Section 1
The Derivative and the Slope of a Graph
Trace the graph and sketch the tangent lines at $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right) .$
Estimate the slope of the graph at the point $(x, y) .$ (Each square on the grid is 1 unit by 1 unit.)
Revenue The graph represents the revenue $R$ (in millions of dollars per year) for Polo Ralph Lauren from 1999 through $2005,$ where $t$ represents the year, with $t=9 \mathrm{cor}-$ responding to 1999 . Estimate the slopes of the graph for the years 2002 and 2004 .
Sales The graph represents the sales $S$ (in millions of dollars per year) for Scotts Miracle-Gro Company from 1999 through $2005,$ where $t$ represents the year, with $t=9$ corresponding to 1999 . Estimate the slopes of the graph for the years 2001 and 2004 .
Consumer Trends The graph shows the number of visitors $V$ to a national park in hundreds of thousands during a one-year period, where $t=1$ corresponds to January. Estimate the slopes of the graph at $t=1,8,$ and 12 .
Athletics Two long distance runners starting out side by side begin a $10,000$ -meter run. Their distances are given by $s=f(t)$ and $s=g(t),$ where $s$ is measured in thousands of meters and $t$ is measured in minutes.(a) Which runner is running faster at $t_{1} ?$(b) What conclusion can you make regarding their rates at $t_{2} ?$(c) What conclusion can you make regarding their rates at $t_{3} ?$(d) Which runner finishes the race first? Explain.
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=6-2 x ;(2,2)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=2 x+4 ;(1,6)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=-1 ;(0,-1)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=6 ;(-2,6)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=x^{2}-1 ;(2,3)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=4-x^{2} ;(2,0)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=x^{3}-x ;(2,6)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=x^{3}+2 x ;(1,3)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=2 \sqrt{x} ;(4,4)$$
Use the limit definition to find the slope of the tangent line to the graph of $f$ at the given point.$$f(x)=\sqrt{x+1} ;(8,3)$$
Use the limit definition to find the derivative of the function.$$f(x)=3$$
Use the limit definition to find the derivative of the function.$$f(x)=-2$$
Use the limit definition to find the derivative of the function.$$f(x)=-5 x$$
Use the limit definition to find the derivative of the function.$$f(x)=4 x+1$$
Use the limit definition to find the derivative of the function.$$g(s)=\frac{1}{3} s+2$$
Use the limit definition to find the derivative of the function.$$h(t)=6-\frac{1}{2} t$$
Use the limit definition to find the derivative of the function.$$f(x)=x^{2}-4$$
Use the limit definition to find the derivative of the function.$$f(x)=1-x^{2}$$
Use the limit definition to find the derivative of the function.$$h(t)=\sqrt{t-1}$$
Use the limit definition to find the derivative of the function.$$f(x)=\sqrt{x+2}$$
Use the limit definition to find the derivative of the function.$$f(t)=t^{3}-12 t$$
Use the limit definition to find the derivative of the function.$$f(t)=t^{3}+t^{2}$$
Use the limit definition to find the derivative of the function.$$f(x)=\frac{1}{x+2}$$
Use the limit definition to find the derivative of the function.$$g(s)=\frac{1}{s-1}$$
Use the limit definition to find an equation of the tangent line to the graph of $f$ at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point.$$f(x)=\frac{1}{2} x^{2} ;(2,2)$$
Use the limit definition to find an equation of the tangent line to the graph of $f$ at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point.$$f(x)=-x^{2} ;(-1,-1)$$
Use the limit definition to find an equation of the tangent line to the graph of $f$ at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point.$$f(x)=(x-1)^{2} ;(-2,9)$$
Use the limit definition to find an equation of the tangent line to the graph of $f$ at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point.$$f(x)=2 x^{2}-1 ;(0,-1)$$
Use the limit definition to find an equation of the tangent line to the graph of $f$ at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point.$$f(x)=\sqrt{x}+1 ;(4,3)$$
Use the limit definition to find an equation of the tangent line to the graph of $f$ at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point.$$f(x)=\sqrt{x+2} ;(7,3)$$
Use the limit definition to find an equation of the tangent line to the graph of $f$ at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point.$$f(x)=\frac{1}{x} ;(1,1)$$
Use the limit definition to find an equation of the tangent line to the graph of $f$ at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point.$$f(x)=\frac{1}{x-1} ;(2,1)$$
Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.$$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=-\frac{1}{4} x^{2}} & {x+y=0}\end{array}$$
Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.$$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=x^{2}+1} & {2 x+y=0}\end{array}$$
Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.$$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=-\frac{1}{2} x^{3}} & {6 x+y+4=0}\end{array}$$
Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.$$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=x^{2}-x} & {x+2 y-6=0}\end{array}$$
Describe the $x$ -values at which the function is differentiable. Explain your reasoning.$$y=|x+3|$$
Describe the $x$ -values at which the function is differentiable. Explain your reasoning.$$y=\left|x^{2}-9\right|$$
Describe the $x$ -values at which the function is differentiable. Explain your reasoning.$$y=(x-3)^{2 / 3}$$
Describe the $x$ -values at which the function is differentiable. Explain your reasoning.$$y=x^{2 / 5}$$
Describe the $x$ -values at which the function is differentiable. Explain your reasoning.$$y=\sqrt{x-1}$$
Describe the $x$ -values at which the function is differentiable. Explain your reasoning.$$y=\frac{x^{2}}{x^{2}-4}$$
Describe the $x$ -values at which the function is differentiable. Explain your reasoning.$$y=\left\{\begin{array}{ll}{x^{3}+3,} & {x<0} \\ {x^{3}-3,} & {x \geq 0}\end{array}\right.$$
Describe the $x$ -values at which the function is differentiable. Explain your reasoning.$$y=\left\{\begin{array}{ll}{x^{2},} & {x \leq 1} \\ {-x^{2},} & {x>1}\end{array}\right.$$
Describe the $x$ -values at which $f$ is differentiable.$$f(x)=\frac{1}{x-1}$$
Describe the $x$ -values at which $f$ is differentiable.$$f(x)=\left\{\begin{array}{ll}{x^{2}-3,} & {x \leq 0} \\ {3-x^{2},} & {x>0}\end{array}\right.$$
Identify a function that has the given characteristics. Then sketch the function.$$f(0)=2 ; f^{\prime}(x)=-3,-\infty<x<\infty$$
Identify a function that has the given characteristics. Then sketch the function.$$\begin{array}{l}{f(-2)=f(4)=0 ; f^{\prime}(1)=0, f^{\prime}(x)<0} \\ {\text { for } x<1 ; f^{\prime}(x)>0 \text { for } x>1}\end{array}$$
Use a graphing utility to graph $f$ on the interval $[-2,2] .$ Complete the table by graphicallyestimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically.$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {-2} & {-\frac{3}{2}} & {-1} & {-\frac{1}{2}} & {0} & {\frac{1}{2}} & {1} & {\frac{3}{2}} & {2} \\ \hline f(x) & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline f^{\prime(x)} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array}$$$$f(x)=\frac{1}{4} x^{3}$$
Use a graphing utility to graph $f$ on the interval $[-2,2] .$ Complete the table by graphicallyestimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically.$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {-2} & {-\frac{3}{2}} & {-1} & {-\frac{1}{2}} & {0} & {\frac{1}{2}} & {1} & {\frac{3}{2}} & {2} \\ \hline f(x) & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline f^{\prime(x)} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array}$$$$f(x)=\frac{1}{2} x^{2}$$
Use a graphing utility to graph $f$ on the interval $[-2,2] .$ Complete the table by graphicallyestimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically.$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {-2} & {-\frac{3}{2}} & {-1} & {-\frac{1}{2}} & {0} & {\frac{1}{2}} & {1} & {\frac{3}{2}} & {2} \\ \hline f(x) & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline f^{\prime(x)} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array}$$$$f(x)=-\frac{1}{2} x^{3}$$
Use a graphing utility to graph $f$ on the interval $[-2,2] .$ Complete the table by graphicallyestimating the slopes of the graph at the given points. Then evaluate the slopes analytically and compare your results with those obtained graphically.$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {-2} & {-\frac{3}{2}} & {-1} & {-\frac{1}{2}} & {0} & {\frac{1}{2}} & {1} & {\frac{3}{2}} & {2} \\ \hline f(x) & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline f^{\prime(x)} & {} & {} & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array}$$$$f(x)=-\frac{3}{2} x^{2}$$
Find the derivative of the given function $f$. Then use a graphing utility to graph $f$ and its derivative in the same viewing window. What does the $x$ -intercept of the derivative indicate about the graph of $f ?$$$f(x)=x^{2}-4 x$$
Find the derivative of the given function $f$. Then use a graphing utility to graph $f$ and its derivative in the same viewing window. What does the $x$ -intercept of the derivative indicate about the graph of $f ?$$$f(x)=2+6 x-x^{2}$$
Find the derivative of the given function $f$. Then use a graphing utility to graph $f$ and its derivative in the same viewing window. What does the $x$ -intercept of the derivative indicate about the graph of $f ?$$$f(x)=x^{3}-3 x$$
Find the derivative of the given function $f$. Then use a graphing utility to graph $f$ and its derivative in the same viewing window. What does the $x$ -intercept of the derivative indicate about the graph of $f ?$$$f(x)=x^{3}-6 x^{2}$$
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The slope of the graph of $y=x^{2}$ is different at every point on the graph of $f .$
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If a function is continuous at a point, then it is differentiable at that point.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If a function is differentiable at a point, then it is continuous at that point.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.A tangent line to a graph can intersect the graph at more than one point.
Writing Use a graphing utility to graph the two function $f(x)=x^{2}+1$ and $g(x)=|x|+1$ in the same viewin window. Use the zoom and trace features to analyze the graphs near the point $(0,1) .$ What do you observe? Whic function is differentiable at this point? Write a short paragraph describing the geometric significance of differentiability at a point.