A baseball is given an initial velocity with magnitude $v_0$ at an angle $\phi$ above the surface of an incline, which is in turn inclined at an angle $\theta$ above the horizontal (Fig. ) (a) Calculate the distance, measured along the incline, from the launch point to where the baseball strikes the incline. Your answer will be in terms of $v_0, 8, \theta$, and $\phi$. (b) What angle $\phi$ gives the maximum range, measured along the incline?

use a calculator to evaluate each function. Round your answers to four decimal places.(Be sure the calculator is in the correct angle mode.) $\cot 79.56^{\circ}$

Writing

$$f ^ { \prime \prime \prime } = \left( f ^ { \prime \prime } \right) ^ { \prime }$$

, consider the third order differential equation

$$f ^ { \prime \prime \prime } - a _ { 1 } f ^ { \prime \prime } - a _ { 2 } f ^ { \prime } - a _ { 3 } f = 0$$

where

$$a _ { 1 } , a _ { 2 }$$

, and

$$a_3$$

are real numbers. Let

$$f _ { 1 } = f, f _ { 2 } = f ^ { \prime } - a _ { 1 } f$$

and

$$f _ { 3 } = f ^ { \prime \prime } - a _ { 1 } f ^ { \prime } - a _ { 2 } f ^ { \prime \prime }$$

. (a) Show that

$$\left[ \begin{array} { l } { f _ { 1 } } \\ { f _ { 2 } } \\ { f _ { 3 } } \end{array} \right]$$

is a solution to the system

$$\left\{ \begin{array} { l } { f _ { 1 } ^ { \prime } = a _ { 1 } f _ { 1 } + f _ { 2 } } \\ { f _ { 2 } ^ { \prime } = a _ { 2 } f _ { 1 } + f _ { 3 } } \\ { f _ { 3 } ^ { \prime } = a _ { 3 } f _ { 1 } } \end{array} \right.$$

, that is

$$\left[ \begin{array} { l } { f _ { 1 } ^ { \prime } } \\ { f _ { 2 } ^ { \prime } } \\ { f _ { 3 } ^ { \prime } } \end{array} \right] = \left[ \begin{array} { l l } { a _ { 1 } } & { 1 } & { 0 } \\ { a _ { 2 } } & { 0 } & { 1 } \\ { a _ { 3 } } & { 0 } & { 0 } \end{array} \right] \left[ \begin{array} { l } { f _ { 1 } } \\ { f _ { 2 } } \\ { f _ { 3 } } \end{array} \right]$$

. (b) Show further that if

$$\left[ \begin{array} { l } { f _ { 1 } } \\ { f _ { 2 } } \\ { f _ { 3 } } \end{array} \right]$$

is any solution to this system, then

$$f=f_1$$

is a solution to

$$( * )$$

. Remark. A similar construction casts every linear differential equation of order n (with constant coefficients) as an

$$n \times n$$

linear system of first order equations. However, the matrix need not be diagonalizable, so other methods have been developed.

Sketch the graph of a function that satisfies all of the given conditions.

$f^{\prime}(x)>0$ if $|x|<2, \quad f^{\prime}(x)<0$ if $|x|>2$, $f^{\prime}(-2)=0, \quad \lim _{x \rightarrow 2}\left|f^{\prime}(x)\right|=\infty, \quad f^{\prime \prime}(x)>0$ if $x \neq 2$

Triangle $A B C$ with vertices $A(3,0), B(3,4)$, and $C(0,0)$ is rotated $90^{\circ}$ clockwise about the origin to its image $\triangle A^{\prime} B^{\prime} C^{\prime}$. Graph both triangles. What are the coordinates of the vertices of $\Delta A^{\prime} B^{\prime} C^{\prime}$ ?

What do basting, brushing, and glazing have in common?

Projectile Motion The path of a projectile thrown at an angle of $45\degree$ with level ground is

$$y=x-\frac{32}{v_0^2}\left(x^2\right)$$

where the initial velocity is $v_0$ feet per second.

(a) Find the $x$-coordinate of the point where the projectile strikes the ground. Use the symmetry of the path of the projectile to locate the $x$-coordinate of the point where the projectile reaches its maximum height.

You randomly choose one of the marbles. Without replacing the first marble, you choose a second marble. a. Name two ways you can find the total number of possible outcomes. b. Find the total number of possible outcomes.

Certain automobile license plates consist of a sequence of three letters followed by three digits.

If no letters and no digits are repeated, how many license plates are possible?

Sketch the graph of a function $f$ having the indicated characteristics.
$f(0)=4, f(6)=0$ $f^{\prime}(x)<0$ if $x<2$ and $x>4$ $f^{\prime}(2)$ does not exist.

$$f^{\prime}(4) =0 \\ f^{\prime}(x) >0 \text { if } 2<x<4 \\ f^{\prime \prime}(x) <0, x \neq 2$$

A ball, when dropped from any height, bounces $\frac{1}{3}$ of the original height. If the ball is dropped, bounces back up, and continues to bounce up and down so that it has traveled $106$ feet when it strikes the ground for the fourth time, what is the original height from which it was dropped?

During a $6.0$-s time interval, a flywheel with a constant angular acceleration turns through 500 radians that acquire an angular velocity of $100 \mathrm{rad} / \mathrm{s}$. (a) What is the angular velocity at the beginning of the $6.0 \mathrm{~s}$ ? (b) What is the angular acceleration of the flywheel?

Suppose that an object is thrown upward with an initial velocity of $64$ feet per second off the edge of a $960$ -foot cliff. The height $h(t)$ in feet of the object after $t$-seconds is given by the function

$$h(t)=-16 t^2+64 t+960$$

Determine how long before the object strikes the ground.

The velocity of a (fast) automobile on a straight highway is given by the function

$$v(t)= \begin{cases}3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45\end{cases}$$

where $t$ is measured in seconds and $v$ has units of $\mathrm{m} / \mathrm{s}$. What is the distance traveled by the automobile in the first $30 \mathrm{~s}$ ?