Rates and differentiation

Interpreting the first and second derivatives of a quantity over time, describing changes in words such as increasing at a decreasing rate, and the meaning of concavity and points of inflection in a model.

What the derivatives say

In words	$Q'$	$Q''$	Graph
Increasing at an increasing rate	$+$	$+$	rising, concave up
Increasing at a decreasing rate	$+$	$-$	rising, concave down
Decreasing, and falling faster and faster	$-$	$-$	falling, concave down
Decreasing, but more and more slowly	$-$	$+$	falling, concave up

Growth that slows down

Many real quantities grow like this. The point of inflection at t = 5 is where the growth is fastest.

Try this

Where is the gradient of the curve largest?

Examples

Example 1●●

The volume of water in a tank is $V = 100 + 20t - t^2$ litres for $0 \le t \le 20$ minutes. Describe how $V$ is changing at $t = 5$, and find when the volume is greatest.

$V' = 20 - 2t$, so $V'(5) = 10$: the volume is increasing at $10$ L/min.

$V'' = -2 < 0$, so the rate of increase is getting smaller.

$V' = 0$ at $t = 10$, which gives the greatest volume.

Example 2●

"The price of petrol is still rising, but more slowly than before." Write this using $P$, the price.

$\dfrac{dP}{dt} > 0$ and $\dfrac{d^2P}{dt^2} < 0$.

Flashcards

Increasing at a decreasing rate

$Q' > 0$ and $Q'' < 0$

Concave up means

The rate of change is increasing.

Point of inflection in a model

Where the rate of change is greatest or least.

Practice

"Increasing at a decreasing rate" means:
"The temperature is falling, but more and more slowly" means:
$V = 100 + 20t - t^2$. Find $\dfrac{dV}{dt}$ at $t = 5$.
When is that volume greatest?
Find $\dfrac{d^2V}{dt^2}$ for the same $V$.
$Q = t^3 - 6t^2 + 20t$. When is $Q$ increasing most slowly?
On an S-shaped growth curve, the growth is fastest at:
A graph of $Q$ against $t$ is concave up. This means:
"Unemployment is falling, and falling faster each month." With $U$ for unemployment: