An initial amplitude k, damping constant c, and frequency f or period p are given. (Recall that frequency and period are related by the equation f=1 / p.)

(a) Find a function that models the damped harmonic motion. Use a function of the form $y=k e^{-c t} \sin \omega t$ in Exercises given below.

(b) Graph the function.

$$k=1, \quad c=1, \quad p=1$$

Graph each equation. $5 y+6 x=11$

Use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. $f(x)=2+3^{x / 2}$

Find the sum. 7 + (-2)

Find the sum. -54 + 40

Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations.

$$y = 2 - \sqrt { x + 1 }$$

Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations.

$$y = \sqrt { x + 4 } - 3$$

Use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. $f(x)=1-2^{x+3}$

In the problem below, the function f is differentiable and $f_x(2,1)=-3, f_y(2,1)=4$, and $f(2,1)=7$.

Find $f_r(2,1)$ and $f_\theta(2,1)$, where $r$ and $\theta$ are polar coordinates, $x=r \cos \theta$ and $y=r \sin \theta$. If $\vec{u}$ is the unit vector in the direction $2 \vec{i}+\vec{j}$, show that $f_{\vec{u}}(2,1)=f_r(2,1)$ and explain why this should be the case.

Find the sum. - 25 + 50.

Solve the equation. $7+3 c=-11$

Is the urinary bladder in the abdominal cavity?

Find the sum. $353.19 +$7,837

Determine the transformations you must perform on the graph of $f(x)=\log _b(x)$ to obtain the graph of $g(x)=a \log _b(x-h)+k$.