Find the instantaneous of change of the volume of a cube with respect to its edge length $x$ as $x$ changes from

$x=5$.

Find the average rate of change of the volume of a cube with respect to its edge length $x$ as $x$ changes from

(i) 5 to 6

(ii) 5 to $5.1$

(iii) 5 to $5.01$

Find the differential $dy$ given $y=\left(x^2+5\right)^3, \quad x=1, d x=0.05$ .

Find an equation of the tangent to the curve at the given point or endpoint of the vector without eliminating the parameter.

$$\mathbf{r}(t)=5 \cos t \mathbf{i}+5 \sin t \mathbf{j} ; \quad 3 \mathbf{i}+4 \mathbf{j}$$

Use Newton's method to find the root of the equation $x^4+x-1=0$ in the interval $[0,1]$ correct to six decimal places.

Find the linear and quadratic approximations to $f(x)=\sqrt{25-x^2}$ near 3.

For what values of $x$ is the linear approximation accurate to within $0.1$ ?

Find the linear and quadratic approximations to $f(x)=\sqrt{25-x^2}$ near 3.

Illustrate part given above by graphing $f$ and both approximations.

Find the linear and quadratic approximations to $f(x)=\sqrt{25-x^2}$ near 3.

Find $d y$ if $y=\left(4-x^2\right)^{3 / 2}$.

A plane flying horizontally at an altitude of $1 \mathrm{mi}$ and a speed of $500 \mathrm{mi} / \mathrm{h}$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $2 \mathrm{mi}$ away from the station.

Find $f^{\prime}(a)$.

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

Differentiate each function.

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

Find the derivative of the function $f(t)=\left(2 t^2-6 t+1\right)^{-8}$.

Find each limit

$$\lim _{\theta \rightarrow 0} \frac{\sin (\cos \theta)}{\sec \theta}$$