The graphs of f, f′ and f″

Calculus, grade 11 (5 units) Lesson 6 of 26Learning

The graphs of f, f′ and f″

What each graph tells

Each derivative describes the rate of change of the function before it, so the links between $\displaystyle f$ and $\displaystyle f'$ hold between $\displaystyle f'$ and $\displaystyle f''$ too:

Graph	What you see	What it says about $\displaystyle f$
$\displaystyle f'$	above / below the $\displaystyle x$ -axis	$\displaystyle f$ rises / falls
$\displaystyle f'$	crosses the $\displaystyle x$ -axis	an extremum point of $\displaystyle f$
$\displaystyle f'$	rises / falls	$\displaystyle f$ is concave up / concave down
$\displaystyle f'$	an extremum point	an inflection point of $\displaystyle f$
$\displaystyle f''$	above / below the $\displaystyle x$ -axis	$\displaystyle f$ is concave up / concave down
$\displaystyle f''$	crosses the $\displaystyle x$ -axis	an inflection point of $\displaystyle f$

An extremum point of $\displaystyle f'$ is an inflection point of $\displaystyle f$ : there the slope of $\displaystyle f$ stops growing and starts shrinking, or the other way round.

Example:

\displaystyle x^3-3x

and its two derivatives

The board below shows three graphs: $\displaystyle {f(x)}={x^3-3x}$ , the derivative $\displaystyle {f'(x)}={3x^2-3}$ and the second derivative $\displaystyle {f''(x)}={6x}$ .

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

\displaystyle y=3x^{2}-3

\displaystyle y=6x

$\displaystyle f'$ crosses the $\displaystyle x$ -axis at $\displaystyle {x}={-1}$ and at $\displaystyle {x}={1}$ – there are the extremum points of $\displaystyle f$ .
$\displaystyle f'$ has a minimum at $\displaystyle {x}={0}$ , and there $\displaystyle f''$ crosses the $\displaystyle x$ -axis – the inflection point of $\displaystyle f$ , $\displaystyle (0,0)$ .
$\displaystyle f''<0$ where $\displaystyle x<0$ : there $\displaystyle f'$ falls and $\displaystyle f$ is concave down. $\displaystyle f''>0$ where $\displaystyle x>0$ : $\displaystyle f'$ rises and $\displaystyle f$ is concave up.

Which graph is which

Given three graphs – of a function, of its derivative and of its second derivative – you tell them apart by asking which graph describes the rate of change of which:

Graph $\displaystyle g$ is the derivative of graph $\displaystyle h$ if $\displaystyle g$ crosses the $\displaystyle x$ -axis exactly where $\displaystyle h$ has an extremum, $\displaystyle g$ is above the axis where $\displaystyle h$ rises and below it where $\displaystyle h$ falls.
Check every pair. The function is the graph that is the derivative of none of the others; the second derivative is the graph that no other graph is the derivative of.
For a polynomial, each derivative lowers the degree by 1: if one of the graphs is a parabola and another a straight line, the line is the derivative of the parabola.

To show that no other order works, it is enough to find, for each other order, a place where the link fails – for example, a graph that rises where its “derivative” is below the axis.

Exercise 1

The boards show the graph of $\displaystyle {f(x)}={x^3-6x^2+9x}$ .

Write the derivative $\displaystyle f'(x)$ on the board. Its graph appears beside the graph of $\displaystyle f$ .

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

Write the second derivative $\displaystyle f''(x)$ on the board.

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

Where does $\displaystyle f'$ cross the $\displaystyle x$ -axis, and what does $\displaystyle f$ have there?
Where does $\displaystyle f''$ cross the $\displaystyle x$ -axis, and what does $\displaystyle f$ have there?

Exercise 2

The board shows the graph of the derivative $\displaystyle {f'(x)}={x^2-4x+3}$ of a function $\displaystyle f$ .
The graph of $\displaystyle f$ itself is not given.

Mark on the graph the points where it crosses the $\displaystyle x$ -axis, and its extremum point.

\displaystyle y=x^{2}-4x+3

At $\displaystyle {x}={1}$ and at $\displaystyle {x}={3}$ , $\displaystyle f$ has extremum points: what is the type of each?
Where is $\displaystyle f$ concave up, and where concave down?
What is the $\displaystyle x$ -coordinate of the inflection point of $\displaystyle f$ ?

The

\displaystyle x

of the inflection point

/ – 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

Exercise 3

The function $\displaystyle f$ can be differentiated twice for every $\displaystyle x$ .
It has only two extremum points: a minimum where $\displaystyle {x}={3}$ , and a maximum where $\displaystyle {x}={5}$ .

What is the sign of $\displaystyle f'$ where $\displaystyle x<3$ , where $\displaystyle 3<x<5$ and where $\displaystyle x>5$ ?

A possible derivative is $\displaystyle {f'(x)}={-(x-3)(x-5)}$ .
Solve the inequality $\displaystyle f'(x)>0$ , and check that the answer fits the part before.

The board shows the graph of $\displaystyle {f'(x)}={-(x-3)(x-5)}$ .
Mark its extremum point.

\displaystyle y=-(x-3)(x-5)

What does $\displaystyle f$ have at the point you marked?
Explain why every function like $\displaystyle f$ – a minimum at 3, a maximum at 5 – has an inflection point between 3 and 5.

Exercise 4

The board shows three graphs: $\displaystyle {f(x)}={x^3-3x^2}$ and two graphs whose formulas are hidden.
One of them is the graph of $\displaystyle f'$ , and the other – of $\displaystyle f''$ .

Write on the board the formula of each hidden graph.

\displaystyle f(x)=x^{3}-3x^{2}

\displaystyle y=\,?

\displaystyle y=\,?

How can you tell from the graphs themselves, with no formulas, which is $\displaystyle f'$ and which is $\displaystyle f''$ ?

Formulas: calculus, grade 11 (5 units)

Open as a page

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