Concepts and Vocabulary
True or False The domain of a function \(f\) and the domain of its derivative function \(f^\prime\) are always equal.
True or False If a function is continuous at a number \(c\), then it is differentiable at \(c\).
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 \(c\).
The instruction “Differentiate \(f\)” means to find the ____ of \(f\).
Skill Building
In Problems 5–10, find the rate of change of each function f at any real number \(c.\)
\(f(x)=10\)
\(f(x)=-4\)
\(f(x)=2x+3\)
\(f(x)=3x-5\)
\(f(x)=2-x^{2}\)
\(f(x)=2x^{2}+4\)
In Problems 11–16, differentiate each function f and determine the domain of \(f^\prime\). Use form (3) on page 154.
\(f(x)=5\)
\(f(x)=-2\)
\(f(x)=3x^{2}+x+5\)
\(f(x)=2x^{2}-x-7\)
\(f(x)=5 \sqrt{x-1}\)
\(f(x)= 4\sqrt{x+3}\)
In Problems 17–22, differentiate each function \(f\). Graph \( y=f( x)\) and \(y=f^\prime ( x)\) on the same set of coordinate axes.
\(f(x)=\dfrac{1}{3}x+1\)
\(f(x)=-4x-5\)
\(f(x)=2x^{2}-5x\)
\(f(x)=-3x^{2}+2\)
\(f(x)=x^{3}-8x\)
\(f(x)=-x^{3}-8\)
In Problems 23–26, for each figure determine if the graphs represent a function f and its derivative \(f^\prime\). If they do, indicate which is the graph of f and which is the graph of \(f^\prime\).
In Problems 27–30, use the graph of f to obtain the graph of \(f^\prime\).
In Problems 31–34, the graph of a function f is given. Match each graph to the graph of its derivative \(f^\prime\) in A–D.
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In Problems 35–44, determine whether each function f has a derivative at c. If it does, what is \(f^\prime ( c) ?\) If it does not, give the reason why.
\(f( x) =x^{2/3}\) at \(c=-8\)
\(f( x) =2x^{1/3}\) at \(c=0\)
\(f(x)=|x^{2}-4|\) at \(c=2\)
\(f(x)=\vert x^{2}-4|\) at \(c=-2\)
\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 2x+3 &\hbox{if} & x < 1 \\ x^{2}+4 &\hbox{if} & x ≥ 1 \end{array} } &\hbox{at} & c=1 \end{array} \right.\)
\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 3-4x &\hbox{if} & x < -1 \\ 2x+9 &\hbox{if} & x ≥ -1 \end{array} } &\hbox{at} & c=-1 \end{array} \right.\)
\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} -4+2x &\hbox{if} & x ≤ \dfrac{1}{2} \\ 4x^{2}-4 &\hbox{if} & x > \dfrac{1}{2} \end{array} } &\hbox{at} & c=\dfrac{1}{2} \end{array} \right.\)
\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 2x^{2}+1 &\hbox{if} & x < -1 \\[3pt] -1-4x &\hbox{if} & x ≥ -1 \end{array} } &\hbox{at} & c=-1 \end{array} \right.\)
\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 2x^{2}+1 &\hbox{if} & x < -1 \\[3pt] 2+2x &\hbox{if} & x ≥ -1 \end{array} } &\hbox{at} & c=-1 \end{array} \right.\)
\(f(x)=\left\{ \begin{array}{@{}l@{\quad}l} {\ \begin{array}{@{}l@{\quad}ll} 5-2x &\hbox{if} & x < 2 \\ x^{2} &\hbox{if} & x ≥ 2 \end{array} } &\hbox{at} & c=2 \end{array} \right.\)
In Problems 45 and 46, use the given points \((c, f(c))\) on the graph of the function f.
Heaviside Functions In Problems 47 and 48:
\(u_{1}( t) =\left\{ \begin{array}{l@{\quad}ll} 0 &\hbox{if} & t < 1 \\ 1 &\hbox{if} & t ≥ 1 \end{array} \right.\) at \(c=1\)
\(u_{3}( t) =\left\{ \begin{array}{l@{\quad}ll} 0 &\hbox{if} & t < 3 \\ 1 &\hbox{if} & t ≥ 3 \end{array} \right.\) at \(c=3\)
In Problems 49–52, find the derivative of each function.
\(f(x)=mx+b\)
\(f(x)=ax^{2}+bx+c\)
\(f(x)=\dfrac{1}{x^{2}}\)
\(f(x)=\dfrac{1}{\sqrt{x}}\)
Applications and Extensions
In Problems 53–60, each limit represents the derivative of a function f at some number c. Determine f and c in each case.
\({\lim\limits_{h\rightarrow 0}\dfrac{(2+h)^{2}-4}{h} }\)
\({\lim\limits_{h\rightarrow 0}\dfrac{(2+h)^{3}-8}{h} }\)
\({\lim\limits_{x\rightarrow 1}\dfrac{x^{2}-1}{x-1}}\)
\({\lim\limits_{x\rightarrow 1}\dfrac{x^{14}-1}{x-1}}\)
\({\lim\limits_{x\rightarrow {\pi /6}}\dfrac{\sin x- \dfrac{1}{2}}{x-\dfrac{\pi }{6}}}\)
\({ \lim\limits_{x\rightarrow {\pi /4}}\dfrac{\cos x-\dfrac{\sqrt{2}}{2}}{ x-\dfrac{\pi }{4}}}\)
\({\lim\limits_{x\rightarrow 0}\dfrac{2(x+2)^{2}-(x+2)-6}{x}}\)
\({\lim\limits_{x \rightarrow 0}\dfrac{3x^{3}-2x}{x}}\)
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:
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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:
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 \(t=5{s}\).)
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.
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.
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.
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.
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 \((1,-1)\).
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:
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:
Use the definition of the derivative to show that \(f( x) =\left\vert x\right\vert\) has no derivative at \(0.\)
Use the definition of the derivative to show that \(f( x) =\sqrt[3]{x}\) has no derivative at \(0.\)
If \(f\) is an even function that is differentiable at \(c\), show that its derivative function is odd. That is, show \(f\prime (-c)=-f\prime (c)\).
If \(f\) is an odd function that is differentiable at \(c\), show that its derivative function is even. That is, show \(f\prime (-c)=f\prime (c).\)
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:
Challenge Problems
Let \(f\) be a function defined for all \(x.\) Suppose \(f\) has the following properties: \[ f(u+v)=f(u)f(v) \qquad f(0)=1 \qquad f\prime (0) \hbox{ exists} \]
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 \(b\).
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 \(p\).