Using, estimate $\lim _{x \rightarrow 0} f(x)$.

$$\begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ \hline-0.1 & 2.875 \\ \hline-0.01 & 2.92 \\ \hline-0.001 & 2.998 \\ \hline 0 & \text { Undefined } \\ \hline 0.001 & 2.9987 \\ \hline 0.01 & 2.865 \\ \hline 0.1 & 2.78145 \\ \hline 0.15 & 2.678 \\ \hline \end{array}$$

For the following exercise,

The volume $V$ of a sphere with respect to its radius $r$ is given by $V=\frac{4}{3} \pi r^3$.

Find the instantaneous rate of change of V when r = 3 cm.

What is the left-hand limit of the function as $x$ approaches $0$ as we discussed earlier?

What is the right-hand limit of the function as $x$ approaches $0$ as we discussed earlier?

For the following exercise,

The volume $V$ of a sphere with respect to its radius $r$ is given by $V=\frac{4}{3} \pi r^3$.

Find the average rate of change of V as r changes from $1$ cm to $2$ cm.

Find an equation that could be represented by as we discussed earlier.

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$s(0)=0, s(5)=0$.

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$$s^{\prime}(3)=-16$$

Find an equation that could be represented by as we discussed earlier.

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$$s(3)=96$$

For the following exercise,

Find the average rate of change $\frac{f(x+h)-f(x)}{h}$.

$$f(x)=\sqrt{x}$$

Allow the speed of light, $c$, to be equal to $1.0$. If the mass, $m$, is $1$, what occurs to $m$ as $v \rightarrow c$? Using the values listed as we discussed earlier, make a conjecture as to what the mass is as $v$ approaches $1.00$.

$$\begin{array}{|c|c|} \hline \boldsymbol{v} & \boldsymbol{m} \\ \hline 0.5 & 1.15 \\ \hline 0.9 & 2.29 \\ \hline 0.95 & 3.20 \\ \hline 0.99 & 7.09 \\ \hline 0.999 & 22.36 \\ \hline 0.99999 & 223.61 \\ \hline \end{array}$$

For the following exercise,

Explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s(t)=-16 t^2+80 t$.

$$s^{\prime}(2)=16$$

For the following exercise,

Find the average rate of change $\frac{f(x+h)-f(x)}{h}$.

$$f(x)=\frac{1}{x^2}$$