Practice: rational functions

Calculus, grade 11 (4 units) Lesson 10 of 20Practice

Practice: rational functions

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

Find the domain of $\displaystyle f$ , and write the equation of its vertical asymptote on the board.

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

Find $\displaystyle f'(x)$ .

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

/ – 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

Solve the equation $\displaystyle {f'(x)}={0}$ .

Write the sign table of $\displaystyle f'$ , and where $\displaystyle f$ rises and falls.

Mark the extremum points of $\displaystyle f$ , and write the kind of each one.

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

\displaystyle x=2

$\displaystyle {f(x)}={\frac{3x}{x^{2}+9}}$ .

Is $\displaystyle f$ even, odd or neither? Write $\displaystyle f(-x)$ and simplify.

$\displaystyle f(x)=\frac{3x}{x^{2}+9}$

/ – 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 $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{3x}{x^{2}+9}$

/ – 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 $\displaystyle {f'(x)}={0}$ .

Mark the extremum points of $\displaystyle f$ , and write the equation of its horizontal asymptote on the board.

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

$\displaystyle {f(x)}={\frac{x+3}{x-1}}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{x+3}{x-1}$

/ – 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 the equation of the tangent to the graph at the point where $\displaystyle {x}={3}$ on the board.

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

\displaystyle x=1

The graph has one more tangent parallel to this one.
Solve an equation that gives the $\displaystyle x$ of its point of contact, and write this tangent too on the board of part b.

$\displaystyle {f(x)}={\frac{ax+2}{x-3}}$ , and $\displaystyle a$ is a parameter.
The graph of $\displaystyle f$ passes through the point $\displaystyle (4,10)$ .

Find $\displaystyle a$ .

Substitute $\displaystyle {a}={2}$ , and find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{2x+2}{x-3}$

/ – 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

Prove that $\displaystyle f$ falls on its whole domain. Does it have extremum points?

On the board – the graph of $\displaystyle {g(x)}={f'(x)}$ , the derivative of a function $\displaystyle f$ defined for every $\displaystyle x$ .

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

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

Find the $\displaystyle x$ of the extremum point of $\displaystyle f$ , and say what kind it is.

\displaystyle 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

On the board – the graph of the function $\displaystyle {f(x)}={\frac{4x+1}{ax^{2}-2x}}$ , and $\displaystyle a$ is a parameter.
$\displaystyle {x}={2}$ is a vertical asymptote of $\displaystyle f$ .

\displaystyle f(x)=\,?

\displaystyle x=2

Find $\displaystyle a$ .

Substitute the $\displaystyle a$ you found. Find the domain of $\displaystyle f$ and all its asymptotes parallel to the axes.

Find $\displaystyle f'(x)$ .

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

/ – 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 $\displaystyle {f'(x)}={0}$ .

Mark the extremum points of $\displaystyle f$ , and write where it rises and falls.

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

\displaystyle x=0

\displaystyle x=2

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 $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$