The cost function C(x) and the selling price of each article, p(x) are given below. Determine the profit function P(x).

$C\left(x\right)=-0.03x^2+70x+75$

$p\left(x\right)=80$

Find the piecewise formula of the graph below.

For the function $g\left(x\right)=\mid 3x-6\mid +2$, what is the largest interval on which $g\left(x\right)$ is increasing?

A telecommunications company needs to install a fiber optic cable from a central office to a new development $3$ miles away on the opposite side of a lake. The lake is $1200$ ft wide. The cost of laying the cable under the lake is \($250$ per foot and \)$150$ per foot along the road. Determine the least expensive point for the cable to reach the development if it is less than or greater than $3000$ ft from the starting point along the road.

Apply the transformations on the graph of $p(x)=\sqrt{x}$ into the graph of $h(x)=4\sqrt{x-3}-6$ . Check your work with the help of a graphing calculator.

Simplify the composite function $a\left(b\left(c\left(x\right)\right)\right)$ given the following:

$a\left(x\right)=-x^3$

$b\left(x\right)=\cos\left(x\right)$

$c\left(x\right)=\sqrt{x}$

Write the logarithm $\log_5120$ in terms of the natural logarithm. Evaluate the value of the new logarithm with the help of a calculator. Write your answer using four decimal places.

Use a graphing calculator to decide if these functions differ by a constant.

$f\left(x\right)=\ln \left(\frac{x+5}{x-5}\right)$ and $g\left(x\right)=\ln \left(\frac{x+5}{x-5}\right)$

Given that $\log_{a}\left(p\right)=0.42$ and $\log_{a}\left(r\right)=0.91$ find the value of $\log_{a}\frac{\sqrt{p}}{\sqrt[4]{r}}$.

For the given function, find the inverse.

$f(x) = -5x + 3$

A runner's position along a track is recorded at various times during a race. The data is presented in a table. Find the runner's average velocity between the time interval of 0 to 2 seconds.

Find the value of $\underset{x\to 0}{\lim }f\left(x\right)$ given that $\underset{x\to 3}{\lim }\left(2x\underset{x\to 0}{\lim }f\left(x\right)\right)=-12$.

A coffee shop charges for Wi-Fi access based on the time used. The graph of the charge function $c$ for $0\le t\le 40$ is given below. Write the intervals of continuity of $c$.

The cost function $C(x)$ and the selling price of each article, $p(x)$ are given below. Determine the profit function $P(x)$.

$C(x)=-0.03x^2+70x+75$

$p(x) = 80 - 0.02x$

Determine the derivative of the function $f\left(x\right)=2x^3$ using the definition:

$f^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}$

Given a function $H\left(t\right)$ that models the height (in centimeters) of a plant $t$ days after it was planted, estimate $H^{\prime }\left(90\right)$ from the provided graph. What does this derivative indicate?

Determine whether the following piecewise function is differentiable at $x=0$.

The graphs of $f$ and $g$ are shown. Find $K^{\prime }\left(6\right)$ if $K=5f-2g$.

Calculate the fourth derivative of the function $f\left(x\right)=5\cos x$.

Determine the derivative of $h\left(x\right)=\frac{x^{\frac{7}{4}}}{x^{\frac{7}{4}}+5}$.

Given that $f(-1)=-1$, $f^{\prime }(-1)=-2$, $g(-1)=-1$, and $g^{\prime }(-1)=5$, find the derivative of $f^4\left(x\right)g\left(x\right)$ at $x=-1$.