🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Calculus Volume 1 ## Gilbert Strang ## Chapter 3 ## Derivatives ## Educators     ### Problem 1 For the following exercises, use Equation 3.3 to find the slope of the secant line between the values$x_{1}$and$x_{2}$for each function$y=f(x)$$$f(x)=4 x+7 ; x_{1}=2, x_{2}=5$$ Erin O. Numerade Educator ### Problem 2 For the following exercises, 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)$$$f(x)=8 x-3 ; x_{1}=-1, x_{2}=3$$ Amy J. Numerade Educator ### Problem 3 For the following exercises, 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)$$$f(x)=x^{2}+2 x+1 ; x_{1}=3, x_{2}=3.5$$ Brian K. Numerade Educator ### Problem 4 For the following exercises, 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)$$$f(x)=-x^{2}+x+2 ; x_{1}=0.5, x_{2}=1.5$$ Amy J. Numerade Educator ### Problem 5 For the following exercises, 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)$$$f(x)=\frac{4}{3 x-1} ; x_{1}=1, x_{2}=3$$ Brian K. Numerade Educator ### Problem 6 For the following exercises, 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)$$$f(x)=\frac{x-7}{2 x+1} ; x_{1}=-2, x_{2}=0$$ Amy J. Numerade Educator ### Problem 7 For the following exercises, 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)$$$f(x)=\sqrt{x} ; x_{1}=1, x_{2}=16$$ Brian K. Numerade Educator ### Problem 8 For the following exercises, 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)$$$f(x)=\sqrt{x-9} ; x_{1}=10, x_{2}=13$$ Amy J. Numerade Educator ### Problem 9 For the following exercises, 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)$$$f(x)=x^{1 / 3}+1 ; x_{1}=0, x_{2}=8$$ Brian K. Numerade Educator ### Problem 10 For the following exercises, 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)$$$f(x)=6 x^{2 / 3}+2 x^{1 / 3} ; x_{1}=1, x_{2}=27$$ Amy J. Numerade Educator ### Problem 11 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=3-4 x, a=2$$ Brian K. Numerade Educator ### Problem 12 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=\frac{x}{5}+6, a=-1$$ Amy J. Numerade Educator ### Problem 13 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=x^{2}+x, a=1$$ Brian K. Numerade Educator ### Problem 14 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=1-x-x^{2}, a=0$$ Amy J. Numerade Educator ### Problem 15 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=\frac{7}{x}, a=3$$ Brian K. Numerade Educator ### Problem 16 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=\sqrt{x+8}, a=1$$ Amy J. Numerade Educator ### Problem 17 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=2-3 x^{2}, a=-2$$ Brian K. Numerade Educator ### Problem 18 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=\frac{-3}{x-1}, a=4$$ Amy J. Numerade Educator ### Problem 19 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=\frac{2}{x+3}, a=-4$$ Brian K. Numerade Educator ### Problem 20 For the following functions, a. use Equation 3.4 to find the slope of the tangent line$m_{\text { tan }}=f^{\prime}(a),$and b. find the equation of the tangent line to$f$at$x=a$. $$f(x)=\frac{3}{x^{2}}, a=3$$ Amy J. Numerade Educator ### Problem 21 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=5 x+4, a=-1$$ Brian K. Numerade Educator ### Problem 22 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=-7 x+1, a=3$$ Amy J. Numerade Educator ### Problem 23 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$x=a$ Brian K. Numerade Educator ### Problem 24 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=3 x^{2}-x+2, a=1$$ Amy J. Numerade Educator ### Problem 25 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=\sqrt{x}, a=4$$ Brian K. Numerade Educator ### Problem 26 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=\sqrt{x-2}, a=6$$ Amy J. Numerade Educator ### Problem 27 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=\frac{1}{x}, a=2$$ Brian K. Numerade Educator ### Problem 28 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=\frac{1}{x-3}, a=-1$$ Amy J. Numerade Educator ### Problem 29 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=\frac{1}{x^{3}}, a=1$$ Brian K. Numerade Educator ### Problem 30 For the following functions$y=f(x),$find$f^{\prime}(a)$using Equation$3.3 .$$$f(x)=\frac{1}{\sqrt{x}}, a=4$$ Amy J. Numerade Educator ### Problem 31 For the following exercises, 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$P .$ITI$f(x)=x^{2}+3 x+4, P(1,8)$(Round to decimal places.) Brian K. Numerade Educator ### Problem 32 For the following exercises, 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 .$$$[\mathrm{T}] f(x)=\frac{x+1}{x^{2}-1}, P(0,-1)$$ Amy J. Numerade Educator ### Problem 33 For the following exercises, 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$P .$[T]$f(x)=10 e^{0.5 x}, P(0,10)$(Round to 4 decimal places.) Brian K. Numerade Educator ### Problem 34 For the following exercises, 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 .$$$[\mathrm{T}] f(x)=\tan (x), P(\pi, 0)$$ Amy J. Numerade Educator ### Problem 35 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. $$s(t)=\frac{1}{3} t+5$$ Brian K. Numerade Educator ### Problem 36 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. $$s(t)=t^{2}-2 t$$ Amy J. Numerade Educator ### Problem 37 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. $$s(t)=2 t^{3}+3$$ Brian K. Numerade Educator ### Problem 38 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. $$s(t)=\frac{16}{t^{2}}-\frac{4}{t}$$ Amy J. Numerade Educator ### Problem 39 Use the following graph to evaluate a.$f^{\prime}(1)$and b.$f^{\prime}(6) .$ Brian K. Numerade Educator ### Problem 40 Use the following graph to evaluate a.$f^{\prime}(-3)$and b.$f^{\prime}(1.5) .$ Amy J. Numerade Educator ### Problem 41 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. $$f(x)=x^{1 / 3}, x=0$$ Brian K. Numerade Educator ### Problem 42 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. $$f(x)=x^{2 / 3}, x=0$$ Carson M. Numerade Educator ### Problem 43 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. $$f(x)=\left\{\begin{array}{l}{1, x<1} \\ {x, x \geq 1}\end{array}, x=1\right.$$ Brian K. Numerade Educator ### Problem 44 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. $$f(x)=\frac{|x|}{x}, x=0$$ Amy J. Numerade Educator ### Problem 45 [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. Brian K. Numerade Educator ### Problem 46 [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. Amy J. Numerade Educator ### Problem 47 Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by$s=f(t)$and$s=g(t)$where$s$is measured in feet and$t$is measured in seconds. a. Which vehicle has traveled farther at$t=2$seconds? b. What is the approximate velocity of each vehicle at$t=3$seconds? c. Which vehicle is traveling faster at$t=4$seconds? d. What is true about the positions of the vehicles at$t-4$seconds? Brian K. Numerade Educator ### Problem 48 [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. Amy J. Numerade Educator ### Problem 49 [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$m_{\text { tan }}=f^{\prime}(a)=0 .$ Norman A. Numerade Educator ### Problem 50 [T] For the function$f(x)=\frac{x}{1+x^{2}}, \quad$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 values of$x=a$for which$m_{\text { tan }}=f^{\prime}(a)=0$ Amy J. Numerade Educator ### Problem 51 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. Brian K. Numerade Educator ### Problem 52 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$f^{\prime}(2.718)$. Amy J. Numerade Educator ### Problem 53$[\mathrm{T}]$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)\$ . Norman A.