A 0.26-kg rock is thrown vertically upward from the top of a cliff that is 32 m high. When it hits the ground at the base of the cliff, the rock has a speed of 29 m/s. Assuming that air resistance can be ignored, find (a) the initial speed of the rock and (b) the greatest height of the rock as measured from the base of the cliff.

A skateboarder starts at point $\mathrm{A}$ in Figure $8-27$ and rises to a height of $2.64 \mathrm{~m}$ above the top of the ramp at point $B$. What was the skateboarder's initial speed at point A?

Meteorite On October 9, 1992, a 27-pound meteorite struck a car in Peekskill, NY, creating a dent about $22 \mathrm{~cm}$ deep. If the initial speed of the meteorite was $550 \mathrm{~m} / \mathrm{~s}$, what was the average force exerted on the meteorite by the car?

Perform the indicated computations. Write the answers in scientific notation.

$$\left(4 \times 10^6\right)^{-1}$$

Find dy/dx. You can use NDER to support your computations.

$$y = log_3 (1+x ln 3)$$

A rock is dropped from a height of 576 ft and falls toward Earth in a straight line. In t seconds the rock drops a distance of $s=16 t^2 \mathrm{ft}$. What is the instantaneous velocity of the rock when it hits the ground?

An astronaut standing on the moon throws a rock upward. The height of the rock is

$$s=-\frac{27}{10}t^2+27t+6$$

where $s$ is measured in feet and $t$ is measured in seconds.

(a) Find expressions for the velocity and acceleration of the rock.

Determine the rectangular coordinates of the vertices of the ellipse whose polar equation is $r=2 /(2-\sin \theta)$.

A rock is dropped from a window and falls to the ground below. The height, $s$ (in meters), of the rock above ground is a function of the time, $t$ (in seconds), since the rock was dropped, so $s = f(t)$.

Sketch a possible graph of $s$ as a function of $t$.

The following alternating series are divergent (but you are not asked to prove this). Show that $a_{n} \rightarrow 0$. Why doesn’t the alternating series test prove (incorrectly) that these series converge? (a) $2-\frac{1}{2}+\frac{2}{3}-\frac{1}{4}+\frac{2}{5}-\frac{1}{6}+\frac{2}{7}-\frac{1}{8} \cdots$ (b) $\frac{1}{\sqrt{2}}-\frac{1}{2}+\frac{1}{\sqrt{3}}-\frac{1}{3}+\frac{1}{\sqrt{4}}-\frac{1}{4}+\frac{1}{\sqrt{5}}-\frac{1}{5} \cdots$

Two rocks are thrown off a cliff. One rock ($1$) is thrown horizontally with a speed of $20 \mathrm{~m} / \mathrm{s}$. The other rock $(2)$ is thrown at an angle $\theta$ relative to the horizontal with a speed of $30 \mathrm{~m} / \mathrm{s}$. While the two rocks are in the air, rock $2$ is always directly above $1$. Find $\theta$.

The inverse of a relation is created by interchanging the x- and y- coordinates of each ordered pair in the relation. The statement “If a relation is a function, then the inverse of the relation is also a function” is sometimes true. Give an example of a relation and its inverse that are both functions. Then give an example of a relation and its inverse that are both not functions.

Complete the details of the following argument: If

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

converges to a finite sum s, then $\frac{1}{2} s=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\cdots$ and $s-\frac{1}{2} s=1+\frac{1}{3}+\frac{1}{5}+\cdots.$ Why does this lead to a contradiction?

2. If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, Babylonian geometers approximated the length of the hypotenuse by the formula $c=b+\left(\frac{a^2}{2b}\right)$. (i) How does this approximation compare to the actual result when a=3 and b=4? When a=5 and b=12? When a=12 and b=5? (ii) Give an algebraic argument demonstrating that this formula results in an approximation that is too large.