1. Limits and Continuity

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

a. $[0, 2]$

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

c. $[0, 1]$

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 

a. s(t)=−16t^2+80t+60 at t=3

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 

c. s(t)=40 sin 2t at t=0

Tangent lines with zero slope

a. Graph the function f(x)=x^2−4x+3.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

a. Graph the position function, for 0≤t≤9.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)?

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.

a. When will the rock strike the ground? 

Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$.

Make two tables, one showing values of $g$ for $t=8.9,8.99$, and $8.999$ and one showing values of $g$ for $t=9.1,9.01$, and $9.001$.

Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$.

Make a conjecture about the value of ${\displaystyle\lim_{t\to9}\frac{t-9}{\sqrt{t}-3}}$.

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$. <IMAGE>

Calculate $g\left(x\right)$ for each value of $x$ in the following table.

Let $g\left(x\right)=\frac{x^3-4x}{8\left|x-2\right|}$. <IMAGE>

Make a conjecture about the values of ${\displaystyle\lim_{x\to2^{-}}g\left(x\right)}$, ${\displaystyle\lim_{x\to2^{+}}g\left(x\right)}$, and ${\displaystyle\lim_{x\to2}g\left(x\right)}$ or state that they do not exist.

Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.

$f\left(x\right)=\frac{x-2}{\ln \mid x-2\mid }$; $a=2$

Use a graph of f to estimate $\underset{x\to a}{\lim }f\left(x\right)$ or to show that the limit does not exist. Evaluate f(x) near $x=a$ to support your conjecture.

$f\left(x\right)=\frac{1-\cos (2x-2)}{{(x-1)}^2};a=1$