🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Calculus Volume 1 ## Gilbert Strang ## Chapter 2 ## Limits ## Educators     + 3 more educators ### Problem 1 For the following exercises, points$P(1,2)$and$Q(x, y)$are on the graph of the function$f(x)=x^{2}+1$IT] Complete the following table with the appropriate values:$y$-coordinate of$Q,$the point$Q(x, y),$and the slope of the secant line passing through points$P$and$Q$. Round your answer to eight significant digits. Norman A. Numerade Educator ### Problem 2 For the following exercises, points$P(1,2)$and$Q(x, y)$are on the graph of the function$f(x)=x^{2}+1$Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to$f$at$x=1$ Amy J. Numerade Educator ### Problem 3 For the following exercises, points$P(1,2)$and$Q(x, y)$are on the graph of the function$f(x)=x^{2}+1$Use the value in the preceding exercise to find the equation of the tangent line at point$P .$Graph$f(x)$and the tangent line. For the following exercises, points$P(1,1)$and$Q(x, y)$are on the graph of the function$f(x)=x^{3}$ Thomas P. Numerade Educator ### Problem 4 For the following exercises, points$P(1,2)$and$Q(x, y)$are on the graph of the function$f(x)=x^{2}+1$[T] Complete the following table with the appropriate values: y-coordinate of$Q,$the point$Q(x, y),$and the slope of the secant line passing through points$P$and$Q .$Round your answer to eight significant digits. Amy J. Numerade Educator ### Problem 5 For the following exercises, points$P(1,2)$and$Q(x, y)$are on the graph of the function$f(x)=x^{2}+1$Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to$f$at$x=1$ Arya P. Numerade Educator ### Problem 6 For the following exercises, points$P(1,2)$and$Q(x, y)$are on the graph of the function$f(x)=x^{2}+1$Use the value in the preceding exercise to find the equation of the tangent line at point$P .$Graph$f(x)$and the tangent line. For the following exercises, points$P(4,2)$and$Q(x, y)$are on the graph of the function$f(x)=\sqrt{x}$. Amy J. Numerade Educator ### Problem 7 [T] Complete the following table with the appropriate values:$y$-coordinate of$Q$, the point$Q(x, y),$and the slope of the secant line passing through points$P$and$Q .$Round your answer to eight significant digits. $$\begin{array}{|l|l|l|l|}\hline \boldsymbol{x} & \boldsymbol{y} & \boldsymbol{Q}(\boldsymbol{x}, \boldsymbol{y}) & \boldsymbol{m}_{\mathrm{sec}} \\\hline 4.1 & \text { a. } & \text { e. } & \text { i. } \\\hline 4.01 & \text { b. } & \text { f. } & \text { j. } \\\hline 4.001 & \text { c. } & \text { g. } & \text { k. } \\\hline 4.0001 & \text { d. } & \text { h. } & \text { l. } \\\hline\end{array}$$ Norman A. Numerade Educator ### Problem 8 For the following exercises, points$P(1,2)$and$Q(x, y)$are on the graph of the function$f(x)=x^{2}+1$Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to$f$at$x=4$. Amy J. Numerade Educator ### Problem 9 For the following exercises, points$P(1,2)$and$Q(x, y)$are on the graph of the function$f(x)=x^{2}+1$Use the value in the preceding exercise to find the equation of the tangent line at point$P .$ Norman A. Numerade Educator ### Problem 10 For the following exercises, points$P(1.5,0)$and$Q(\phi, y)$are on the graph of the function$f(\phi)=\cos (\pi \phi)$[T] Complete the following table with the appropriate values:$y$-coordinate of$Q,$the point$Q(x, y),$and the slope of the secant line passing through points$P$and$Q .$Round your answer to eight significant digits. Amy J. Numerade Educator ### Problem 11 For the following exercises, points$P(1.5,0)$and$Q(\phi, y)$are on the graph of the function$f(\phi)=\cos (\pi \phi)$Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the tangent line to$f$at$x=4 .$ Yuou S. Numerade Educator ### Problem 12 For the following exercises, points$P(1.5,0)$and$Q(\phi, y)$are on the graph of the function$f(\phi)=\cos (\pi \phi)$Use the value in the preceding exercise to find the equation of the tangent line at point$P .$For the following exercises, points$P(-1,-1)$and$Q(x, y)$are on the graph of the function$f(x)=\frac{1}{x}$ Amy J. Numerade Educator ### Problem 13 For the following exercises, points$P(1.5,0)$and$Q(\phi, y)$are on the graph of the function$f(\phi)=\cos (\pi \phi)$[T] Complete the following table with the appropriate values:$y$-coordinate of$Q,$the point$Q(x, y),$and the slope of the secant line passing through points$P$and$Q$. Round your answer to eight significant digits. Norman A. Numerade Educator ### Problem 14 For the following exercises, points$P(1.5,0)$and$Q(\phi, y)$are on the graph of the function$f(\phi)=\cos (\pi \phi)$Use the values in the right column of the table in the preceding exercise to guess the value of the slope of the line tangent to$f$at$x=-1$ Amy J. Numerade Educator ### Problem 15 For the following exercises, points$P(1.5,0)$and$Q(\phi, y)$are on the graph of the function$f(\phi)=\cos (\pi \phi)$Use the value in the preceding exercise to find the equation of the tangent line at point P. Norman A. Numerade Educator ### Problem 16 For the following exercises, the position function of a ball dropped from the top of a 200 -meter tall building is given by$s(t)=200-4.9 t^{2},$where position$s$is measured in meters and time$t$is measured in seconds. Round your answer to eight significant digits. [T] Compute the average velocity of the ball over the given time intervals. a.$[4.99,5]$b.$[5,5.01]$C.$[4.999,5]$d.$[5,5.001]$ Amy J. Numerade Educator ### Problem 17 For the following exercises, the position function of a ball dropped from the top of a 200 -meter tall building is given by$s(t)=200-4.9 t^{2},$where position$s$is measured in meters and time$t$is measured in seconds. Round your answer to eight significant digits. Use the preceding exercise to guess the instantaneous velocity of the ball at$t=5 \mathrm{sec} .$ Joseph L. Numerade Educator ### Problem 18 For the following exercises, consider a stone tossed into the air from ground level with an initial velocity of 15$\mathrm{m} / \mathrm{sec}$. Its height in meters at time$t$seconds is$h(t)=15 t-4.9 t^{2}$[TI Compute the average velocity of the stone over the given time intervals.$
\begin{array}{ll}{\text { a. }} & {[1,1.05]} \\ {\text { b. }} & {[1,1.01]} \\ {\text { c. }} & {[1,1.005]} \\ {\text { d. }} & {[1,1.001]}\end{array}
$ Amy J. Numerade Educator ### Problem 19 For the following exercises, consider a stone tossed into the air from ground level with an initial velocity of 15$\mathrm{m} / \mathrm{sec}$. Its height in meters at time$t$seconds is$h(t)=15 t-4.9 t^{2}$Use the preceding exercise to guess the instantaneous velocity of the stone at$t=1 \mathrm{sec} .$ Norman A. Numerade Educator ### Problem 20 For the following exercises, consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by$h(t)=600+78.4 t-4.9 t^{2},$where$t$is measured in seconds. [T] Compute the average velocity of the rocket over the given time intervals. a.$[9,9.01]$b.$[8.99,9]$c.$[9,9.001]$d.$[8.999,9]$ Amy J. Numerade Educator ### Problem 21 For the following exercises, consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by$h(t)=600+78.4 t-4.9 t^{2},$where$t$is measured in seconds. Use the preceding exercise to guess the instantaneous velocity of the rocket at$t=9$sec. Norman A. Numerade Educator ### Problem 22 For the following exercises, consider an athlete running a$40-\mathrm{m}$dash. The position of the athlete is given by$d(t)=\frac{t^{3}}{6}+4 t, \quad$where$d$is the position in meters and$t$is the time elapsed, measured in seconds. IT] Compute the average velocity of the runner over the given time intervals. a.$[1.95,2.05]$b.$[1.995,2.005]$c.$[1.9995,2.0005]$d.$[2,200001]$ Amy J. Numerade Educator ### Problem 23 For the following exercises, consider an athlete running a$40-\mathrm{m}$dash. The position of the athlete is given by$d(t)=\frac{t^{3}}{6}+4 t, \quad$where$d$is the position in meters and$t$is the time elapsed, measured in seconds. Use the preceding exercise to guess the instantaneous velocity of the runner at$t=2 \mathrm{sec} .$ Barsha R. Numerade Educator ### Problem 24 For the following exercises, consider the function$f(x)=|x| .$Sketch the graph of$f$over the interval$[-1,2]$and shade the region above the$x$-axis. Amy J. Numerade Educator ### Problem 25 For the following exercises, consider the function$f(x)=|x| .$Use the preceding exercise to find the exact value of the area between the$x$-axis and the graph of$f$over the interval$[-1,2]$using rectangles. For the rectangles, use the square units, and approximate both above and below the lines. Use geometry to find the exact answer. Norman A. Numerade Educator ### Problem 26 For the following exercises, consider the function$f(x)=\sqrt{1-x^{2}}$. (Hint: This is the upper half of a circle of radius 1 positioned at$(0,0) . )$Sketch the graph of$f$over the interval$[-1,1]$ Amy J. Numerade Educator ### Problem 27 For the following exercises, consider the function$f(x)=\sqrt{1-x^{2}}$. (Hint: This is the upper half of a circle of radius 1 positioned at$(0,0) . )$Use the preceding exercise to find the exact area between the$x$-axis and the graph of$f$over the interval$[-1,1]$using rectangles. For the rectangles, use squares 0.4 by 0.4 units, and approximate both above and below the lines. Use geometry to find the exact answer. Carson M. Numerade Educator ### Problem 28 For the following exercises, consider the function$f(x)=-x^{2}+1$Sketch the graph of$f$over the interval$[-1,1]$ Amy J. Numerade Educator ### Problem 29 For the following exercises, consider the function$f(x)=-x^{2}+1$Approximate the area of the region between the$x$-axis and the graph of$f$over the interval$[-1,1] .\$ Carson M.