The tangent to a rational function's graph

Calculus, grade 11 (4 units) Lesson 7 of 20Learning

The tangent to a rational function's graph

The equation of the tangent

The tangent to the graph at the point $\displaystyle (x_1,y_1)$ is the line through the point whose slope is the derivative there – as for polynomials in grade 10:

\displaystyle m

\displaystyle =

\displaystyle f'(x_1)

\displaystyle y-y_1

\displaystyle =

\displaystyle m(x-x_1)

$\displaystyle {y_1}={f(x_1)}$ – substitute in $\displaystyle f$ , not in $\displaystyle f'$ ;
$\displaystyle {m}={f'(x_1)}$ – substitute in $\displaystyle f'$ ;
substitute in the equation of the line, and rewrite it as $\displaystyle {y}={mx+b}$ .

Example

$\displaystyle {f(x)}={\frac{x+1}{x-2}}$ .
Find the equation of the tangent to the graph at the point where $\displaystyle {x}={3}$ .

In the solution below we work out $\displaystyle y_1$ and the slope, and substitute in the equation of the line:

$\displaystyle {y_1}={f(3)}={\frac{4}{1}}={4}$ – the point $\displaystyle (3,4)$ ;
$\displaystyle {f'(x)}={\frac{-3}{(x-2)^{2}}}$ (from the previous lesson), so $\displaystyle {m}={f'(3)}={\frac{-3}{1}}={-3}$ ;
$\displaystyle {y-4}={-3(x-3)}$ , that is $\displaystyle {y}={-3x+13}$ .

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

\displaystyle y=-3x+13

\displaystyle x=2

\displaystyle A=(3,4)

A tangent with a given slope

When the slope of the tangent is given and the point is not, look for the $\displaystyle x$ where the derivative equals the slope – solve the equation $\displaystyle {f'(x)}={m}$ :

a tangent parallel to the $\displaystyle x$ -axis: the slope is 0, $\displaystyle {f'(x)}={0}$ ;
a tangent parallel to the line $\displaystyle {y}={mx+b}$ : parallel lines have the same slope, $\displaystyle {f'(x)}={m}$ .

Example: $\displaystyle {f(x)}={x+\frac{4}{x}}$ , and $\displaystyle {f'(x)}={1-\frac{4}{x^{2}}}$ .
$\displaystyle {1-\frac{4}{x^{2}}}={0}$ gives $\displaystyle {x^{2}}={4}$ , that is $\displaystyle {x}={2}$ or $\displaystyle {x}={-2}$ .
$\displaystyle {f(2)}={4}$ and $\displaystyle {f(-2)}={-4}$ , so the tangents parallel to the $\displaystyle x$ -axis are $\displaystyle {y}={4}$ and $\displaystyle {y}={-4}$ (the dashed ones).

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

\displaystyle y=4

\displaystyle y=-4

\displaystyle x=0

Exercise 1

$\displaystyle {f(x)}={\frac{2x}{x+1}}$ .
The point $\displaystyle (1,1)$ is on the graph.

Work out the slope of the tangent at this point.

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

/ – 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

Write the equation of the tangent on the board.

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

\displaystyle (1,1)

Exercise 2

$\displaystyle {f(x)}={1-\frac{1}{(x-5)^{2}}}$ .

Find the domain of $\displaystyle f$ and its asymptotes parallel to the axes.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=1-\frac{1}{(x-5)^{2}}$

/ – 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 $\displaystyle {x}={4}$ on the board.

\displaystyle f(x)=1-\frac{1}{(x-5)^{2}}

\displaystyle x=5

\displaystyle y=1

\displaystyle (4,0)

On the board – the tangent and the asymptotes. Mark the point where the tangent meets the horizontal asymptote.

\displaystyle f(x)=1-\frac{1}{(x-5)^{2}}

\displaystyle x=5

\displaystyle y=1

\displaystyle y=-2x+8

Where does the tangent meet each of the two asymptotes?

Exercise 3

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

Find $\displaystyle f'(x)$ .

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

/ – 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

Find the points of the graph where the tangent is parallel to the line $\displaystyle {y}={-x+5}$ : solve the equation $\displaystyle {f'(x)}={-1}$ .

Write the equations of the two tangents on the board.

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

\displaystyle x=1

Exercise 4

$\displaystyle {f(x)}={\frac{x^{2}}{x-1}}$ , and $\displaystyle {f'(x)}={\frac{x^{2}-2x}{(x-1)^{2}}}$ .

Where is the tangent to the graph parallel to the $\displaystyle x$ -axis? Solve the equation $\displaystyle {f'(x)}={0}$ .

Write the equations of the tangents parallel to the $\displaystyle x$ -axis.

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