Consider an object whose position is given by the function $p\left(t\right)=5t^2-10t$, where $p$ is in meters and $t$ is in seconds. Find the velocity and acceleration of the object at $t=4$.

Calculate the second derivative of $5x^3+9y^3=19$.

Two capacitors, $C_1$ and $C_2$, are connected in series in an electrical circuit, forming an equivalent capacitance $C$, described by the following equation:

$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}$

If $C_1$ is increasing at a rate of $0.2\text{ F/sec}$ and $C_2$ is decreasing at a rate of $0.1\text{ F/sec}$, at what rate is $C$ changing when $C_1$ is $40\text{ F}$ and $C_2$ is $60\text{ F}$?

A diver jumps from a cliff into the sea, and the height $h$ (in feet) of the diver above sea level $t$ seconds after jumping is given by the equation $h\left(t\right)=50-16t+4t^2$. Estimate the change in the diver's height over the interval $1\le t\le 1.6$.

The function $C\left(t\right)=\frac{80t}{t+4}$ (in $\text{ppm}$) represents the concentration of a solution, where $t\ge 0$ is the time in minutes. Determine the average rate at which the concentration changes over the interval $[0,5]$.

What value of $c$ makes $h\left(z\right)=3z^2+\frac{c}{z}$ have a point of inflection at $z=-2$?

Determine the absolute minimum and maximum values of the function $f\left(x\right)=\sqrt{16-x^2}$ on $[-1,4]$ and identify where they occur.

Find all critical points and domain endpoints for the function $h\left(x\right)=2x^5-\frac{5}{x}$.

Analyze the function $p\left(x\right)=(x+1)(x-6{)}^2$ to determine the intervals of concavity.

Determine the inflection points and intervals of concavity for the function $y=3x-\sin x$, where $-\pi \le x\le \pi$.

Draw a graph of the following function: $f\left(x\right)=x^3-6x^2+9x$.

A bakery is designing a custom cake box using a $20$-inch by $30$-inch sheet of cardboard. The cardboard is folded in half to form a $20$-inch by $15$-inch rectangle. To create the sides of the box, four identical squares with side length $x$ inches are cut from each corner of the folded rectangle. The cardboard is then unfolded, and the six flaps formed by the cuts are folded upward to create the sides and lid of the box. Let $V\left(x\right)$ represent the volume of the finished cake box in terms of $x$. Determine the domain of $V$.

Determine the derivative of the function $f\left(x\right)=\ln \left(\frac{5x}{{(3x^2+4)}^5}\right)$.

Find the derivative of the function $y={\left(2x\right)}^{3\sqrt{x-2}}$.

For the function $j\left(x\right)=\cot \left(21x\right)$, determine the derivative of the inverse function at the point $(1,\frac{\pi }{84})$ without calculating the inverse.

Given $P\left(x\right)=5x^2-3x+8$ and $Q\left(x\right)=5x^2-3x-12$, are $P\left(x\right)$ and $Q\left(x\right)$ antiderivatives of the same function?

If the indefinite integral of f(x) = 6x^2 is F(x) = 2x^3 + C, verify this by differentiating F(x). What should the derivative be?

Evaluate the integral: $\int {\tan }^{3}x{\sec }^{7}xdx$