Use the following limit definition to determine the slope of the line tangent to the graph of $f$ at $P$, where $f\left(x\right)=\frac{6}{\sqrt{x}}$ and $P(36,1)$:

$m_{\text{tan}}=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$

A rocket is launched vertically up, and its height above the ground is given by the function $h\left(t\right)=400t-50t^2$, where $h$ is in meters and $t$ is in seconds. The graph below shows $h\left(t\right)$ along with its velocity function $v\left(t\right)=\frac{dh}{dt}$ and the acceleration function $a\left(t\right)=\frac{d^{2}h}{d{t}^2}$ for the time interval $0\le t\le 8$ $\text{s}$.

Using the graph, determine the time when the rocket changes direction.

Find the limit by converting it first to a derivative at $x=3$.

$\underset{x\to 3}{\lim }\frac{x^8-6561}{x-3}$

For the function $h(x)=-\(\frac{12}{x^3}\)$, identify the intervals of $x$-values where the function increases and decreases as $x$ increases.

If the current in a circuit is related to the voltage, $V$, by the formula: $I=\frac{kV}{V^2+R^2}$, where $k$ and $R$ are constants, find $\frac{dI}{dV}$.

Given a function $H\left(t\right)$ that models the height (in centimeters) of a plant $t$ days after it was planted, estimate $H^{\prime }\left(90\right)$ from the provided graph. What does this derivative indicate?

If a function $k$ has a derivative at the point $c$, what can be said about the continuity of $k$ at $c$?