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%MODULE 1: Intro to SAGE
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%BUILD  TITLE PAGE 
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\begin_layout Title
Related Rates
\end_layout

\begin_layout Subtitle
The derivative expresses the 
\emph on
instantaneous rate of change
\emph default
.
 We can use this fact to solve problems involving rates of change that are
 related to each other.
\end_layout

\begin_layout BeginFrame
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{
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Defining the problem
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}
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\backslash
begin{question}
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\end_inset

 
\end_layout

\begin_layout Itemize
The radius of a ripple in a pond grows at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 How fast is the circumference changing at 3
\begin_inset space ~
\end_inset

seconds? 
\end_layout

\begin_layout Itemize
The radius of a ripple in a pond grows at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 How fast is the area changing at 3
\begin_inset space ~
\end_inset

seconds? 
\end_layout

\begin_layout Itemize
A camera is directed at the launch of a rocket.
 The camera is 3
\begin_inset space ~
\end_inset

km from the rocket.
 Once the rocket rises 1
\begin_inset space ~
\end_inset

km, it has a speed of 1500
\begin_inset space ~
\end_inset

km/hr.
 How fast must the angle of the camera change to remain fixed on the rocket?
 
\end_layout

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end{question}
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{
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Leibniz notation
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}
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\begin_layout Standard
\noindent
In most modules we have favored Newton's notation 
\begin_inset Formula $\ec f'$
\end_inset

 for the derivative of a function 
\begin_inset Formula $\ec f$
\end_inset

.
 In this module we will use Leibniz notation 
\begin_inset Formula $\ec df/dt$
\end_inset

 instead.
 
\emph on
Why?
\emph default
 Clarity of expression.
 All the problems can be solved using Newton's notation, but Newton's notation
 does not indicate the independent variable.
 Problems in related rates usually have several variables that depend on
 time, and the derivatives are always 
\emph on
with respect to time
\emph default
, which the Leibniz notation 
\begin_inset Formula $\ec df/dt$
\end_inset

 emphasizes.
\end_layout

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{
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\end_inset

First example: Experimentation
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 The radius of a ripple in a pond increasing at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 How fast is the circumference changing at 3 seconds?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The problem asks us to find 
\begin_inset Formula $\ec dC/dt$
\end_inset

, given that 
\begin_inset Formula $\ec dr/dt=5$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
You can visualize this using the SAGElet 
\begin_inset Quotes eld
\end_inset

The Ripple Effect: Circumference
\begin_inset Quotes erd
\end_inset

.
 It animates the ripple, and approximates 
\begin_inset Formula $\ec dC/dt$
\end_inset

 using the secant lines.
 Experiment with different values of 
\begin_inset Formula $\ec dr/dt$
\end_inset

 (
\begin_inset Quotes eld
\end_inset

change in radius per second
\begin_inset Quotes erd
\end_inset

) and see if you can determine a pattern.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Interestingly, any fixed choice of 
\begin_inset Formula $\ec dr/dt$
\end_inset

 gives a constant value for 
\begin_inset Formula $\ec dC/dt$
\end_inset

, 
\emph on
regardless of the value of 
\begin_inset Formula $\ec r$
\end_inset


\emph default
.
 Here are some values that we found:
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ec
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\begin_inset Tabular
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\begin_inset Formula $\frac{dr}{dt}$
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0 
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1 
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5 
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10 
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\begin_inset Formula $\frac{dC}{dt}$
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\end_layout

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</cell>
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\begin_layout Plain Layout
0 
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</cell>
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\begin_inset Formula $2\pi$
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</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $10\pi$
\end_inset

 
\end_layout

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</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
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\begin_layout Plain Layout
\begin_inset Formula $20\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

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}
\end_layout

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\begin_layout Standard
This suggests the relationship 
\begin_inset Formula $\ec dC/dt=2\pi(dr/dt)$
\end_inset

.
 Since 
\begin_inset Formula $\ec dr/dt$
\end_inset

 is constant, so is 
\begin_inset Formula $\ec dC/dt$
\end_inset

.
 But can we 
\emph on
prove
\emph default
 it?
\end_layout

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{
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First Example: Concept
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 The radius of a ripple in a pond increasing at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 How fast is the circumference changing at 3 seconds?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The problem asks us to find 
\begin_inset Formula $\ec dC/dt$
\end_inset

, given that 
\begin_inset Formula $\ec dr/dt=5$
\end_inset

.
\end_layout

\begin_layout Standard
If we had a relationship between 
\begin_inset Formula $\ec r$
\end_inset

 and 
\begin_inset Formula $\ec C$
\end_inset

, we could differentiate it with respect to 
\begin_inset Formula $\ec t$
\end_inset

.
 Since the radius 
\begin_inset Formula $\ec r$
\end_inset

 and the circumference 
\begin_inset Formula $\ec C$
\end_inset

 depend on time 
\begin_inset Formula $\ec t$
\end_inset

, the Chain Rule would give us quantities 
\begin_inset Formula $\ec dr/dt$
\end_inset

 and 
\begin_inset Formula $\ec dC/dt$
\end_inset

.
 We would end up with a relationship between the rates!
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
This is the general procedure for related rates:
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
find an equation that relates the quantities; 
\end_layout

\begin_layout Itemize
differentiate with respect to time; and 
\end_layout

\begin_layout Itemize
solve for the unknown rate.
 
\end_layout

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end{method}
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{
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First Example: Solution
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}
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\end_layout

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{
\backslash
textwidth}{6in}
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\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 The radius of a ripple in a pond increasing at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 How fast is the circumference changing at 3 seconds?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1-2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
The problem asks us to find 
\begin_inset Formula $dC/dt$
\end_inset

, given that 
\begin_inset Formula $dr/dt=5$
\end_inset

.
\end_layout

\begin_layout Standard
This is the general procedure for related rates:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
find an equation that relates the quantities; 
\end_layout

\begin_layout Itemize
differentiate with respect to time; and 
\end_layout

\begin_layout Itemize
solve for the unknown rate.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset


\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset

An equation that relates 
\begin_inset Formula $\ec r$
\end_inset

 and 
\begin_inset Formula $\ec C$
\end_inset

 is 
\begin_inset Formula 
\begin{align*}
\ec C & {\ec=2\pi r}.
\end{align*}

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset

Differentiating with respect to time gives 
\begin_inset Formula 
\begin{align*}
\dims C & {\dims=2\pi r}\\
\ec\frac{dC}{dt} & {\ec=2\pi\frac{dr}{dt}}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

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<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Substituting 
\begin_inset Formula $\ec dr/dt=5$
\end_inset

, we have 
\begin_inset Formula 
\begin{align*}
\dims\frac{dC}{dt} & {\dims=2\pi\frac{dr}{dt}}\\
\ec\frac{dC}{dt} & {\ec=10\pi}.
\end{align*}

\end_inset

 Since 
\begin_inset Formula $\ec t$
\end_inset

 does not appear in the equation, 
\begin_inset Formula $\ec dC/dt=10\pi$
\end_inset

 for 
\emph on
all
\emph default
 values of 
\begin_inset Formula $\ec t$
\end_inset

.
 At 3
\begin_inset space ~
\end_inset

seconds, therefore, the circumference is increasing at 
\begin_inset Formula $\ec10\pi$
\end_inset


\begin_inset space ~
\end_inset

cm/s, approximately.
 This agrees with the answer found by experimentation.
\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Second Example: Experimentation
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 The radius of a ripple in a pond increasing at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 How fast is the area changing at 3 seconds?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The problem asks us to find 
\begin_inset Formula $\ec dA/dt$
\end_inset

, given that 
\begin_inset Formula $\ec dr/dt=5$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
You can visualize this using the SAGElet 
\begin_inset Quotes eld
\end_inset

The Ripple Effect: Area
\begin_inset Quotes erd
\end_inset

.
 It animates the ripple, and approximates 
\begin_inset Formula $\ec dA/dt$
\end_inset

 using the secant lines of 
\begin_inset Formula $\ec A$
\end_inset

.
 Experiment with different values of 
\begin_inset Formula $\ec dr/dt$
\end_inset

 (
\begin_inset Quotes eld
\end_inset

change in radius per second
\begin_inset Quotes erd
\end_inset

) and see if you can determine a pattern.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
Unlike the circumference problems, fixed values of 
\begin_inset Formula $\ec dr/dt$
\end_inset

 do 
\emph on
not
\emph default
 result in a constant 
\begin_inset Formula $\ec dA/dt$
\end_inset

, depending on the value of 
\begin_inset Formula $\ec r$
\end_inset

.
 Here are some values that we found with 
\begin_inset Formula $\ec r'=5$
\end_inset

:
\end_layout

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\align center
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

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\begin_inset ERT
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\backslash
ec
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2 
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3 
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4 
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5 
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5 
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</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
15 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
20 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
25 
\end_layout

\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $dA/dt$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $25\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $75\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $125\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $175\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $225\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Notice that 
\begin_inset Formula $\ec dA/dt\approx125\pi$
\end_inset

 when 
\begin_inset Formula $\ec t=3$
\end_inset

.
 It seems to obey the pattern 
\begin_inset Formula 
\[
{\ec\frac{dA}{dt}=2\pi r\cdot\frac{dr}{dt}-\left(\frac{dr}{dt}\right)^{2}\pi}.
\]

\end_inset

 (This pattern is not obvious! Don't worry if you couldn't guess it.) But
 can we 
\emph on
prove
\emph default
 it?
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Second Example: Concept
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 The radius of a ripple in a pond increasing at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 How fast is the area changing at 3 seconds?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The problem asks us to find 
\begin_inset Formula $\ec dA/dt$
\end_inset

, given that 
\begin_inset Formula $\ec dr/dt=5$
\end_inset

.
\end_layout

\begin_layout Standard
As in the previous example, if we had a relationship between 
\begin_inset Formula $\ec r$
\end_inset

 and 
\begin_inset Formula $\ec A$
\end_inset

, then we could differentiate it with respect to 
\begin_inset Formula $\ec t$
\end_inset

 and obtain a relationship between the rates 
\begin_inset Formula $\ec dr/dt$
\end_inset

 and 
\begin_inset Formula $\ec dC/dt$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Recall the general procedure for related rates:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
find an equation that relates the quantities; 
\end_layout

\begin_layout Itemize
differentiate with respect to time; and 
\end_layout

\begin_layout Itemize
solve for the unknown rate.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Second Example: Solution
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 The radius of a ripple in a pond increasing at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 How fast is the area changing at 3 seconds?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The problem asks us to find 
\begin_inset Formula $dA/dt$
\end_inset

, given that 
\begin_inset Formula $dr/dt=5$
\end_inset

.
\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1-3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Recall the general procedure for related rates:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{method}
\end_layout

\end_inset

 
\end_layout

\begin_layout Itemize
find an equation that relates the quantities; 
\end_layout

\begin_layout Itemize
differentiate with respect to time; and 
\end_layout

\begin_layout Itemize
solve for the unknown rate.
 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{method}
\end_layout

\end_inset


\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
An equation that relates 
\begin_inset Formula $\ec r$
\end_inset

 and 
\begin_inset Formula $\ec A$
\end_inset

 is 
\begin_inset Formula 
\begin{align*}
\ec A & {\ec=\pi r^{2}}.\\
\phantom{\frac{dA}{dt}} & \phantom{=2\pi r\cdot\frac{dr}{dt}.}
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Differentiating with respect to time gives 
\begin_inset Formula 
\begin{align*}
\dims A & {\dims=\pi r^{2}}\\
\ec\frac{dA}{dt} & {\ec=2\pi r\cdot\frac{dr}{dt}}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3->
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Substituting 
\begin_inset Formula $\ec dr/dt=5$
\end_inset

, we have 
\begin_inset Formula 
\begin{align*}
\dims\frac{dA}{dt} & {\dims=2\pi r\cdot\frac{dr}{dt}}\\
\ec\frac{dA}{dt} & {\ec=10\pi r}.
\end{align*}

\end_inset

 But what is 
\begin_inset Formula $\ec r$
\end_inset

?
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
It can happen when solving a related-rate problem that you need to determine
 the value of a quantity that was not given.
 In this case, we know that the radius increases at a rate of 5
\begin_inset space ~
\end_inset

cm/s.
 This is a linear relationship, so 
\begin_inset Formula $\ec r=5t$
\end_inset

.
 At this particular moment (3
\begin_inset space ~
\end_inset

seconds) 
\begin_inset Formula $\ec t=3$
\end_inset

, so 
\begin_inset Formula $\ec r=15$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Standard
\align center
Hence 
\begin_inset Formula ${\displaystyle \ec\frac{dA}{dt}=150\pi}$
\end_inset

.
\end_layout

\end_deeper
\begin_layout BeginFrame
Second Example: Comparison to approximation
\end_layout

\begin_layout Standard
\noindent
Unlike the example with circumference, the approximation here was somewhat
 off.
 Using secant lines, the SAGElet approximated 
\begin_inset Formula $\ec dA/dt\approx125\pi$
\end_inset

.
 The correct answer was instead 
\begin_inset Formula $\ec dA/dt=150\pi$
\end_inset

.
 The error is due to a term in the pattern we identified with the approximation,
 
\begin_inset Formula $\ec dA/dt\approx2\pi r(dr/dt)-{\hilite(dr/dt)^{2}\pi}$
\end_inset

.
 Increasing the value of 
\begin_inset Quotes eld
\end_inset

units per second
\begin_inset Quotes erd
\end_inset

 in the SAGElet obtains better approximations, because the secant lines
 are computed closer to 
\begin_inset Formula $\ec t=3$
\end_inset

.
 
\emph on
(Try it!)
\emph default
 For example, we obtained the following results with different values of
 
\begin_inset Quotes eld
\end_inset

units per second
\begin_inset Quotes erd
\end_inset

: 
\end_layout

\begin_layout Standard
\align center
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ec
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="10">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $\ec t$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
2 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
2.25 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
2.5 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
2.75 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 3 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
3.25 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
3.5 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
3.75 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
4 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $n=1$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $75\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $\hilite125\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $175\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $n=2$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $87.5\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $112.5\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $\hilite137.5\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $162.5\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $187.5\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $n=4$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $93.75\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $106.25\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $118.75\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $131.25\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $\hilite143.75\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $156.25\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $168.75\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $181.25\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size tiny
\begin_inset Formula $193.75\pi$
\end_inset

 
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
 Values of 
\begin_inset Formula $\ec dA/dt$
\end_inset

 for different values 
\begin_inset Formula $\ec n$
\end_inset

 of 
\begin_inset Quotes eld
\end_inset

units per second
\begin_inset Quotes erd
\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset VSpace -2ex
\end_inset

As 
\begin_inset Formula $\ec n$
\end_inset

 increases, the accuracy of the approximation of 
\begin_inset Formula $\ec dA/dt$
\end_inset

 increases.
\end_layout

\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Third Example
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout OverlayArea
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

 A camera is directed at the launch of a rocket.
 The camera is 3
\begin_inset space ~
\end_inset

km from the rocket.
 Once the rocket rises 1
\begin_inset space ~
\end_inset

km, it has a speed of 1500
\begin_inset space ~
\end_inset

km/hr.
 How fast must the angle of the camera change to remain fixed on the rocket?
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The problem asks us to find 
\begin_inset Formula $\ec d\alpha/dt$
\end_inset

, given that 
\begin_inset Formula $\ec dh/dt=1500$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
How can we relate the two rates? The quantities 
\begin_inset Formula $\ec\alpha$
\end_inset

 and 
\begin_inset Formula $\ec h$
\end_inset

 are related by a right triangle: 
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/rocket_diagram.pdf
	scale 10

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Notice 
\begin_inset Formula $\ec h$
\end_inset

 lies on the side opposite 
\begin_inset Formula $\ec\alpha$
\end_inset

, and the side adjacent 
\begin_inset Formula $\ec\alpha$
\end_inset

 is constant at 3
\begin_inset space ~
\end_inset

km.
 We can relate the quantities using trigonometry:
\begin_inset Formula 
\[
{\ec\tan\alpha=\frac{h}{3}}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<3>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
This allows us to differentiate and relate the rates:
\begin_inset Formula 
\begin{align*}
{\dims\tan\alpha} & {\dims=\frac{h}{3}}\\
\ec\sec^{2}\alpha\cdot\frac{d\alpha}{dt} & {\ec=\frac{dh/dt}{3}}.
\end{align*}

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{explanation}
\end_layout

\end_inset

Remember from the Chain Rule that, since we differentiate with respect to
 
\begin_inset Formula $t$
\end_inset

,
\begin_inset Newline newline
\end_inset

 the derivatives must be multiplied by 
\begin_inset Formula $dh/dt$
\end_inset

 and 
\begin_inset Formula $d\alpha/dt$
\end_inset

 as appropriate.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{explanation}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<4-5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
We can substitute known values at this instant:
\begin_inset VSpace -2ex
\end_inset


\begin_inset Formula 
\begin{align*}
{\dims\tan\alpha} & {\dims=\frac{h}{3}}\\
\dims\sec^{2}\alpha\cdot\frac{d\alpha}{dt} & {\dims=\frac{dh/dt}{3}}\\
\ec\sec^{2}\alpha\cdot\frac{d\alpha}{dt} & {\ec=\frac{1500}{3}=500}.
\end{align*}

\end_inset

 
\end_layout

\begin_layout Standard
\align center
But what is the value of 
\begin_inset Formula $\ec\alpha$
\end_inset

?
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<5>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset VSpace -2ex
\end_inset

Since 
\begin_inset Formula $\ec\tan\alpha=h/3$
\end_inset

, then 
\begin_inset Formula $\ec\alpha=\arctan(h/3)$
\end_inset

.
 Thus
\begin_inset VSpace -2ex
\end_inset


\begin_inset Formula 
\[
{\ec\alpha=\arctan\frac{1}{3}}.
\]

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<6>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Thus:
\begin_inset Formula 
\begin{align*}
{\dims\tan\alpha} & {\dims=\frac{h}{3}}\\
\dims\sec^{2}\alpha\cdot\frac{d\alpha}{dt} & {\dims=\frac{dh/dt}{3}}\\
\dims\sec^{2}\alpha\cdot\frac{d\alpha}{dt} & {\dims=\frac{1500}{3}=500}\\
\ec\sec^{2}\left(\arctan\frac{1}{3}\right)\cdot\frac{d\alpha}{dt} & {\ec=500}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<7>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Thus:
\begin_inset Formula 
\begin{align*}
{\dims\tan\alpha} & {\dims=\frac{h}{3}}\\
\dims\sec^{2}\alpha\cdot\frac{d\alpha}{dt} & {\dims=\frac{dh/dt}{3}}\\
\dims\sec^{2}\alpha\cdot\frac{d\alpha}{dt} & {\dims=\frac{1500}{3}=500}\\
\dims\sec^{2}\left(\arctan500\right)\cdot\frac{d\alpha}{dt} & {\dims=500}\\
\ec\frac{d\alpha}{dt} & {\ec=\frac{500}{\sec^{2}\left(\arctan\frac{1}{3}\right)}}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<8>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Thus:
\begin_inset VSpace -2ex
\end_inset


\begin_inset Formula 
\begin{align*}
{\dims\tan\alpha} & {\dims=\frac{h}{3}}\\
 & \dims\vdots\\
\ec\frac{d\alpha}{dt} & {\ec=\frac{500}{\sec^{2}\left(\arctan\frac{1}{3}\right)}}.
\end{align*}

\end_inset

 From trigonometry, 
\begin_inset Formula $\ec\sec^{2}\left(\arctan\frac{1}{3}\right)=\frac{\sqrt{10}}{3}$
\end_inset

.
 Hence
\begin_inset Formula 
\[
{\ec\frac{d\alpha}{dt}=\frac{500}{\frac{\sqrt{10}}{3}}=\frac{1500}{\sqrt{3}}\approx866}\mathrm{\ radians\ per\ hour}.
\]

\end_inset


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{
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Conclusion
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Problems that involve 
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{
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\backslash
hilite
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 related rates
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}
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 can be solved by taking derivatives.
 
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\begin_layout Itemize
The method of solution is to 
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\begin_deeper
\begin_layout Itemize
find an equation that relates the quantities; 
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differentiate with respect to time; and 
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solve for the unknown rate.
 
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\end_deeper
\begin_layout Itemize
Quantities that are constant should not be substituted in before taking
 the derivative.
 
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Sometimes you will have to go back and re-examine an equation in order to
 determine the value of a quantity.
 
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\end_document
