The region bounded by $y=4x$, $y=2$ and the $y$-axis is revolved about the $y$-axis. What is the volume?

Find the derivative of $p(x)=\ln \left(-4x+2\right)$, on an interval where the argument is positive.

Use the Direct Comparison Test with ${\sum }_{n=1}^{\infty }\frac{1}{2^n}$ to decide ${\sum }_{n=1}^{\infty }\frac{1}{2^n+6}$.

A spherical balloon is inflated so that its radius grows at $5$ centimeters per second, with $V=\frac{4}{3}\pi r^3$. How fast is the volume changing when the radius is $7$ centimeters? Give the answer in cubic centimeters per second.

Find $\int 5x{e}^{4x}dx$.

A particle has position $r(t)=⟨e^{-t},t^2-3t⟩$. Find its VELOCITY vector.

Let $\frac{dy}{dx}=y+1$ with $y(0)=2$. Use Euler's Method with TWO steps of size $h=0.25$ to approximate $y(0.5)$.

Which of these p-series CONVERGES?

Let $\frac{dy}{dx}=x-y$ with $y(1)=4$. Use Euler's Method with ONE step of size $h=0.5$ to approximate $y(1.5)$.

Evaluate ${\int }_4^7\frac{1}{x\left(x-1\right)}dx$.

Evaluate ${\lim }_{x\to \infty }\frac{2x^2+9x}{9x^2+9}$.

Find $\int \frac{4}{\left(x+3\right)\left(x+4\right)}dx$.

On which set is $r(x)=2x^3-54x$ CONCAVE UP?

Use the definition $w^{\prime }(x)={\lim }_{h\to 0}\frac{w(x+h)-w(x)}{h}$ to find $w^{\prime }(x)$ for $w(x)=4x^2-6x-8$.

Apply the Limit Comparison Test to ${\sum }_{n=1}^{\infty }\frac{4n+5}{7n^2-n}$ using $b_n=\frac{1}{n}$. What is the limit, and what does it establish?

A particle has velocity $v(t)=⟨5t,12⟩$. What is its SPEED at $t=1$?

A particle moves with $x=2t^2-1$ and $y=t^2+2t$. What is its SPEED at $t=2$?

Evaluate ${\lim }_{x\to \infty }\frac{2x+2}{x^2+5}$.

A rectangle has perimeter $20$ centimeters, so a side of length $x$ gives area $A(x)=x\left(10-x\right)$ for $0\le x\le 10$. What is the greatest possible AREA, in square centimeters?

A particle has position $s(t)=2t^3-3t^2-72t$ meters at time $t$ seconds. Find the acceleration at $t=-1$.

Evaluate ${\lim }_{x\to 4}\frac{x^2-16}{x-4}$.

Find $\frac{dy}{dx}$ for the relation $5x^2-4y^2=-8$.

Find the equation of the line tangent to $g(x)=-3x^2-8x$ at $x=6$.

Evaluate ${\int }_0^{\infty }4e^{-5x}dx$.

Find $\int 3x\cos xdx$, with $x$ in radians.

A curve is given by $x=2t^2-1$ and $y=t^4$. Find $\frac{dy}{dx}$.

Solve $\frac{dy}{dx}=6xy$ with $y(0)=7$, giving $y$ explicitly.

Find every critical point of $r(x)=2x^3+6x^2$.

Find the area of ONE petal of $r=2\sin \left(5\theta \right)$, with $\theta$ in radians.

Find the derivative of $f(x)=-6x^{5/3}$.

A particle has position $s(t)=2t^3+9t^2$ at time $t$. Is it speeding up or slowing down at $t=1$?

Find the EXACT length of $y=\frac{2}{3}{x}^{3/2}$ from $x=0$ to $x=8$.

For $\frac{dP}{dt}=0.2P\left(1-\frac{P}{800}\right)$, with $P$ in deer and $t$ in years, at what population is the population growing FASTEST, and how fast is it growing then?

What is the INTERVAL of convergence of ${\sum }_{n=1}^{\infty }\frac{{\left(x-1\right)}^n}{n^2{2}^n}$? Give your answer in interval notation.

Find the value of $c$ the Mean Value Theorem guarantees for $q(x)=x^2+3x$ on $\left[2,8\right]$.

Find the derivative of $w(x)={\left(2x^4+7\right)}^7$.

What does the nth-Term Test tell you about ${\sum }_{n=1}^{\infty }\frac{2n^2+6}{9n^2-7n}$?

Evaluate ${\lim }_{x\to 2}\frac{x^2-4}{x^2+3x-10}$ using L'Hôpital's Rule.

For $x=2t+1$ and $y=t^2+2t$, find $\frac{dy}{dx}$ at $t=-3$.

Find the derivative of $v(x)=e^{-7x^3-2}$.

Find the derivative of $s(x)=\left(5x+4\right)\cos x$.

What is the coefficient of $x^3$ in the Maclaurin series for $f(x)=\ln \left(1+x\right)$?

Find $\int x\sqrt{x^2+3}dx$.

Classify every critical point of $r(x)=2x^3-9x^2$.

Find the value of $c$ the Mean Value Theorem guarantees for $f(x)=x^2-4x$ on $\left[-5,5\right]$.

What is the average value of $f(x)=x^2$ on $\left[1,4\right]$?

A particle has velocity $v(t)=⟨20t^3,21t^2⟩$. What is its SPEED at $t=1$?

A particle has position $s(t)=2t^3-3t^2-12t$ at time $t$. Is it speeding up or slowing down at $t=-2$?

The FOURTH-degree Maclaurin polynomial for a function $f$ is used to approximate $f\left(0.4\right)$. Taking $M=1$, what is the Lagrange error bound, rounded to four decimal places?

Solve $\frac{dy}{dx}=\frac{x}{y}$ with $y(0)=5$, giving $y$ explicitly.

Classify every critical point of $g(x)=2x^3-15x^2+36x$.

Find the derivative of $g(x)=6x^{8/5}$.

A particle has position $r(t)=⟨\cos t,t^2⟩$. Find its VELOCITY vector.

Find the area of the region bounded by $y=x^3+1$, the $x$-axis, $x=1$ and $x=5$.

What is the THIRD-degree Taylor polynomial for $f(x)=\ln x$ about $x=4$?

Evaluate ${\int }_0^{\infty }6e^{-x}dx$.

Let $\frac{dy}{dx}=2y$ with $y(0)=3$. Use Euler's Method with TWO steps of size $h=0.5$ to approximate $y(1)$.

A particle has position $s(t)=2t^3+6t^2$ meters at time $t$ seconds. Find the velocity at $t=3$.

Find the area of ONE petal of $r=4\sin \left(3\theta \right)$, with $\theta$ in radians.

Use the Direct Comparison Test with ${\sum }_{n=1}^{\infty }\frac{1}{n}$ to decide ${\sum }_{n=1}^{\infty }\frac{1}{n-5\sqrt{n}}$, for $n$ large enough that every term is positive.