Suppose that a function f is differentiable at $x_0$ and that $f^{\prime}\left(x_0\right)>0$. Prove that there exists an open interval containing $x_0$ such that if $x_1$ and $x_2$ are any two points in this interval with $x_1<x_0<x_2$, then $f\left(x_1\right)<f\left(x_0\right)<f\left(x_2\right)$.

The side of a cube is measured to be 25 cm, with a possible error of $\pm 1 \mathrm{~cm}$.

Use differentials to estimate the error in the calculated volume.

On a certain clock the minute hand is 4 in long and the hour hand is 3 in long. How fast is the distance between the tips of the hands changing at 9 0'clock?

Use a graphing utility to make rough estimates of the locations of all horizontal tangent lines, and then find their exact locations by differentiating.

$$y=\frac{x^2+9}{x}$$

Show the following function which is given by

$$f(x)= \begin{cases}x^2 \sin (1 / x), & x \neq 0 \\ 0, & x=0\end{cases}$$

is continuous and differentiable at $x=0$. Sketch the graph of $f$ near $x=0$.

The side of a square is measured to be 10 ft, with a possible error of $\pm 0.1 \mathrm{ft}$.

Estimate the percentage errors in the side and the area.

The side of a square is measured to be 10 ft, with a possible error of $\pm 0.1 \mathrm{ft}$.

Use differentials to estimate the error in the calculated area.

Show that if k is a constant, then the following derivative which is given by

$$\frac{d}{d x}[\cos k x]=-k \sin k x$$

Use a graphing utility to make rough estimates of the locations of all horizontal tangent lines, and then find their exact locations by differentiating.

$$y=\frac{1}{3} x^3-\frac{3}{2} x^2+2 x$$

Show the following function which is given by

$$f(x)= \begin{cases}x \sin (1 / x), & x \neq 0 \\ 0, & x=0\end{cases}$$

is continuous but not differentiable at $x=0$. Sketch the graph of $f$ near $x=0$. (See Figure $2.6 .6$ and the remark following Example 5 in Section 2.6.)

A meteor enters the Earth's atmosphere and burns up at a rate that, at each instant, is proportional to its surface area. Assuming that the meteor is always spherical, show that the radius decreases at a constant rate.

Use the differential dy to approximate $\Delta y$ when x changes as indicated.

$y=x \sqrt{8 x+1}$; from x=3 to x=3.05

Without using any trigonometric identities, find the limit which is given by

$$\lim _{x \rightarrow 0} \frac{\tan (x+y)-\tan y}{x}$$

[Hint: Relate the given limit to the definition of the derivative of an appropriate function of $y$.]

Water is stored in a cone-shaped reservoir (vertex down). Assuming the water evaporates at a rate proportional to the surface area exposed to the air, show that the depth of the water will decrease at a constant rate that does not depend on the dimensions of the reservoir.