Eliminate the parameter for the following. What do these parametric equations represent in the $x-y$ plane?

$x = \sin(t) \quad$ $y = \cos(t)$

Identify the second derivative,$\dfrac{dy^2}{d^2x}$ for the parametric equations below.

$x = \sin(3t) \quad$ $y = 4\cos(3t)$

Identify where there exist horizontal tangents for the parametric equations below.

$x = \ln(t^2) − 6t^3 \qquad$ $y = 4t^2 − 5\ln(t)$

Identify the velocity and acceleration functions of the particle moving according to the position function below.

$s(t) = 4t^3 + 3\ln(t)$

Let $s(t)$ represent the position of a particle with respect to time $t$. Find the distance traveled by the particle from $t = 0$ to $t = 4$

$s(t) = t^2 − 2t − 3$

Determine the displacement of the particle with position function $s(t)$ from $t = 2$ to $t = 5$

$s(t) = −\sin\left(\dfrac{π}{2}t\right)$

Identify the speed of the particle at the given time. Do not worry about units.

$s(t) = −4t^2, \quad$ $t = 4$

Identify the position function for the following. It is known that $s(0) = 1$

$v(t) = \sqrt{\tan^2(t) + 1}$

Sand is pouring out of a bag at a constant rate of $4 \text{ cm}^3 \text{ s}^{-1}$, which forms a conical pile on the ground beneath the bag. The cone has a radius that is equal to its height. Identify the rate of change of the radius of the cone when its radius is $2 \text{ cm}$.

A $12$-foot ladder slides down a wall at a rate of $2$ feet per second. At what rate is the foot of the ladder sliding away from the base of the wall when the ladder is $4$ feet off the ground?

Find the volume of the solid that is the result of the revolution of the curve $y = x^2$ around the $x$-axis on the domain $0 ≤ x ≤ 3$.

Find the volume of the solid formed by the revolution of the curve $y = x^3 − 3x − 1$ bounded by $x = 0$, $x = 2$, and $y = 1$ around the $y$-axis in cm$^3$.

Find the volume in cubic inches of the solid formed by the revolution of the region bounded between the curves $y = x$ and $y = x^2$ around the $x$-axis.

Find the area of the solid formed by the function $f(x)$ from $0 ≤ x ≤ 4$, bounded by the $y$-axis, and with square cross sections. The side of each cross section is equal to the height of the function above the $x$-axis.

$f(x) = − (x − 2)^2 + 4$

Identify the integral necessary to calculate the volume formed by the revolution of the curve made up of the given cross section. Do not evaluate the integral. The cross sections are bounded by the curve and the $x$-axis, where the radius of the circle is the height of the function.

$y = e^x$ $\text{with}$ $\text{semi-circular}$ $\text{cross}$ $\text{sections.}$ $−1 ≤ x ≤ 1$