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

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.

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 $t$.

people/minute^2

people/minute

people

minutes/person

4\pi r^2

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

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

3\pi r^2

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.

-200

500

1000

-100

No impact as time does not depend on coefficients within derivative expressions linearly with respect to $t$.

Time cannot be determined without knowing initial water volume $V(0)$.

It doubles exactly since only $t$'s coefficient was reduced by half.

It increases by less than double but more than original time.