Estimate the change in the volume of a cylinder when its radius changes from $r=2\text{ m}$ to $r=2.02\text{ m}$, given the height $h=5\text{ m}$. Note: $V\left(r\right)=\pi r^{2}h$.

Provide the differential expression $dy=f^{\prime }\left(x\right)dx$ for the following function:

$f\left(x\right)=2{\sin }^2\left(2x\right)$

Using Newton's Method, estimate the solution to the equation $x=e^{-x}$ in the interval $(0,1)$. Starting with an initial guess of $x_0=0.5$, what is the first approximation $\left(x_1\right)$ in two decimal places?

A money worth $\$3000$ is deposited in a $4-year$ savings account that pays a fixed annual interest rate $r$ (expressed as a decimal) with quarterly compounding. The amount in the account after $4$ years is given by $a\(\left\)(r\(\right\))=3000\(\left\)(1+\(\frac{r}{4}\)\(\right\))^{16}$. Find the value of $r$ (using Newton’s method) needed for the account to grow to $\$4000$ after $4$ years.

Write the differential of the surface area of a cone with a slant height $l$, when the slant height changes from $l$ to $l+dl$.

The length of a rectangular prism is measured to be $120\pm 0.5\text{ cm}$, width $80\text{ cm}$, and height $60\text{ cm}$. Estimate the percentage error in the volume calculation based on the uncertainty in the length measurement.