The rate of change of a rate

Calculus, grade 11 (5 units) Lesson 7 of 26Learning

The rate of change of a rate

Distance, speed and acceleration

When a body moves along a straight line, $\displaystyle s(t)$ is its position at time $\displaystyle t$ . The speed is the rate of change of the position, and the acceleration is the rate of change of the speed:

\displaystyle a(t)=v'(t)=s''(t)

In the example below a body moves by $\displaystyle {s(t)}={t^3-6t^2+9t}$ (metres, seconds):

$\displaystyle {v(t)}={s'(t)}={3t^2-12t+9}={3(t-1)(t-3)}$ .
$\displaystyle {a(t)}={s''(t)}={6t-12}$ .

The board shows $\displaystyle s$ and $\displaystyle v$ , with $\displaystyle t$ on the $\displaystyle x$ -axis:

\displaystyle s(x)=x^{3}-6x^{2}+9x

\displaystyle v(x)=3x^{2}-12x+9

$\displaystyle v>0$ where $\displaystyle t<1$ and where $\displaystyle t>3$ : the body moves forwards. $\displaystyle v<0$ where $\displaystyle 1<t<3$ : the body moves back.
$\displaystyle a<0$ where $\displaystyle t<2$ and $\displaystyle a>0$ where $\displaystyle t>2$ : at $\displaystyle {t}={2}$ , $\displaystyle v$ has a minimum and $\displaystyle s$ an inflection point.
A positive acceleration says that the speed rises – not always that the body speeds up. Where $\displaystyle 2<t<3$ the speed is negative and rises towards 0: the body moves back and slows down.

Prices that rise more slowly

In a debate on the cost of living, one speaker showed a graph of the price index $\displaystyle P(t)$ and said: “Prices keep rising”. Another speaker showed the same graph and said: “The fight against rising prices is working – the rise is slowing down”. The board shows such a graph:

\displaystyle P(x)=100+20\sqrt{x}

Both are right, each about a different derivative:

The first speaker talks about $\displaystyle P'$ : $\displaystyle P'(t)>0$ , prices rise.
The second speaker talks about $\displaystyle P''$ : $\displaystyle P''(t)<0$ , the graph is concave down – the rate of the rise keeps shrinking.

The fastest growth

When a quantity grows, the moment it grows fastest is the maximum point of $\displaystyle f'$ . There $\displaystyle f''$ changes sign from + to −, so it is an inflection point of $\displaystyle f$ : up to it the growth speeds up (concave up), and after it the growth slows down (concave down).

A familiar example – filling a vessel with water at a constant rate:

In a narrow part of the vessel the water level rises fast, in a wide part slowly.
In a vessel that widens upwards, the level rises more and more slowly: the graph of the height against time is concave down.
In a vessel that narrows upwards, the level rises faster and faster: the graph is concave up.

Exercise 1

A body moves along a straight line by $\displaystyle {s(t)}={t^3-6t^2+9t}$ , with $\displaystyle t\geq0$ (metres, seconds).

Work out the speed of the body after 2 seconds.

$\displaystyle s(t)=t^{3}-6t^{2}+9t$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find the acceleration $\displaystyle {a(t)}={s''(t)}$ .

$\displaystyle s(t)=t^{3}-6t^{2}+9t$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the inequality $\displaystyle a(t)>0$ : when does the speed rise?

The speed after 2 seconds is negative, and the acceleration after 2 seconds is 0.
What does this say about the motion at that moment?

Exercise 2

The number of bacteria in a culture (in thousands) $\displaystyle t$ hours after the start of an experiment is $\displaystyle {P(t)}={-t^3+12t^2+100}$ , with $\displaystyle 0\leq t\leq8$ .

Find $\displaystyle P''(t)$ .

$\displaystyle P(t)=-t^{3}+12t^{2}+100$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {P''(t)}={0}$ .

Work out the rate at which the culture grows at the moment you found.

$\displaystyle P(t)=-t^{3}+12t^{2}+100$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The board shows the graph of $\displaystyle P$ , with $\displaystyle t$ on the $\displaystyle x$ -axis.
Mark the point where the culture grows fastest.

\displaystyle P(x)=-x^{3}+12x^{2}+100

Exercise 3

The chart shows the length of the day in Tel Aviv over a year:

When is the longest day of the year?
Over which period does the length of the day grow?

Work out by how much the length of the day grows from each month to the next, from January to June.
When does the length of the day grow at the fastest rate?
Which point of the graph is that?

Exercise 4

Water is poured at a constant rate into a glass that grows wider from the bottom up.
Is the graph of the water’s height against time concave up or concave down? Give your reason.

Exercise 5

A news report says: “The number of unemployed is still rising, but more slowly”.
What are the signs of the first and of the second derivative of the number of unemployed against time?

Formulas: calculus, grade 11 (5 units)

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Rules of differentiation and the tangent line

The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a product
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The tangent line at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle y-y_1=f'(x_1)(x-x_1)$

The second derivative

The second derivative – the derivative of the derivative
$\displaystyle f''(x)$ $\displaystyle =$ $\displaystyle \left(f'(x)\right)'$
The function is concave up ( $\displaystyle \cup$ )
$\displaystyle f''(x)>0$
The function is concave down ( $\displaystyle \cap$ )
$\displaystyle f''(x)<0$
A possible inflection point – check that the sign of $\displaystyle f''$ changes there
$\displaystyle f''(x_0)=0$
A minimum point
$\displaystyle f'(x_0)=0,\ f''(x_0)>0$
A maximum point
$\displaystyle f'(x_0)=0,\ f''(x_0)<0$
Acceleration – the derivative of the speed
$\displaystyle a(t)=v'(t)=s''(t)$

Asymptotes

A vertical asymptote – $\displaystyle f(x)$ grows or falls without end as $\displaystyle x$ approaches $\displaystyle a$
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle f(x)$ approaches $\displaystyle b$ as $\displaystyle x\to\infty$ or as $\displaystyle x\to-\infty$
$\displaystyle y=b$

Roots

The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Radians and periods

Half a turn in radians
$\displaystyle \pi$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The length of an arc of $\displaystyle \alpha$ radians in a circle of radius $\displaystyle R$
$\displaystyle l$ $\displaystyle =$ $\displaystyle \alpha R$
The period of $\displaystyle \sin x$ and of $\displaystyle \cos x$ is $\displaystyle 2\pi$
$\displaystyle \sin(x+2\pi)$ $\displaystyle =$ $\displaystyle \sin x$
The period of $\displaystyle \tan x$ is $\displaystyle \pi$
$\displaystyle \tan(x+\pi)$ $\displaystyle =$ $\displaystyle \tan x$
The limit behind the derivative of $\displaystyle \sin x$ ( $\displaystyle x$ in radians)
$\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1$

Trigonometric identities

The Pythagorean identity
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =$ $\displaystyle 1$
Tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$
Sine – an odd function
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
Cosine – an even function
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
The sine of the supplement to $\displaystyle \pi$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The cosine of the complement to $\displaystyle \frac{\pi}{2}$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The sine of a double angle
$\displaystyle \sin2\alpha$ $\displaystyle =$ $\displaystyle 2\sin\alpha\cdot\cos\alpha$
The cosine of a double angle
$\displaystyle \cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha$
The difference of two sines
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =$ $\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}$
The sine of a sum and of a difference
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta$

Derivatives of the trigonometric functions

The derivative of sine ( $\displaystyle x$ in radians)
$\displaystyle (\sin x)'$ $\displaystyle =$ $\displaystyle \cos x$
The derivative of cosine
$\displaystyle (\cos x)'$ $\displaystyle =$ $\displaystyle -\sin x$
The derivative of tangent
$\displaystyle (\tan x)'$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2x}$