Contextual Applications of Differentiation

The graph of $f(x)$ has a corner or cusp at $x = c$.

The graph of the derivative, $f'(x)$, has a vertical asymptote at $x = c$.

The function $f(x)$ is not continuous at $x = c$.

The limit of $f'(x)$ as $x$ approaches $c$ does not exist.

liters/second, negative

liters/second, positive

seconds/liter, positive

seconds/liter, negative

people/minute^2

people/minute

people

minutes/person

Water is being removed from the tank at an accelerating rate.

The rate at which water is being added to the tank is increasing.

There is no change in the volume of water in the tank.

The volume of water in the tank is decreasing.

4\pi r^2

$\frac{4}{3} \pi r^2$

$\frac{4}{9} \pi r^2$

3\pi r^2

It must be less than zero as $P(t)$ is decreasing with time.

It must equal zero since $P(t)$ and $t$ are inversely related.

It must be greater than zero since $P(t)$ decreases when $t$ increases.

It varies depending on the value of $t$.

The student's computer is plugged in and charging. They turned on low-power mode to increase the rate at which the battery charges.

The student's computer is unplugged; battery is draining. But, they turned on low-power mode to decrease the rate at which the battery drains.

The student's computer is unplugged; battery is draining. They turned their brightness all the way up which increases the rate at which battery drains.

The student's computer is plugged in and charging. They turned their brightness all the way up which decreased the rate at which the battery charges.

$f'(x_0) < 0$

$f'(x_0) = 0$

$f'(x_0) > 0$

$f'(x_0)$ is undefined.

Finding a limit as time approaches ten minutes.

Solving a differential equation that models cooling over time.

Calculating a definite integral from zero to ten minutes.

Computing an instantaneous rate of change at ten minutes.

$\frac{d}{dt}(3t^2 + 5)$ evaluated from $t=0$ to $t=4$

$(3(4)^2 + 5) - (3(0)^2 + 5)$

$\int_{0}^{4}(3t^2 + 5) ,dt$

$\int_{0}^{4}R(t) ,dt$ without incorporating the specific rate function