Rational functions: parameters and tangents

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

Rational functions: parameters and tangents

A family with a parameter

A letter in the formula besides $\displaystyle x$ – a parameter – makes a family of functions, one for each value of the letter. A fact about one member of the family is an equation in the parameter. The example below works with the family $\displaystyle {f(x)}={\frac{x^2+a}{x}}$ :

$\displaystyle {f'(x)}={\frac{2x\cdot x-(x^2+a)}{x^2}}={\frac{x^2-a}{x^2}}$ .
For an extremum point at $\displaystyle {x}={2}$ : $\displaystyle {f'(2)}={0}$ , so $\displaystyle {\frac{4-a}{4}}={0}$ and $\displaystyle {a}={4}$ .
For $\displaystyle {a}={4}$ : $\displaystyle {f(x)}={\frac{x^2+4}{x}}$ , with a minimum point $\displaystyle (2,4)$ and a maximum point $\displaystyle (-2,-4)$ .
For $\displaystyle a<0$ : $\displaystyle x^2-a>0$ for every $\displaystyle x$ , so $\displaystyle f'(x)>0$ throughout the domain – the function rises, and has no extremum points.

The tangent at a point of the graph

The tangent at the point of the graph where $\displaystyle {x}={x_1}$ is the line through $\displaystyle (x_1,f(x_1))$ with the slope $\displaystyle f'(x_1)$ . The example below finds the tangent to $\displaystyle {f(x)}={\frac{4}{x}}$ at the point where $\displaystyle {x}={2}$ :

$\displaystyle {f(2)}={2}$ : the point $\displaystyle (2,2)$ ;
$\displaystyle {f'(x)}={-\frac{4}{x^2}}$ , so $\displaystyle {f'(2)}={-1}$ ;
$\displaystyle {y-2}={-1\cdot(x-2)}$ , that is $\displaystyle {y}={-x+4}$ .

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

\displaystyle y=-x+4

\displaystyle (2,2)

Tangents with a given slope

When the slope of the tangent is given and the point is not, the equation $\displaystyle {f'(x)}={m}$ gives the points. The example below finds the tangents to $\displaystyle {f(x)}={\frac{x}{x-1}}$ whose slope is $\displaystyle -1$ :

$\displaystyle {f'(x)}={\frac{(x-1)-x}{(x-1)^2}}={\frac{-1}{(x-1)^2}}$ ;
$\displaystyle {\frac{-1}{(x-1)^2}}={-1}$ gives $\displaystyle {(x-1)^2}={1}$ , so $\displaystyle {x}={0}$ or $\displaystyle {x}={2}$ ;
the points $\displaystyle (0,0)$ and $\displaystyle (2,2)$ , and the tangents $\displaystyle {y}={-x}$ and $\displaystyle {y}={-x+4}$ .

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

\displaystyle x=1

\displaystyle y=1

\displaystyle y=-x

\displaystyle y=-x+4

The two tangents are parallel: the graph of $\displaystyle f$ is symmetric about the point $\displaystyle (1,1)$ where its asymptotes meet.

The asymptotes of a moved graph

The asymptotes move with the graph:

$\displaystyle {g(x)}={f(x-p)+k}$ : the graph moves $\displaystyle p$ to the right and $\displaystyle k$ up, so an asymptote $\displaystyle {x}={a}$ becomes $\displaystyle {x}={a+p}$ , and $\displaystyle {y}={b}$ becomes $\displaystyle {y}={b+k}$ ;
$\displaystyle {g(x)}={|f(x)|}$ : the parts of the graph under the $\displaystyle x$ -axis flip above it; a vertical asymptote stays, and $\displaystyle {y}={b}$ becomes $\displaystyle {y}={|b|}$ .

Exercise 1

The family is $\displaystyle {f(x)}={\frac{ax}{x^2+4}}$ , with $\displaystyle a\neq0$ .
One of the functions has a maximum point at $\displaystyle (2,1)$ .

Find $\displaystyle f'(x)$ .

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

/ – 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

Explain why every function in the family has extremum points at $\displaystyle {x}={\pm2}$ .

$\displaystyle {f(2)}={\frac{2a}{8}}$ .
Find $\displaystyle a$ .

Mark the extremum points of the function you found.

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

Exercise 2

The function is $\displaystyle {f(x)}={\frac{x+1}{x-1}}$ .

Find $\displaystyle f'(x)$ .

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

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

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

\displaystyle x=1

\displaystyle (3,2)

At which points of the graph is the slope of the tangent $\displaystyle -2$ ?

Write on the board the equations of the two tangents whose slope is $\displaystyle -2$ .

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

\displaystyle x=1

Exercise 3

The family is $\displaystyle {f(x)}={\frac{x^2+a}{x-1}}$ .

Find $\displaystyle f'(x)$ .

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

/ – 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

One function of the family has an extremum point at $\displaystyle {x}={3}$ .
$\displaystyle {f'(3)}={\frac{9-6-a}{4}}$ .
Find $\displaystyle a$ .

$\displaystyle f$ has no extremum points when the numerator of $\displaystyle f'$ , $\displaystyle x^2-2x-a$ , has no roots apart from a root it shares with the denominator.
Its discriminant is $\displaystyle 4+4a$ .
Solve the inequality.

What is the graph of $\displaystyle f$ for $\displaystyle {a}={-1}$ ? Explain why it has no extremum points.

Exercise 4

The function is $\displaystyle {f(x)}={\frac{2}{x}}$ .
$\displaystyle {g(x)}={f(x-1)+3}$ .

Write on the board the equations of the asymptotes of $\displaystyle g$ .

\displaystyle g(x)=\frac{2}{x-1}+3

What are the asymptotes of $\displaystyle {h(x)}={|f(x)-3|}$ ? Explain.

Formulas: calculus, grade 11 (5 units)

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