Evaluate the derivative of the following function.

$f(x) = \ln(2x) + 6$

Identify the equation of the line tangent to the point at $x = 0$ for the function below.

$f(x) = 4\arctan(2x)$

$\displaystyle\int [\tan(x) + sec^2(x)]dx =$

$\displaystyle\int_0^1 \dfrac{4}{1 + x^2}dx =$

Identify the time where the position function below reaches its minimum velocity, $t ≥ 0$.

$s(t) = t^4 − 3t^3 − 6t^2 + 27t$

A plane in horizontal (level) flight is travelling at a speed of 500 feet per second. The plane passes 100 feet directly above the eyes of a boy who watches the plane as it flies away from him.

At what rate is the angle of the boy’s eyes changing when the plane has travelled 100 feet horizontally from the point where it passed directly overhead?

$\dfrac{d}{dx} \left[\displaystyle\int_0^{x^2 + 7x} \sin(t) \cos(t)dt\right] =$

$\lim\limits_{x\to \infty} ⁡\dfrac{x^3}{\ln(x)} =$

$\displaystyle\int \dfrac{2 + 2x}{1 − x^2}dx =$

Use the following table to calculate the trapezoidal sum of the area under the curve on the interval $[0, 4], (Δx = 1)$.

Which of the following answer choices is a solution to the differential equation below?

$y”’ − 6y” + 11y’ − 6y = 0$

Determine the average value under the function

$f(x) = 3\cos⁡(x)$ on the interval $\left[0, \dfrac{π}{6}\right]$.

$\displaystyle\int \sin^7(x) \cot(x)dx =$

$\displaystyle\int_2^3 x^2\sqrt{x^3 − 7}dx =$

Identify extrema for the following function.

$f(x) = 3x^3 + \dfrac{45}{2}x^2 + 54x + 3$

Which value of $c$ below satisfies the Mean Value Theorem for the function below on $[3, 5]$?

$f(x) = x^2 − 6x + 15$

$\displaystyle\int \dfrac{2x}{\sqrt{16 − x^4}}dx =$

Solve the following differential equation given the initial value.

$\dfrac{dy}{dx} = \tan(x), \quad$ $y(0) = 1$

If $f(x) = \sqrt{x}$ and $g(x) = x^2 + 3$, identify the following limit.

$\lim\limits_{x\to −4} ⁡f(g(x))$

Which of the following properly identifies the tangent line to the curve $f(x) = \sec⁡(2x)$ at $x = \frac{π}{8}$?

$\displaystyle\int x^2\cos(x^3)dx =$

Which of the following integrals properly identifies the volume of the solid formed by the revolution of the area between the curves $y = x^3$ and $y = x$ around the $x$-axis?

$xy = 2x^2 + 3y$.

Solve for $\dfrac{dy}{dx}$

Solve for $x$.

$\ln(x^3) − \ln(\sqrt{x}) = 4$

The slope field above is of which of the following differential equations?

Identify the $x$-values where critical points exist for the function below on $0 ≤ x ≤ \sqrt{π}$

$f(x) = \cos(3x^2)$

Identify the total distance traveled by a particle that moves according to the function $s(x) = \cos(x)$ from $t = 0$ to $t = π$

Which of the following is the value of the derivative of the inverse of the function $f(x) = x^2$ at the mirror point to $(8, 64)$?

Which of the following properly identifies the correct acceleration function to the position function below?

$s(t) = \ln(\cos(x^2))$

$\displaystyle\int \tan^5(x)dx =$