1. Limits and Continuity

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\left\lbrack1,4\right\rbrack$

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-16t^2+128t$. Find the average velocity of the object over the following intervals.

$\left\lbrack1,2\right\rbrack$

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$. Find the average velocity of the object over the following intervals.

$\left\lbrack0,3\right\rbrack$

The position of an object moving vertically along a line is given by the function $s\left(t\right)=-4.9t^2+30t+20$. Find the average velocity of the object over the following intervals.

$\left\lbrack0,h\right\rbrack$, where $h\gt{0}$ is a real number

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 1^{-}}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 1^{+}}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

a. f(1)

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

d. ${\lim }_{x\to 1}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

h. ${\lim }_{x\to 3}f\left(x\right)$

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

l. ${\lim }_{x\to 2}f\left(x\right)$

Consider the position function s(t) =−16t^2+100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.

Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1. <IMAGE>

Consider the position function s(t)=−16t^2+100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3. <IMAGE>

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

$h\left(2\right)$

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

$h\left(4\right)$