0. Functions

Evaluate the triple integral ${\int }_{}^{}{\int }_{}^{}{\int }_{}^{}z{d}_V$, where E is the region bounded by the parabolic cylinder $y=x^2$, the planes $z=0$, $z=4$, and $y=4$.

Which of the following best describes the contour lines (level curves) of the function $f(x,y)=x^2+7y^2$?

Consider the function $z=e^x\cos \left(y\right)$. Which of the following best describes the domain of this function?

If $f\left(x\right)=2x^2+1$, what is the value of $f\left(3\right)$?

Given the function $f\left(x\right)=x^2$ defined on the interval $0\le x\le 2$, determine the y-coordinate of the centroid of the region bounded by the graph of $f\left(x\right)$, the x-axis, and the lines $x=0$ and $x=2$.

Find the third-degree Taylor polynomial, $t_3\left(x\right)$, for the function $f\left(x\right)=\frac{1}{x}$ centered at $a=2$.

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)