Concavity: how the rate of change changes

Calculus, grade 11 (5 units) Lesson 2 of 26Learning

Concavity: how the rate of change changes

A slope that grows and a slope that shrinks

The derivative says how fast the function changes. That rate can change too: as you move along the graph from left to right, the slope of the tangent can grow or shrink. The board below shows the graphs of $\displaystyle {f(x)}={x^2}$ and $\displaystyle {g(x)}={-x^2}$ :

\displaystyle f(x)=x^{2}

\displaystyle g(x)=-x^{2}

The slopes of the tangents at a few points:

$\displaystyle x$	$\displaystyle -2$	$\displaystyle -1$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$
$\displaystyle {f'(x)}={2x}$	$\displaystyle -4$	$\displaystyle -2$	$\displaystyle 0$	$\displaystyle 2$	$\displaystyle 4$
$\displaystyle {g'(x)}={-2x}$	$\displaystyle 4$	$\displaystyle 2$	$\displaystyle 0$	$\displaystyle -2$	$\displaystyle -4$

The slope of $\displaystyle f$ grows all the time: the graph bends upwards, like $\displaystyle \cup$ .
The slope of $\displaystyle g$ shrinks all the time: the graph bends downwards, like $\displaystyle \cap$ .

Concave up, concave down

The way the graph bends depends on whether its slope grows or shrinks:

A function is concave up ( $\displaystyle \cup$ ) on an interval where the derivative $\displaystyle f'$ rises – the rate of change keeps growing.
A function is concave down ( $\displaystyle \cap$ ) on an interval where the derivative $\displaystyle f'$ falls – the rate of change keeps shrinking.
These intervals are the intervals of concavity of the function.

Concavity does not depend on whether the function rises or falls: a rising function can be concave up (rising faster and faster) or concave down (rising more and more slowly).

Tangents and chords

A chord is a segment joining two points on the graph. The board below shows the graph of $\displaystyle {f(x)}={x^2}$ , the tangent $\displaystyle {y}={2x-1}$ at the point $\displaystyle (1,1)$ , and the chord from $\displaystyle (0,0)$ to $\displaystyle (1,1)$ , which lies on the line $\displaystyle {y}={x}$ :

\displaystyle f(x)=x^{2}

\displaystyle y=2x-1

\displaystyle y=x

For a function that is concave up, the graph is above its tangents, and the chords are above the graph.
For a function that is concave down – the other way round: the graph is below its tangents, and the chords are below the graph.

A fall that grows steeper

Careful: when a function falls and grows steeper, its slope shrinks. For example, the slopes $\displaystyle -1$ , $\displaystyle -3$ , $\displaystyle -6$ keep getting smaller. So a fall that grows steeper is concave down, and a fall that grows gentler is concave up.

Example:

\displaystyle x^3

The example below finds the intervals of concavity of $\displaystyle {f(x)}={x^3}$ :

$\displaystyle {f'(x)}={3x^2}$ – a parabola that falls where $\displaystyle x<0$ and rises where $\displaystyle x>0$ .
So $\displaystyle f$ is concave down where $\displaystyle x<0$ , and concave up where $\displaystyle x>0$ .

\displaystyle f(x)=x^{3}

$\displaystyle f$ rises everywhere, but in two different ways: left of 0 the rise grows gentler, and right of 0 it grows steeper.
At the point $\displaystyle (0,0)$ the graph passes from concave down to concave up.

Exercise 1

A car sets off.
The table gives the distance $\displaystyle s$ it has covered (in metres) after $\displaystyle t$ seconds:

$\displaystyle t$	0	1	2	3	4	5
$\displaystyle s$	0	1	4	9	16	25

Work out the average speed between $\displaystyle {t}={1}$ and $\displaystyle {t}={2}$ .

/ – 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

Work out the average speed between $\displaystyle {t}={3}$ and $\displaystyle {t}={4}$ .

/ – 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

Is the graph of $\displaystyle s$ against $\displaystyle t$ concave up or concave down? Give your reason.

Exercise 2

The function is $\displaystyle {f(x)}={x^3-3x}$ .
Its derivative, $\displaystyle {f'(x)}={3x^2-3}$ , falls where $\displaystyle x<0$ and rises where $\displaystyle x>0$ .

Mark on the graph the point where the graph passes from concave down to concave up.

\displaystyle f(x)=x^{3}-3x

Write the intervals of concavity of $\displaystyle f$ .
On the interval $\displaystyle 0<x<1$ the function falls. Does the fall grow steeper or gentler there?

Exercise 3

The function is $\displaystyle {f(x)}={\frac{1}{x}}$ , on the interval $\displaystyle x>0$ .

Work out the slope of the tangent to the graph where $\displaystyle {x}={1}$ .

$\displaystyle f(x)=\frac{1}{x}$

/ – 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

Work out the slope of the tangent to the graph where $\displaystyle {x}={2}$ .

$\displaystyle f(x)=\frac{1}{x}$

/ – 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

$\displaystyle {f'(x)}={-\frac{1}{x^2}}$ .
Does $\displaystyle f'$ rise or fall on the interval $\displaystyle x>0$ ?
Is $\displaystyle f$ concave up or concave down there?

Exercise 4

The function is $\displaystyle {f(x)}={4-x^2}$ .
The point $\displaystyle (1,3)$ is on its graph.

Write on the board the equation of the tangent to the graph at the point $\displaystyle (1,3)$ .

\displaystyle f(x)=4-x^{2}

\displaystyle (1,3)

Is the tangent above the graph or below it?
How does that fit the concavity of $\displaystyle f$ ?

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
$\displaystyle (\tan x)'$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2x}$