = \frac{3}{5}$ \pause , which means} $$ {\small r = \frac{3}{5}h = \frac{3}{5}2 = \frac{6}{5}.} $$ {\small This means that} $$ {\small V = \frac{1}{3}\left(\frac{6}{5}\right)^2\pi h = \frac{12\pi}{25}h.} $$ \end{column} \end{columns} \pause {\small Taking the derivative with respect to time} $$ {\small \frac{dV}{dt} = \frac{12\pi}{25}\frac{dh}{dt} = \frac{12\pi}{25}\frac{1}{10} = \frac{6\pi}{125} \frac{m^3}{s}.} $$ \end{frame} \begin{frame} {\small Water is leaking out of a tank shaped like a right circular cone with height $5$ m and top radius $3$ m. When the water level in the cone is $2$ m, the water level is decreasing at a rate of $0.1 \frac{m}{s}$. How fast is the water leaking out of the cone?} \begin{columns} \begin{column}{0.45\textwidth} \begin{center} \includegraphics[width=5cm]{problem2.png} \end{center} \end{column} \begin{column}{0.55\textwidth} {\small The volume of the water in the cone is $V = \frac{1}{3}\pi r^2h$ and using the figure above and similar triangles $\frac{r}{h} = \frac{3}{5}$, which means} $$ {\small r = \frac{3}{5}h \ \ \ \ \xcancel{= \frac{3}{5}2 = \frac{6}{5}.}} $$ {\small This means that} $$ {\small V = \frac{1}{3}\left(\textcolor{red}{\frac{3}{5}h}\right)^2\pi h = \textcolor{red}{\frac{3\pi}{25}h^3}.} $$ \end{column} \end{columns} {\small Taking the derivative with respect to time} $$ {\small \frac{dV}{dt} = \frac{3\pi}{25}\textcolor{red}{3h^2}\frac{dh}{dt} = \frac{3\pi}{25}\textcolor{red}{3(2)^2}\frac{\tr{-1}}{10} = \textcolor{red}{\frac{-36\pi}{250}} \frac{m^3}{s}.} $$ \end{frame} \begin{frame} \large \begin{enumerate}[a)] \item A streetlight hangs 5 meters above the ground. Regina, who is 1.5 meters tall, walks away from the point under the light at a rate of 2 meters per second. How fast is her shadow lengthening when she is 7 meters away from the point under the light?\\ (Hint: Use similar triangles.) \vskip 25pt \item Suppose Regina has the ability to magically shrink herself. At what rate must she do this to keep her shadow a constant length of 3 meters? Write this as a function of only her distance from the point under the light. \end{enumerate} \end{frame} \begin{frame} \large A revolving beacon from a light house shines on the straight shore, and the closest point on the shore is a pier one half mile from the lighthouse. Let $\theta$ denote the angle between the lighthouse, pier, and point on the shore where the light shines. \begin{enumerate} \item Write the distance from the pier to the point of light as a function of $\theta$. \item What is the rate of change of the distance from the pier to the point of light with respect to $\theta$. \item Suppose $\theta$ is a function of time $t$. Give an expression for the rate of change of distance with respect to time $t$. \item Suppose that the light makes 1 revolution per minute. How fast is the light traveling along the straight beach at the instant it pases over a shorepoint 1 mile away from the shorepoint nearest the searchlight? \end{enumerate} \end{frame} \begin{frame} \large Given that a spherical raindrop evaporates at a rate proportional to its surface area, how fast does the radius shrink? \vskip 60pt The minute hand on a watch is $8$ mm long and the hour hand is $4$ mm long. How fast is the distance between the tips of the hands changing at one o'clock? \end{frame} \begin{frame} \large \begin{enumerate}[a)] \item Find the shortest line segment with endpoints on the $x$ and $y$ axes going through the point $( 1,8)$. \vskip 15pt \item What is the area of the triangle formed by the shortest line segment? \vskip 15pt \item What is the rate of change of area with respect to the $x$-coordinate of the point on the $x$-axis? \vskip 15pt \item For which $x$ is the area increasing? \end{enumerate} \end{frame} \begin{frame} \large The speed limit on a stretch of highway is 55 mph. Highway patrol officer, Sgt. Miguel, stations himself at a point, out of view of the motorists, 50 feet off the highway. Miguel is equipped with a radar gun which measures the speed at which a car approaches {\bf his position}. \vskip 15pt He takes a reading of suspected speeders by pointing his radar gun at a point on the highway 120 feet from the point on the highway closest to him. The radar gun picks up a reading of 48 feet/sec for a green Chevy driven by Alyssa. How fast is she traveling? Is Alyssa speeding? \end{frame} \begin{frame} \large At a certain moment, ship $ A $ is 6 miles south and 8 miles west of ship $ B $ . Ship $ A $ at that moment is steaming east at 12 mph, while ship $ B $ is steaming north at 15 mph. \vskip 15pt Are the ships approaching each other or separating from each other? At what rate? \end{frame} \begin{frame} \large Particle $A$ moves along the positive horizontal axis, and particle $B$ along the graph of $$ f( x) = -\sqrt 3x,\;x\leq 0.$$ At a certain time, $A$ is at the point $( 5,0)$ and moving with speed $3$ units/sec; and $B$ is at a distance of $3$ units from the origin moving with speed $4$ units/sec. At what rate is the distance between $A$ and $B$ changing? \end{frame} \end{document}