Calculate the volume formed by the revolution of the function below around the $y$-axis. The function is bounded by the $y$-axis, $y = 4$, and $x = 3.5$.

$f(x) = \dfrac{3}{(x + 1)^2} + e^x$

Identify the acceleration of the particle given by the position function below at time $t = 7.6$.

$s(t) = \ln(t^2) + \ln(\sin(t))$

Solve for $y$ given $\frac{dy}{dx} = tan^2⁡ (x), y(0) = 1$

Find the bounded area between the two functions below.

$f(x) = \sqrt{x}$

$g(x) = x^2$

How many critical points does the function below contain?

$f(x) = $ $\dfrac{1}{5}x^5 + \dfrac{1}{3}x^3 − \dfrac{7}{3}x^3 − \dfrac{1}{2}x^2 + 6x$

At what time(s) does the velocity of the particle given by the parametric equations below equal $0$ within the first $25$ seconds?

$y = 20 \sin(\sqrt{t})$

$x = 3t$

The average value of $f(x) = e^x$ on the interval $[1, 5]$ is equal to which of the following?

Identify the acceleration of a particle moving along the curve $s(x) = \ln (x^3 + \cos(x))$ at $x = 3$.

$\lim\limits_{n\to \infty} ⁡\dfrac{n^2}{2^n}$

Identify the equation of the tangent line to the function $f(x) = x \cdot \ln⁡(10x) \cdot \sin⁡(x)$ at $x = 3$.

A balloon being filled with air expands at $4$ cm$^3$ per minute. Assuming the balloon fills in a spherical shape, how fast is the radius of the spherical balloon increasing in cm per minute after $2.25$ minutes?

A soldier drops a bomb out of an airplane at $537$ feet of altitude traveling parallel to the ground. The bomb falls at a rate of $32$ feet per second due to gravity, while the plane travels away from the bomb at a rate of $731$ feet per second. At what rate is the distance between the plane and the bomb changing when the bomb hits the ground?

Find the position of the particle at time $t = 3$ for the particle given by the velocity function below. The particle is at $0$ when $t = 1$.

$v(t) = \sin((\ln(t))^2)$

Find the displacement of a particle traveling along the curve below from $t = 1$ to $t = 4.3$.

$s(t) = 4^x$

Find the slope of the line tangent to the parametric curve at $t = 6$.

$x = 4\sin(t^2)$

$y = e^t + t^2$