0. Functions

Given the Laplace transform $F\left(s\right)=\frac{1}{s^2+1}$, find the corresponding function $f\left(t\right)$.

If $f\left(x\right)=x^3$, which of the following is the value of $f(-2)$?

Let $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$, where $f$ is the function whose graph is shown. Which of the following statements is true about $g\left(x\right)$?

Given the sphere defined by the equation $x^2+y^2+z^2=9$, what is the intersection of this sphere with the yz-plane?

Find the exact length of the curve $y=\frac{1}{4}{x}^2-\frac{1}{2}\ln \left(x\right)$ for $1\le x\le 4$.

Evaluate the vector integral from $t=0$ to $t=4$ of the vector function $sec\left(t\right)tan\left(t\right)\mathbf{i}+tcos\left(2t\right)\mathbf{j}+{sin}^2\left(2t\right)cos\left(2t\right)\mathbf{k}$ with respect to $t$.

Evaluate the line integral of the vector field $F(x,y)=(x+7y,x^2)$ along the curve $C$, where $C$ is the line segment from $(0,0)$ to $(1,2)$. Which of the following is the value of the integral ${\int }_C^{}(x+7y)dx+x^{2}dy$?

What is the area enclosed by the curves $y=x^3-8x^2+18x-5$ and $y=x+5$?

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = tan(2x - π)

Evaluate the line integral of the vector field $F(x,y)=(2x,3y)$ along the path $C$ given by $r\left(t\right)=(t,t^2)$ for $0\le t\le 1$.

Consider the geometric series with first term $2$ and common ratio $\frac{1}{2}$. Write out the first four terms of the series: $2,1,\frac{1}{2},\frac{1}{4}$. What is the sum of the infinite series?

Evaluate the integral $\int \frac{5x^2\sqrt{x}-4}{x^2} dx$.

Which of the following is a power series representation for the function $f\left(x\right)=x^8\arctan \left(x^3\right)$ centered at $x=0$?

If $f\left(x\right)=x^2$, which of the following is the value of $f(-3)$?

Express the radius of a sphere as a function of the sphere’s surface area. Then express the surface area as a function of the volume.