Sketch the graph of the function $g\left(x\right)=2\sqrt{\mid x-7\mid }$ and its vertical tangent line.

Suppose $g\left(x\right)=2\sqrt{x+2}$. Use limits to determine the exact value of $g^{\prime }\left(2\right)$.

Given a function f(x) with a jump discontinuity at x = -1, what feature will the derivative graph f'(x) exhibit at this point?

Determine the values of $x\in (-3,3)$ at which $f$ is not differentiable using the following graph:

Find the derivative of the function $f\left(x\right)=5x^{\sqrt{7}}$.

Determine the sixth derivative of the given function.

$f\left(x\right)=\frac{8x^5}{15}$

Calculate the derivative of the given function.

$f\left(x\right)=\frac{3e^x+2}{3e^x-2}$

Find the derivative of $y=\sec \left(\tan x\right)$.

Find the derivative of $g\left(f\left(x\right)\right)$ when $x=-2$ using the table given below.

Let $R$ be the region bounded by $y=\frac{1}{2}\sqrt{9-x^2}$, $y=0$, $x=0$, and $x=2$. Find the volume of the solid generated when $R$ is revolved about the $x$-axis.

A chemical plant produces a compound at a variable rate given by

,

where $t$ is measured in hours and $R$ in kilograms. How many kilograms are produced in the first $10$ hours?

If it takes $25\text{ N}$ to hold a spring stretched $0.1\text{ m}$ beyond its equilibrium, how much work is required to stretch the spring $0.2\text{ m}$ from equilibrium?

What is the integral of 6 * 4^x dx using the constant multiple rule?

Evaluate the integral \( \int \left( \frac{4}{x} + \frac{1}{x^3} \right) \, dx \) by splitting it into simpler integrals.

Evaluate ∫ 5/√(1 - x^2) dx using the constant multiple rule.

What is the integral of \( \int x^0 \, dx \) using integration by parts?

What condition must be met for a rational function to be decomposed into partial fractions?

If we want to receive $\$6000$ per year forever from a continuously compounded account with a $9\%$ annual interest rate, how much should we deposit now? Write your answer as a whole number.