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True or False The domain of a function \(f\) and the domain of its derivative function \(f^\prime\) are always equal.

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True or False If a function is continuous at a number \(c\), then it is differentiable at \(c\).

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Multiple Choice If \(f\) is continuous at a number \(c\) and if \( \lim\limits_{x\rightarrow c}\dfrac{f( x) -f( c) }{x-c}\) is infinite, then the graph of \(f\) has [(a) a horizontal, (b) a vertical, (c) no] tangent line at \(c\).

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The instruction “Differentiate \(f\)” means to find the ____ of \(f\).

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\(f(x)=10\)

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\(f(x)=-4\)

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\(f(x)=2x+3\)

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\(f(x)=3x-5\)

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

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\(f(x)=2x^{2}+4\)

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\(f(x)=5\)

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\(f(x)=-2\)

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\(f(x)=3x^{2}+x+5\)

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\(f(x)=2x^{2}-x-7\)

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\(f(x)=5 \sqrt{x-1}\)

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\(f(x)= 4\sqrt{x+3}\)

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\(f(x)=\dfrac{1}{3}x+1\)

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\(f(x)=-4x-5\)

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\(f(x)=2x^{2}-5x\)

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

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

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

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161

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\(f( x) =x^{2/3}\) at \(c=-8\)

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\(f( x) =2x^{1/3}\) at \(c=0\)

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\(f(x)=|x^{2}-4|\) at \(c=2\)

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\(f(x)=\vert x^{2}-4|\) at \(c=-2\)

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\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 2x+3 &\hbox{if} & x < 1 \\ x^{2}+4 &\hbox{if} & x ≥ 1 \end{array} } &\hbox{at} & c=1 \end{array} \right.\)

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\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 3-4x &\hbox{if} & x < -1 \\ 2x+9 &\hbox{if} & x ≥ -1 \end{array} } &\hbox{at} & c=-1 \end{array} \right.\)

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\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} -4+2x &\hbox{if} & x ≤ \dfrac{1}{2} \\ 4x^{2}-4 &\hbox{if} & x > \dfrac{1}{2} \end{array} } &\hbox{at} & c=\dfrac{1}{2} \end{array} \right.\)

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\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 2x^{2}+1 &\hbox{if} & x < -1 \\[3pt] -1-4x &\hbox{if} & x ≥ -1 \end{array} } &\hbox{at} & c=-1 \end{array} \right.\)

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\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 2x^{2}+1 &\hbox{if} & x < -1 \\[3pt] 2+2x &\hbox{if} & x ≥ -1 \end{array} } &\hbox{at} & c=-1 \end{array} \right.\)

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\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 5-2x &\hbox{if} & x < 2 \\ x^{2} &\hbox{if} & x ≥ 2 \end{array} } &\hbox{at} & c=2 \end{array} \right.\)

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\(u_{1}( t) =\left\{ \begin{array}{l@{\quad}ll} 0 &\hbox{if} & t < 1 \\ 1 &\hbox{if} & t ≥ 1 \end{array} \right.\) at \(c=1\)

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\(u_{3}( t) =\left\{ \begin{array}{l@{\quad}ll} 0 &\hbox{if} & t < 3 \\ 1 &\hbox{if} & t ≥ 3 \end{array} \right.\) at \(c=3\)

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\(f(x)=mx+b\)

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\(f(x)=ax^{2}+bx+c\)

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

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\(f(x)=\dfrac{1}{\sqrt{x}}\)

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\({\lim\limits_{h\rightarrow 0}\dfrac{(2+h)^{2}-4}{h} }\)

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\({\lim\limits_{h\rightarrow 0}\dfrac{(2+h)^{3}-8}{h} }\)

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\({\lim\limits_{x\rightarrow 1}\dfrac{x^{2}-1}{x-1}}\)

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\({\lim\limits_{x\rightarrow 1}\dfrac{x^{14}-1}{x-1}}\)

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\({\lim\limits_{x\rightarrow {\pi /6}}\dfrac{\sin x- \dfrac{1}{2}}{x-\dfrac{\pi }{6}}}\)

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\({ \lim\limits_{x\rightarrow {\pi /4}}\dfrac{\cos x-\dfrac{\sqrt{2}}{2}}{ x-\dfrac{\pi }{4}}}\)

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\({\lim\limits_{x\rightarrow 0}\dfrac{2(x+2)^{2}-(x+2)-6}{x}}\)

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\({\lim\limits_{x \rightarrow 0}\dfrac{3x^{3}-2x}{x}}\)

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For the function \(f(x)=\left\{ \begin{array}{l@{\quad}ll} x^{3} &\hbox{if} & x ≤ 0 \\ x^{2} &\hbox{if} & x>0 \end{array} \right.\), determine whether:

1. \(f\) is continuous at 0.
2. \(f^\prime (0)\) exists.
3. Graph the function \(f\) and its derivative \(f^\prime .\)

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For the function \(f(x)=\left\{ \begin{array}{l@{\quad}ll} 2x &\hbox{if} & x ≤ 0 \\ x^{2} &\hbox{if} & x>0 \end{array} \right. \), determine whether:

1. \(f\) is continuous at 0.
2. \(f^\prime (0)\) exists.
3. Graph the function \(f\) and its derivative \(f^\prime .\)

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Velocity The distance \(s\) (in feet) of an automobile from the origin at time \(t\) (in seconds) is given by \[ s=s( t) =\left\{ \begin{array}{l@{\quad}ll} t^{3} &\hbox{if} & 0 ≤ t < 5 \\ 125 &\hbox{if} & t ≥ 5 \end{array} \right. \] (This could represent a crash test in which a vehicle is accelerated until it hits a brick wall at \(t=5{s}\).)

1. Find the velocity just before impact (at \(t=4.99{s}\)) and just after impact (at \(t=5.01{s})\).
2. Is the velocity function \(v=s\prime ( t)\) continuous at \(t=5?\)
3. How do you interpret the answer to (b)?

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Population Growth A simple model for population growth states that the rate of change of population size \(P\) with respect to time \(t\) is proportional to the population size. Express this statement as an equation involving a derivative.

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Atmospheric Pressure Atmospheric pressure \(p\) decreases as the distance \(x\) from the surface of Earth increases, and the rate of change of pressure with respect to altitude is proportional to the pressure. Express this law as an equation involving a derivative.

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Electrical Current Under certain conditions, an electric current \(I\) will die out at a rate (with respect to time \(t\)) that is proportional to the current remaining. Express this law as an equation involving a derivative.

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Tangent Line Let \(f(x)=x^{2}+2\). Find all points on the graph of \(f\) for which the tangent line passes through the origin.

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Tangent Line Let \(f(x)=x^{2}-2x+1\). Find all points on the graph of \(f\) for which the tangent line passes through the point \((1,-1)\).

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Area and Circumference of a Circle A circle of radius \(r\) has area \(A=\pi r^{2}\) and circumference \(C=2\pi r.\) If the radius changes from \(r\) to \(r + \Delta r\), find the:

1. Change in area
2. Change in circumference
3. Average rate of change of area with respect to radius
4. Average rate of change of circumference with respect to radius
5. Rate of change of circumference with respect to radius

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Volume of a Sphere The volume \(V\) of a sphere of radius \(r\) is \(V=\dfrac{4\pi r^{3}}{3}.\) If the radius changes from \(r\) to \(r+\Delta r\), find the:

1. Change in volume
2. Average rate of change of volume with respect to radius
3. Rate of change of volume with respect to radius

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Use the definition of the derivative to show that \(f( x) =\left\vert x\right\vert\) has no derivative at \(0.\)

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Use the definition of the derivative to show that \(f( x) =\sqrt[3]{x}\) has no derivative at \(0.\)

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If \(f\) is an even function that is differentiable at \(c\), show that its derivative function is odd. That is, show \(f\prime (-c)=-f\prime (c)\).

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If \(f\) is an odd function that is differentiable at \(c\), show that its derivative function is even. That is, show \(f\prime (-c)=f\prime (c).\)

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Tangent Lines and Derivatives Let \(f\) and \(g\) be two functions, each with derivatives at \(c\). State the relationship between their tangent lines at \(c\) if:

1. \(f\prime (c)=g\prime (c)\)
2. \({f}\prime (c)=-{\dfrac{1}{g\prime (c)}}\)
3. \(g\prime(c)\neq 0\)

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Let \(f\) be a function defined for all \(x.\) Suppose \(f\) has the following properties: \[ f(u+v)=f(u)f(v) \qquad f(0)=1 \qquad f\prime (0) \hbox{ exists} \]

1. Show that \(f\prime (x)\) exists for all real numbers \(x.\)
2. Show that \(f\prime (x)=f\prime (0)f(x)\).

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A function \(f\) is defined for all real numbers and has the following three properties: \[ f(1)=5\qquad f(3)=21 \qquad f(a+b)-f(a)=kab+2b^{2} \] for all real numbers \(a\) and \(b\) where \(k\) is a fixed real number independent of \(a\) and \(b\).

1. Use \(a=1\) and \(b=2\) to find \(k\).
2. Find \(f\prime (3)\).
3. Find \(f\prime (x)\) for all real \(x\).

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A function \(f\) is periodic if there is a positive number \(p\) so that \(f(x+p)=f(x)\) for all \(x\). Suppose \(f\) is differentiable. Show that if \(f\) is periodic with period \(p\), then \(f\prime\) is also periodic with period \(p\).