The derivative's graph, parity and a parameter

Calculus, grade 11 (4 units) Lesson 9 of 20Learning

The derivative's graph, parity and a parameter

What the derivative’s graph shows

The value of $\displaystyle f'$ at each $\displaystyle x$ is the slope of $\displaystyle f$ there. So the behaviour of $\displaystyle f$ is read from the graph of $\displaystyle f'$ :

the graph of $\displaystyle f'$ above the $\displaystyle x$ -axis ( $\displaystyle f'>0$ ) – $\displaystyle f$ rises;
the graph of $\displaystyle f'$ below the $\displaystyle x$ -axis ( $\displaystyle f'<0$ ) – $\displaystyle f$ falls;
the graph of $\displaystyle f'$ crosses the $\displaystyle x$ -axis – $\displaystyle f$ has an extremum point there: a maximum when it crosses from above to below, and a minimum when it crosses from below to above;
$\displaystyle f'$ has the same domain as $\displaystyle f$ : near a point outside the domain of $\displaystyle f$ , $\displaystyle f'$ has a vertical asymptote too.

Example

$\displaystyle {f(x)}={x+\frac{4}{x}}$ , and its derivative is $\displaystyle {g(x)}={f'(x)}={1-\frac{4}{x^{2}}}$ – the graph in the second colour.

\displaystyle f(x)=x+\frac{4}{x}

\displaystyle g(x)=1-\frac{4}{x^{2}}

\displaystyle x=0

$\displaystyle g$ crosses the $\displaystyle x$ -axis at $\displaystyle {x}={-2}$ from above to below – $\displaystyle f$ has a maximum there, $\displaystyle (-2,-4)$ ;
$\displaystyle g$ crosses the $\displaystyle x$ -axis at $\displaystyle {x}={2}$ from below to above – $\displaystyle f$ has a minimum there, $\displaystyle (2,4)$ ;
between $\displaystyle -2$ and $\displaystyle 2$ the graph of $\displaystyle g$ is below the $\displaystyle x$ -axis – $\displaystyle f$ falls there, on both sides of $\displaystyle {x}={0}$ ;
$\displaystyle {x}={0}$ is in the domain of neither function.

Exercise 1

On the board – the graph of $\displaystyle {g(x)}={f'(x)}$ , the derivative of a function $\displaystyle f$ .
The domain of $\displaystyle f$ : $\displaystyle x\neq0$ .

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

On which intervals does $\displaystyle f$ rise, and on which does it fall?

At which values of $\displaystyle x$ does $\displaystyle f$ have extremum points, and what kind is each one?

Even and odd functions

As for polynomials, parity is checked by substituting $\displaystyle -x$ :

\displaystyle f(-x)

\displaystyle =

\displaystyle f(x)

\displaystyle f(-x)

\displaystyle =

\displaystyle -f(x)

even – $\displaystyle {f(-x)}={f(x)}$ for every $\displaystyle x$ in the domain: the graph is symmetric about the $\displaystyle y$ -axis;
odd – $\displaystyle {f(-x)}={-f(x)}$ for every $\displaystyle x$ in the domain: the graph is symmetric about the origin;
the domain of an even or odd function is symmetric about 0: if $\displaystyle {x}={1}$ is not in the domain, neither is $\displaystyle {x}={-1}$ .

On the board: $\displaystyle {f(x)}={\frac{3}{x^{2}-1}}$ is even, since $\displaystyle {f(-x)}={\frac{3}{(-x)^{2}-1}}={\frac{3}{x^{2}-1}}$ ; and $\displaystyle {g(x)}={\frac{2x}{x^{2}+1}}$ is odd, since $\displaystyle {g(-x)}={\frac{-2x}{x^{2}+1}}={-g(x)}$ .

\displaystyle f(x)=\frac{3}{x^{2}-1}

\displaystyle g(x)=\frac{2x}{x^{2}+1}

\displaystyle x=1

\displaystyle x=-1

The graph of $\displaystyle f$ is the same on both sides of the $\displaystyle y$ -axis, and the graph of $\displaystyle g$ comes back onto itself when turned half a turn about $\displaystyle (0,0)$ .

Exercise 2

Even, odd or neither?
In parts a and b – write $\displaystyle f(-x)$ , simplify, and end with $\displaystyle f(x)$ or $\displaystyle -f(x)$ .

$\displaystyle f(x)=\frac{x^{2}}{x^{2}+4}$

/ – 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

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

/ – 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

$\displaystyle {f(x)}={\frac{x}{x-2}}$ . Explain with the domain.

A parameter

A parameter is a number written as a letter, for example $\displaystyle a$ . It is found from one given fact: substitute the fact, and solve an equation in $\displaystyle a$ .

Example: $\displaystyle {f(x)}={\frac{4x+1}{ax^{2}-2x}}$ , and $\displaystyle f$ has the vertical asymptote $\displaystyle {x}={2}$ .
In the solution below we use the fact that the denominator is 0 at $\displaystyle {x}={2}$ :

$\displaystyle {a\cdot2^{2}-2\cdot2}={0}$ , that is $\displaystyle {4a-4}={0}$ , and $\displaystyle {a}={1}$ ;
the numerator at $\displaystyle {x}={2}$ : $\displaystyle {4\cdot2+1}={9\neq0}$ , so $\displaystyle {x}={2}$ really is an asymptote;
$\displaystyle {f(x)}={\frac{4x+1}{x^{2}-2x}}$ – the graph on the board, with its two vertical asymptotes $\displaystyle {x}={0}$ and $\displaystyle {x}={2}$ .

\displaystyle f(x)=\frac{4x+1}{x^{2}-2x}

\displaystyle x=0

\displaystyle x=2

Exercise 3

$\displaystyle {f(x)}={x+\frac{a}{x}}$ , and $\displaystyle a$ is a parameter.
$\displaystyle f$ has an extremum point where $\displaystyle {x}={3}$ .

Find $\displaystyle f'(x)$ (the answer depends on $\displaystyle a$ ).

$\displaystyle f(x)=x+\frac{a}{x}$

/ – 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

At the extremum point $\displaystyle {f'(3)}={0}$ . Write an equation in $\displaystyle a$ and solve it.

Find the extremum point where $\displaystyle {x}={3}$ , and say what kind it is.

Exercise 4

$\displaystyle {f(x)}={\frac{a}{x-1}}$ , and $\displaystyle a$ is a parameter.
The slope of the tangent to the graph at the point where $\displaystyle {x}={3}$ is $\displaystyle -2$ .

Find $\displaystyle f'(x)$ .

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

/ – 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

Write an equation in $\displaystyle a$ and solve it.

Formulas: calculus, grade 11 (4 units)

Open as a page

Transformations of the graph of $f$

A horizontal shift: $\displaystyle p$ units to the right ( $\displaystyle p>0$ ) or to the left ( $\displaystyle p<0$ )
$\displaystyle y=f(x-p)$
A vertical shift: $\displaystyle k$ units up ( $\displaystyle k>0$ ) or down ( $\displaystyle k<0$ )
$\displaystyle y=f(x)+k$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A vertical stretch ( $\displaystyle a>1$ ) or squeeze ( $\displaystyle 0<a<1$ )
$\displaystyle y=a\cdot f(x)$
Absolute value: the part below the $\displaystyle x$ -axis is reflected above it
$\displaystyle y=|f(x)|$

The rational function

The horizontal asymptote when the numerator and the denominator have the same degree – $\displaystyle a$ and $\displaystyle b$ the coefficients of the highest power
$\displaystyle y=\frac{a}{b}$
The horizontal asymptote when the numerator’s degree is lower than the denominator’s
$\displaystyle y=0$
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
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$

Derivatives and the tangent

The derivative of a power
$\displaystyle \left(x^{n}\right)'=nx^{n-1}$
The derivative of a product
$\displaystyle \left(u\cdot v\right)'$ $\displaystyle =$ $\displaystyle u'\cdot v+u\cdot v'$
The derivative of a quotient
$\displaystyle \left(\frac{u}{v}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}$
The derivative of $\displaystyle \frac{c}{x}$
$\displaystyle \left(\frac{c}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{c}{x^{2}}$
The derivative of a root – $\displaystyle u$ a function of $\displaystyle x$
$\displaystyle \left(\sqrt{u}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'}{2\sqrt{u}}$
The equation of the tangent at the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The slope of the tangent
$\displaystyle m$ $\displaystyle =$ $\displaystyle f'(x_1)$

Investigating a function with its derivative

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A possible extremum point – check the sign of $\displaystyle f'$ on each side of it
$\displaystyle f'(x)=0$

Integrals and areas

The integral of a power
$\displaystyle \int x^{n}\,dx$ $\displaystyle =$ $\displaystyle \frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
The integral of $\displaystyle \frac{c}{(ax+b)^{2}}$
$\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx$ $\displaystyle =$ $\displaystyle -\frac{c}{a(ax+b)}+C$
The definite integral – $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx$ $\displaystyle =$ $\displaystyle F(b)-F(a)$
The area between two graphs, from $\displaystyle {x}={a}$ to $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$