The graph of a function and the graph of its derivative

Calculus, grade 10 (5 units) Lesson 14 of 26Learning

The graph of a function and the graph of its derivative

What the graph of the derivative tells

The value of $\displaystyle f'(x)$ is the slope of the tangent to the graph of $\displaystyle f$ . So every feature of the graph of the derivative is a feature of the graph of the function:

The graph of the derivative above the $\displaystyle x$ -axis ( $\displaystyle f'>0$ ) – the function rises.
The graph of the derivative below the $\displaystyle x$ -axis ( $\displaystyle f'<0$ ) – the function falls.
The graph of the derivative crosses the $\displaystyle x$ -axis – the function has an extremum point: upwards (from − to +) – a minimum; downwards – a maximum.
The graph of the derivative touches the $\displaystyle x$ -axis without crossing it – the tangent is horizontal, but there is no extremum.

The derivative of a polynomial of degree $\displaystyle n$ is a polynomial of degree $\displaystyle n-1$ : the derivative of a function of degree three is a parabola, and the derivative of a parabola is a straight line.

Example:

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

The first board shows the graph of $\displaystyle {f(x)}={x^3-3x}$ :

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

The second board shows the graph of its derivative, $\displaystyle {f'(x)}={3x^2-3}$ :

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

$\displaystyle f$ rises where $\displaystyle x<-1$ and where $\displaystyle x>1$ – there the parabola is above the $\displaystyle x$ -axis.
$\displaystyle f$ falls where $\displaystyle -1<x<1$ – there the parabola is below the axis.
At the extremum points of $\displaystyle f$ , $\displaystyle {x}={-1}$ and $\displaystyle {x}={1}$ , the parabola crosses the $\displaystyle x$ -axis.
At $\displaystyle {x}={0}$ the fall of $\displaystyle f$ is steepest, and there the parabola has its minimum, $\displaystyle {f'(0)}={-3}$ .

From the graph of the derivative to the function

Even when only the graph of the derivative is known, it tells where the function rises and where it has an extremum. The board below shows the graph of the derivative $\displaystyle {f'(x)}={(x+1)^2(x-3)}$ of a function $\displaystyle f$ :

\displaystyle y=(x+1)^{2}(x-3)

$\displaystyle x$	$\displaystyle x<-1$	$\displaystyle -1$	$\displaystyle -1<x<3$	$\displaystyle 3$	$\displaystyle x>3$
$\displaystyle f'(x)$	−	0	−	0	+
$\displaystyle f(x)$	falls	no extremum	falls	minimum	rises

At $\displaystyle {x}={-1}$ the graph of the derivative touches the $\displaystyle x$ -axis – $\displaystyle f$ has a horizontal tangent there with no extremum.
At $\displaystyle {x}={3}$ it crosses the axis upwards – $\displaystyle f$ has a minimum there.
The graph of the derivative gives the $\displaystyle x$ -coordinate of the minimum, but not its $\displaystyle y$ -coordinate.

An antiderivative

An antiderivative of $\displaystyle f$ is a function $\displaystyle F$ whose derivative is $\displaystyle f$ : $\displaystyle {F'(x)}={f(x)}$ . You find it by going backwards through the rule of differentiation – raise the power by 1 and divide by the new power:

\displaystyle \left(\frac{x^{n+1}}{n+1}\right)'

\displaystyle =

\displaystyle x^n

For example, an antiderivative of $\displaystyle {f(x)}={3x^2}$ is $\displaystyle {F(x)}={x^3}$ . But $\displaystyle x^3+5$ and $\displaystyle x^3-2$ are antiderivatives too: the derivative of a constant number is 0.

All the antiderivatives of $\displaystyle f$ have the same slopes at every $\displaystyle x$ , so their graphs are the same graph, shifted up or down.
So the antiderivative is not unique: the antiderivatives are all $\displaystyle F(x)+C$ , where $\displaystyle C$ is any number.

A function from its derivative and a point

Of all the antiderivatives, only one passes through a given point. The point gives an equation for $\displaystyle C$ .

The example below finds $\displaystyle f$ , given that $\displaystyle {f'(x)}={6x^2-4x}$ and that the graph of $\displaystyle f$ passes through the point $\displaystyle (1,5)$ :

The antiderivatives: $\displaystyle {f(x)}={2x^3-2x^2+C}$ .
$\displaystyle {f(1)}={5}$ , so $\displaystyle {2-2+C}={5}$ , and $\displaystyle {C}={5}$ .
$\displaystyle {f(x)}={2x^3-2x^2+5}$ .

Exercise 1

The board below shows the graph of $\displaystyle {f(x)}={x^3-6x^2}$ .

Add the graph of $\displaystyle f'$ to the board: write $\displaystyle f'(x)$ in a row of your own.

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

Where is the graph you wrote above the $\displaystyle x$ -axis?
What does the graph of $\displaystyle f$ do in that interval?

Exercise 2

The board below shows the graph of the derivative $\displaystyle {f'(x)}={x^2-2x-3}$ of a function $\displaystyle f$ .

Mark the points where the graph of the derivative meets the $\displaystyle x$ -axis.

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

Solve the inequality $\displaystyle f'(x)>0$ : where does $\displaystyle f$ rise?

At which $\displaystyle x$ does $\displaystyle f$ have a maximum, and at which a minimum?
Explain by the graph of the derivative.

maximum

/ – 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

minimum

/ – 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 derivative of a function $\displaystyle f$ is $\displaystyle {f'(x)}={x(x-2)^2}$ .

Solve the inequality $\displaystyle f'(x)>0$ .

What are the extremum points of $\displaystyle f$ , and of what kind?
What does $\displaystyle f$ have at $\displaystyle {x}={2}$ ?

Exercise 4

The function is $\displaystyle {f(x)}={4x^3-6x+1}$ .

Write three different antiderivatives of $\displaystyle f$ .

Check one of them: write $\displaystyle \left(F(x)\right)'$ with the function you chose, and differentiate.

/ – 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 5

The derivative of a function $\displaystyle f$ is $\displaystyle {f'(x)}={3x^2-4}$ .
The graph of $\displaystyle f$ passes through the point $\displaystyle (1,2)$ .

Write all the antiderivatives of $\displaystyle f'$ .

Substitute the point and find $\displaystyle C$ .

Write $\displaystyle f(x)$ in a row of your own on the board, and check that the graph passes through the point.

\displaystyle A=(1,2)

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$

The properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units right ( $\displaystyle p<0$ – left)
$\displaystyle y=f(x-p)$
A vertical stretch ( $\displaystyle |a|>1$ ) or squeeze ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$
A horizontal squeeze by $\displaystyle a$ ( $\displaystyle a>1$ ) or stretch ( $\displaystyle 0<a<1$ )
$\displaystyle y=f(ax)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
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 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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$