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Related rates A spherical snowball is melting at the rate of \(2\) \({\rm{cm}}^3/\min\). How fast is the surface area changing when the radius is \(5\) cm?

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Related rates A lighthouse is \(3\, {\rm km}\) from a straight shoreline. Its light makes one revolution every \(8\) seconds. How fast is the light moving along the shore when it makes an angle of \(30^{ {\circ }}\) with the shoreline?

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Related rates Two planes at the same altitude are approaching an airport, one from the north and one from the west. The plane from the north is flying at \(250\, {\rm mph}\) and is \(30\) mi from the airport. The plane from the west is flying at \(200 \,{\rm mi}/{\rm h}\) and is \(20 \,{\rm mi}\) from the airport. How fast are the planes approaching each other at that instant?

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Critical Numbers \(f(x)=\dfrac{ x^{2}}{2x-1}\)

1. Find all the critical numbers of \(f\).
2. Find the local extrema of \(f.\)

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Critical Numbers Find all the critical numbers of \(f(x)=\cos (2x) \) on the closed interval \([0,\pi ]\).

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\(f(x) =x-\!\sin (2x) \) on \([0,2\pi]\)

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\(f(x) =\) \(\dfrac{3}{2}x^{4}-2x^{3}-6x^{2}+5\) on \([-2, 3]\)

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Rolle’s Theorem Verify that the hypotheses for Rolle’s Theorem are satisfied for the function \(f(x)=x^{3}-4x^{2}+4x\) on \([0,2].\) Find the coordinates of the point(s) at which there is a horizontal tangent line to the graph of \(f\).

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Mean Value Theorem Verify that the hypotheses for the Mean Value Theorem are satisfied for the function \(f(x)=\dfrac{2x-1}{x}\) on the interval \([1,4].\) Find a point on the graph of \(f\) that has a tangent with a slope equal to that of the secant line joining \((1,1) \) to \( \left( 4,\dfrac{7}{4}\right).\)

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Mean Value Theorem Does the Mean Value Theorem apply to the function \(f(x)=\sqrt{x}\) on the interval \([0,9]\)? If not, why not? If so, find the number \(c\) referred to in the theorem.

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\(f(x)=x^{3}-x^{2}-8x+1\)

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\(f(x)=x^{2}-24x^{2/3}\)

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\(f(x)=x^{4}e^{-2x}\)

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Rectilinear Motion The distance \(s\) of an object from the origin at time \(t\) is given by \(s=s(t) =t^{4}+2t^{3}-36t^{2}\). Draw figures to illustrate the motion of the object and its velocity.

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\(f(x)=-x^{3}-x^{2}+2x\)

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\(f(x)=x^{1/3}(x^{2}-9)\)

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\(f(x)=xe^{x}\)

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\(f(x) =\dfrac{x-3}{x^{2}-4}\)

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\(f(x)=x\sqrt{x-3}\)

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\(f(x) =x^{3}-3\ln\;x\)

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\(f(x)=x^{4}+12x^{2}+36x-11\)

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\(f(x)=3x^{4}-2x^{3}-24x^{2}-7x+2\)

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If \(y\) is a function and \(y\prime >0\) for all \(x\) and \( y{\prime \prime} <0\) for all \(x\), which of the following could be part of the graph of \(y=f(x)\) ? See illustrations (A) through (D).

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Sketch the graph of a function \(f\) that has the following properties:

\(f(-3)=2;~f(-1)=-5;~f(2)=-4;\)

\(f(6)=-1~f'(-3) = f^\prime (6)=0\)

\(\lim\limits_{x\rightarrow 0^{-}}f(x)=-\infty;~\lim\limits_{x \rightarrow 0^{+}}f(x)=\infty;\)

\(f^{\prime \prime} (x)>0 \hbox{ if } x<-3 \hbox{ or } 0<x<4\)

\(f^{\prime\prime} (x)<0 \hbox{ if } -\!3<x<0 \hbox{ or } 4<x\)

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Sketch the graph of a function \(f\) that has the following properties:

\(f(-2)=2; f(5)=1; f(0)=0\)

\(f^\prime (x)>0 \hbox{ if } x<-2 \hbox{ or } 5<x\)

\(f^\prime (x)<0 \hbox{ if }\;-2<x<2 \hbox{or } 2<x<5\)

\(f^{\prime \prime} (x)>0 \hbox{ if } x<0 \hbox{ or } 2<x\;{\rm {and}}\;f^{\prime \prime} (x)<0 \hbox{ if } 0<x<2\)

\(\lim\limits_{x\rightarrow 2^{-}}f(x)=-\infty \lim\limits_{x\rightarrow 2^{+}}f(x)=\infty\)

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Mean Value Theorem For the function \(f(x)=x\sqrt{x+1},\) \(0\leq x\leq b\), the number \(c\) satisfying the Mean Value Theorem is \(c=3\). Find \(b\).

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Maximizing Volume An open box is to be made from a piece of cardboard by cutting squares out of each corner and folding up the sides. If the size of the cardboard is \(2 \,{\rm ft}\) by \(3 \,{\rm ft}\), what size squares (in inches) should be cut out to maximize the volume of the box?

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Minimizing Distance Find the point on the graph of \(2y=x^{2}\) nearest to the point \((4,1)\).

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\(f(x) =0\)

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\(f(x) =x^{1/2}\)

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\(f(x) =\cos x\)

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\(f(x) =\sec x\tan x\)

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\(f(x) =\dfrac{2}{x}\)

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\(f(x) =-2x^{-3}\)

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\(f(x)=4x^{3}-9x^{2}+10x-3\)

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\(f(x)=e^{x}+\dfrac{4}{x}\)

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Velocity A box moves down an inclined plane with an acceleration \(a(t) = \) \(t^{2}(t-3)\)\({\rm{cm/s}}^2 \). It covers a distance of \(10 \,{\rm cm}\) in \(2 \,{\rm seconds}\). What was the original velocity of the box?

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Free Fall Two objects begin a free fall from rest at the same height \(1\) second apart. How long after the first object begins to fall will the two objects be \(10 \,{\rm m}\) apart?

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\(\dfrac{xe^{3x}-x}{1-\cos (2x) }\)

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\(\left( \dfrac{1}{x}\right) ^{\tan x}\)

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\(\dfrac{1}{x^{2}}-\dfrac{1}{x^{2}\sec x}\)

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\(\dfrac{\tan x-x}{x-\!\sin x}\)

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\(\lim\limits_{x\rightarrow {\pi }/{2}}\dfrac{\sec ^{2}x}{\sec ^{2}(3x) }\)

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\(\lim\limits_{x\rightarrow 0}\left[ \dfrac{2}{\sin ^{2}x}-\dfrac{1}{1-\cos x}\right] \)

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\(\lim\limits_{x\rightarrow 0}\dfrac{e^{x}-e^{-x}}{\sin x} \)

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\(\lim\limits_{x\rightarrow 0^{-}}x\;\cot \left( \pi x\right) \)

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\(\lim\limits_{x\rightarrow 0}\dfrac{\tan x+\sec x-1}{\tan x-\sec x+1}\)

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\(\lim\limits_{x\rightarrow a}\dfrac{ax-x^{2}}{a^{4}-2a^{3}x+2ax^{3}-x^{4}}\)

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\(\lim\limits_{x\rightarrow 0}\dfrac{x-\!\sin x}{x^{3}}\)

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\(\lim\limits_{x\rightarrow 0}\dfrac{\tan x-\!\sin x}{\sin ^{3}x}\)

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\(\lim\limits_{x\rightarrow \infty }(1+4x)^{2/x}\)

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\(\lim\limits_{x\rightarrow 1}\left[ \dfrac{2}{x^{2}-1}-\dfrac{1}{x-1}\right] \)

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\(\lim\limits_{x\rightarrow 4}\dfrac{x^{2}-16}{x^{2}+x-20}\)

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\(\lim\limits_{x\rightarrow 0^{+}}(\cot x)^{x}\)

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\(\dfrac{\textit{dy}}{\textit{dx}}=e^{x}\), when \(x=0\), \(y=2\)

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\(\dfrac{\textit{dy}}{\textit{dx}}=\dfrac{1}{2}\sec x\tan x,\) when \(x=0,\) \(y=7\)

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\(\dfrac{\textit{dy}}{\textit{dx}}=\dfrac{2}{x}\), when \(x=1,\) then \(y=4\)

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\(\dfrac{d^{2}y}{dx^{2}}=x^{2}-4,\) when \(x=3\), \(y=2,\) when \(x=2,\) \(y=2\)

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Maximizing Profit A manufacturer has determined that the cost \(C\) of producing \(x\) items is given by \(C(x) =200+35x+ 0.02x^{2}\) dollars. Each item can be sold for \({\$}78\). How many items should she produce to maximize profit?

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Optimization The sales of a new stereo system over a period of time are expected to follow the logistic curve \[ f(x)=\frac{5000}{1+5e^{-x}} \qquad x\geq 0 \]

where \(x\) is measured in years. In what year is the sales rate a maximum?

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Maximum Area Find the area of the rectangle of largest area in the fourth quadrant that has vertices at \((0,0)\), \((x,0)\), \(x>0\), and \((0,y)\), \(y<0\). The fourth vertex is on the graph of \(y=\ln x\).

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Differential Equation A motorcycle accelerates at a constant rate from \(0\) to \(72 \,{\rm km}/{\rm h}\) in 10 seconds. How far has it traveled in that time?