Chapter 3

Use Equation 3.3 to find the slope of the secant line between the values \(x_{1}\) and \(x_{2}\) for each function \(y=f(x)\). $$ f(x)=x^{1 / 3}+1 ; x_{1}=0, x_{2}=8 $$

For the following functions, a. use Equation 3.4 to find the slope of the tangent line \(m_{\tan }=f^{\prime}(a),\) and b. find the equation of the tangent line to \(f\) at \(x=a\). $$ f(x)=\frac{x}{5}+6, a=-1 $$

For the following functions, a. use Equation 3.4 to find the slope of the tangent line \(m_{\tan }=f^{\prime}(a),\) and b. find the equation of the tangent line to \(f\) at \(x=a\). $$ f(x)=x^{2}+x, a=1 $$

Consider the below function, $$ f(x)=x^{2}+9 x, a=2 $$ Find the slope of the tangent line \(f^{\prime}(a)\) and the equation of the tangent line at \(x=a\)

Given the function \(y=f(x)\), a. find the slope of the secant line \(P Q\) for each point \(Q(x, f(x))\) with \(x\) value given in the table. b. Use the answers from a. to estimate the value of the slope of the tangent line at \(P\). c. Use the answer from b. to find the equation of the tangent line to \(f\) at point \(P\). \(f(x)=x^{2}+3 x+4, P(1,8)\) (Round to 6 decimal places.) $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & \begin{array}{l} \text { Slope } \\ \boldsymbol{m}_{P Q} \end{array} & \boldsymbol{x} & \begin{array}{l} \text { Slope } \\ \boldsymbol{m}_{P Q} \end{array} \\ \hline 1.1 & \text { (i) } & 0.9 & \text { (vii) } \\ \hline 1.01 & \text { (ii) } & 0.99 & \text { (viii) } \\ \hline 1.001 & \text { (iii) } & 0.999 & \text { (ix) } \\ \hline 1.0001 & \text { (iv) } & 0.9999 & \text { (x) } \\ \hline 1.00001 & \text { (v) } & 0.99999 & \text { (xi) } \\ \hline 1.000001 & \text { (vi) } & 0.999999 & \text { (xii) } \\ \hline \end{array} $$

Given the function \(y=f(x)\), a. find the slope of the secant line \(P Q\) for each point \(Q(x, f(x))\) with \(x\) value given in the table. b. Use the answers from a. to estimate the value of the slope of the tangent line at \(P\). c. Use the answer from b. to find the equation of the tangent line to \(f\) at point \(P\). $$ f(x)=\frac{x+1}{x^{2}-1}, P(0,-1) $$ $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & \begin{array}{l} \text { Slope } \\ \boldsymbol{m}_{P Q} \end{array} & \boldsymbol{x} & \begin{array}{c} \text { Slope } \\ \boldsymbol{m}_{P Q} \end{array} \\ \hline 0.1 & \text { (i) } & -0.1 & \text { (vii) } \\ \hline 0.01 & \text { (ii) } & -0.01 & \text { (viii) } \\ \hline 0.001 & \text { (iii) } & -0.001 & \text { (ix) } \\ \hline 0.0001 & \text { (iv) } & -0.0001 & \text { (x) } \\ \hline 0.00001 & \text { (v) } & -0.00001 & \text { (xi) } \\ \hline 0.000001 & \text { (vi) } & -0.000001 & \text { (xii) } \\ \hline \end{array} $$

Given the function \(y=f(x)\), a. find the slope of the secant line \(P Q\) for each point \(Q(x, f(x))\) with \(x\) value given in the table. b. Use the answers from a. to estimate the value of the slope of the tangent line at \(P\). c. Use the answer from b. to find the equation of the tangent line to \(f\) at point \(P\). $$ f(x)=\tan (x), P(\pi, 0) $$ $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \text { Slope } \boldsymbol{m}_{\boldsymbol{P Q}} \\ \hline 3.1 & \text { (i) } \\ \hline 3.14 & \text { (ii) } \\ \hline 3.141 & \text { (iii) } \\ \hline 3.1415 & \text { (iv) } \\ \hline 3.14159 & \text { (v) } \\ \hline 3.141592 & \text { (vi) } \\ \hline \end{array} $$

IT] For the following position functions \(y=s(t), \quad\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\mathrm{i}) h=0.1, \quad\) (ii) \(h=0.01\) (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and c. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. $$ s(t)=\frac{1}{3} t+5 $$

For the following position functions \(y=s(t),\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h ;\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\) i) \(h=0.1\) (ii) \(h=0.01\), (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and C. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. \(s(t)=\frac{1}{3} t+5\)

IT] For the following position functions \(y=s(t), \quad\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\mathrm{i}) h=0.1, \quad\) (ii) \(h=0.01\) (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and c. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. $$ s(t)=t^{2}-2 t $$

For the following position functions \(y=s(t),\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h ;\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\) i) \(h=0.1\) (ii) \(h=0.01\), (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and C. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. $$ s(t)=t^{2}-2 t $$

IT] For the following position functions \(y=s(t), \quad\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\mathrm{i}) h=0.1, \quad\) (ii) \(h=0.01\) (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and c. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. $$ s(t)=2 t^{3}+3 $$

For the following position functions \(y=s(t),\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h ;\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\) i) \(h=0.1\) (ii) \(h=0.01\), (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and C. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. $$ s(t)=2 t^{3}+3 $$

IT] For the following position functions \(y=s(t), \quad\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\mathrm{i}) h=0.1, \quad\) (ii) \(h=0.01\) (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and c. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. $$ s(t)=\frac{16}{t^{2}}-\frac{4}{t} $$

For the following position functions \(y=s(t),\) an object is moving along a straight line, where \(t\) is in seconds and \(s\) is in meters. Find a. the simplified expression for the average velocity from \(t=2\) to \(t=2+h ;\) b. the average velocity between \(t=2\) and \(t=2+h, \quad\) where \((\) i) \(h=0.1\) (ii) \(h=0.01\), (iii) \(h=0.001,\) and (iv) \(h=0.0001 ;\) and C. use the answer from a. to estimate the instantaneous velocity at \(t=2\) second. $$ s(t)=\frac{16}{t^{2}}-\frac{4}{t} $$

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=x^{1 / 3}, x=0 $$

Use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=x^{1 / 3}, x=0 $$

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=x^{2 / 3}, x=0 $$

Use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=x^{2 / 3}, x=0 $$

Use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=\left\\{\begin{array}{l} 1, x<1 \\ x, x \geq 1 \end{array}, x=1\right. $$

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=\frac{|x|}{x}, x=0 $$

Use the limit definition of derivative to show that the derivative does not exist at \(x=a\) for each of the given functions. $$ f(x)=\frac{|x|}{x}, x=0 $$

[T] The position in feet of a race car along a straight track after \(t\) seconds is modeled by the function $$ s(t)=8 t^{2}-\frac{1}{16} t^{3} $$ a. Find the average velocity of the vehicle over the following time intervals to four decimal places: $$ \begin{array}{l}{\text { i. }[4,4.1]} \\ {\text { ii. }[4,4.01]} \\ {\text { iii. }[4,4.001]} \\ {\text { iv. }[4,4,0001]}\end{array} $$ b. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at \(t=4\) seconds.

The position in feet of a race car along a straight track after \(t\) seconds is modeled by the function \(s(t)=8 t^{2}-\frac{1}{16} t^{3}\). a. Find the average velocity of the vehicle over the following time intervals to four decimal places: i. [4,4.1] ii. [4,4.01] iii. [4,4.001] iv. [4,4.0001] b. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at \(t=4\) seconds.

[T] The distance in feet that a ball rolls down an incline is modeled by the function \(s(t)=14 t^{2}, \quad\) where \(t\) is seconds after the ball begins rolling. a. Find the average velocity of the ball over the following time intervals: $$ \begin{array}{ll}{\text { i. }} & {[5,5.1]} \\ {\text { ii. }} & {[5,5.01]} \\\ {\text { iii. }} & {[5,5.001]} \\ {\text { iv. }} & {[5,5.0001]}\end{array} $$ b. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at \(t=5\) seconds.

The distance in feet that a ball rolls down an incline is modeled by the function \(s(t)=14 t^{2},\) where \(t\) is seconds after the ball begins rolling. a. Find the average velocity of the ball over the following time intervals: i. [5,5.1] ii. [5,5.01] iii. \(\quad[5,5.001]\) iv. [5,5.0001] b. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at \(t=5\) seconds.

[T] The total cost \(C(x),\) in hundreds of dollars, to produce \(x\) jars of mayonnaise is given by \(C(x)=0.000003 x^{3}+4 x+300\) a. Calculate the average cost per jar over the following intervals: i. \([100,100.1]\) ii. \([100,100.01]\) iii. \([100,100.001]\) iv. \([100,100.0001]\) b. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.

The total cost \(C(x)\), in hundreds of dollars, to produce \(x\) jars of mayonnaise is given by \(C(x)=0.000003 x^{3}+4 x+300\) a. Calculate the average cost per jar over the following intervals: i. [100,100.1] ii. [100,100.01] iii. [100,100.001] iv. [100,100.0001] b. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.

[T] For the function \(f(x)=x^{3}-2 x^{2}-11 x+12\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the ZOOM feature on the calculator to approximate the two values of \(x=a\) for which \(m_{\text { tan }}=f^{\prime}(a)=0 .\)

For the function \(f(x)=x^{3}-2 x^{2}-11 x+12\), do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the ZOOM feature on the calculator to approximate the two values of \(x=a\) for which \(m_{\tan }=f^{\prime}(a)=0\)

Suppose that \(N(x)\) computes the number of gallons of gas used by a vehicle traveling \(x\) miles. Suppose the vehicle gets 30 \(\mathrm{mpg.}\) a. Find a mathematical expression for \(N(x)\) b. What is \(N(100) ?\) Explain the physical meaning. c. What is \(N^{\prime}(100) ?\) Explain the physical meaning.

ITI For the function \(f(x)=x^{4}-5 x^{2}+4,\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate \(f^{\prime}(-2), f^{\prime}(-0.5), f^{\prime}(1.7),\) and \(f^{\prime}(2.718)\) .

For the function \(f(x)=x^{4}-5 x^{2}+4,\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the nDeriv function, which numerically finds the derivative, on a graphing calculator to estimate \(f^{\prime}(-2), f^{\prime}(-0.5), f^{\prime}(1.7),\) and \(f^{\prime}(2.718)\)

\([\mathrm{T}]\) For the function \(f(x)=\frac{x^{2}}{x^{2}+1}, \quad\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the nDeriv function on a graphing calculator to find \(f^{\prime}(-4), f^{\prime}(-2), f^{\prime}(2),\) and \(f^{\prime}(4)\) .

For the function \(f(x)=\frac{x^{2}}{x^{2}+1},\) do the following. a. Use a graphing calculator to graph \(f\) in an appropriate viewing window. b. Use the nDeriv function on a graphing calculator to find \(f^{\prime}(-4), f^{\prime}(-2), f^{\prime}(2),\) and \(f^{\prime}(4)\).

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=6$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=6 $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=2-3 x$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=2-3 x $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=\frac{2 x}{7}+1$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=\frac{2 x}{7}+1 $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=4 x^{2}$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=4 x^{2} $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=5 x-x^{2}$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=5 x-x^{2} $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=\sqrt{2 x}$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=\sqrt{2 x} $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=\sqrt{x-6}$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=\sqrt{x-6} $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=\frac{9}{x}$$