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

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

$p(x) = 80 - 0.02x$

A tree is planted. The maximum possible height of the tree is $12.5$ feet. The height of the tree is given by the following equation.

$y=\frac{500}{40+2500e^{-0.75x}}$, where $x$ is in months and $y$ is in feet.

Draw the graph of the function.

Find the value of $f^{\prime }\left(a\right)$ for the function $f\left(x\right)=\sqrt{4x-1}$ when $a=2$.

Use the following limit definition to determine the slope of the line tangent to the graph of $f$ at $P$, where $f\left(x\right)=\frac{4}{3x-2}$ and $P(0,-2)$:

$m_{\text{tan}}=\underset{h\to 0}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$

The function $k$ and point $V$ are given. Determine all points $P$ on the graph of $k$ such that the line tangent to $k$ at $P$ passes through $V$.

$k\left(x\right)=x^2+2x+1$; $V(4,9)$

Simplify the difference quotient $\frac{f\left(x\right)-f\left(a\right)}{x-a}$ for the function $f\left(x\right)=2x^3-\frac{1}{x}$.

Determine the tangent and normal lines to the function below at a specified point.

${(x+y)}^2=4x+6$, $(1,3)$