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${\displaystyle \tan \left(t\right)=\frac{1}{4}}$
Find ${\displaystyle \sin \left(t\right)}$, if terminal point of t is in Quadrant III.

Find the inverse the function and the domain and the range of f and ${\displaystyle f^{-1}}$, if the function is ${\displaystyle f\left(x\right)=\frac{2x}{x+5}}$ and it's one-to-one.

$\tan 28°=\cot$___

${\displaystyle \sin (2{\cos }^{-1}\cdot \left(\frac{\sqrt{2}}{2}\right)}$

${\displaystyle C\left(x\right)=5,000+20x-\frac{1}{4}{x}^2}$
is a cost function.
Calculate the marginal average cost when 20

(a), use your result to estimate the total cost when $x=21$

(b) and calculate the percentage error between this estimate and the actual cost when $x=21$

Find the differential of the following functions:
1. ${\displaystyle y=\tan \left(\sqrt{3t}\right)}$
2. ${\displaystyle y=5-\frac{v^2}{5}+v^2}$

if $x=2,f(x)=1$
if $x=3,f(x)=4$
if $x=5,f(x)=-2$
if $x=8,f(x)=3$
if $x=13,f(x)=6$
f is twice variable for all real numbers.
1. Find $f^{\prime }(4)$
2. Approximate ${\displaystyle {\underset{2}{\int }}^{13}{f}^{\prime }\left(x\right)dx}$
2. Find the value of ${\displaystyle {\underset{2}{\int }}^8(3-f^{\prime }\left(x\right))dx}$

If ${\displaystyle y=3\sqrt{x}+\frac{1}{\sqrt{x}}}$, find the differential dy.

Is the following function form ${\displaystyle \mathbb{Z}}$ to ${\displaystyle \mathbb{Z}}$ one-to-one?
1) ${\displaystyle f\left(n\right)=n-1}$
2) ${\displaystyle f\left(n\right)=n^2+1}$
3) ${\displaystyle f\left(n\right)=n^3}$
4) ${\displaystyle f\left(n\right)=\lceil n∕2\rceil }$

Calculate the average rate of change for the function ${\displaystyle f\left(x\right)=x^2+x-12}$, over the interval:
1. [0,3]
2. [-1,10]

${\displaystyle \cot 2\left(\sin \left(\theta \right)\right)}$
Find the derivative.

A ring lies in the xy plane, centered at the origin. It has a radius of R and a uniformly distributed total charge Q. Due to the ring on the z-axis, as a function of z, what is the potential V(z)?

Find the arc length of the astroid ${\displaystyle x\left(t\right)={\cos }^{3}t,y\left(t\right)={\sin }^{3}t}$.
Calculate the arc length of ${\displaystyle \frac{1}{4}}$ of the astroid ${\displaystyle (0\le t\le \frac{\pi }{2})}$

When can and can't the two-sided limits exist?

${\displaystyle f\left(x\right)=\frac{2}{19+x}}$
Find the power series representation and the interval of convergence.

Name the two assumptions frequently made while estimating a cost function.

${\displaystyle \frac{\tan \alpha +\cot \alpha }{\sin \alpha -\cos \alpha }={\sec }^2\alpha +{\csc }^2\alpha }$

${\displaystyle \sin u=-\frac{24}{25}}$ (${\displaystyle u\in }$ Quadrant III)
${\displaystyle \cos v=-\frac{8}{17}}$ (${\displaystyle v\in }$ Quadrant II)
Evaluate the trigonometric function. Find ${\displaystyle \cos u,\sin v,\text{and}\sin (u-v)}$.

Find the exact value of each expression.
${\displaystyle \left(\tan \right)\frac{17\pi }{12}}$

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