A full investigation of a rational function

Calculus, grade 11 (5 units) Lesson 10 of 26Learning

A full investigation of a rational function

The steps of a full investigation

A full investigation of a rational function takes the steps of grade 10 and adds the second derivative:

the domain, asymptotes and holes, even or odd, the points on the axes;
the first derivative: a table of signs of $\displaystyle f'$ – where $\displaystyle f$ rises and falls, and its extremum points;
the second derivative: a table of signs of $\displaystyle f''$ – where $\displaystyle f$ is concave up and concave down;
inflection points: points of the domain where the sign of $\displaystyle f''$ changes;
a sketch: the asymptotes, then the points, then the shape of each piece by its concavity.

In both tables the sign may change at a point where $\displaystyle f$ is not defined, so those points go into the tables too.

Example:

\displaystyle \frac{x}{x^2+1}

The example below investigates $\displaystyle {f(x)}={\frac{x}{x^2+1}}$ in full.

The domain: every $\displaystyle x$ , since $\displaystyle x^2+1>0$ ; no vertical asymptote.
The numerator’s degree is lower: the horizontal asymptote $\displaystyle {y}={0}$ .
$\displaystyle {f(-x)}={\frac{-x}{x^2+1}}={-f(x)}$ : the function is odd, and its graph is symmetric about the origin.
The points on the axes: only $\displaystyle (0,0)$ .

The first derivative:

f'(x)=\frac{1\cdot(x^2+1)-x\cdot2x}{(x^2+1)^2}=\frac{1-x^2}{(x^2+1)^2}

$\displaystyle {f'(x)}={0}$ where $\displaystyle {x}={\pm1}$ ; $\displaystyle f$ falls where $\displaystyle x<-1$ and where $\displaystyle x>1$ , and rises where $\displaystyle -1<x<1$ .
$\displaystyle \left(-1,-\frac{1}{2}\right)$ – a minimum point; $\displaystyle \left(1,\frac{1}{2}\right)$ – a maximum point.

The second derivative, by the quotient rule again and taking out the common factor $\displaystyle x^2+1$ :

f''(x)=\frac{-2x(x^2+1)^2-(1-x^2)\cdot2(x^2+1)\cdot2x}{(x^2+1)^4}=\frac{2x(x^2-3)}{(x^2+1)^3}

The denominator is positive, so $\displaystyle f''$ has the sign of $\displaystyle 2x(x^2-3)$ , which is 0 at $\displaystyle {x}={0}$ and $\displaystyle {x}={\pm\sqrt{3}}$ :

$\displaystyle x$	$\displaystyle x<-\sqrt{3}$	$\displaystyle -\sqrt{3}<x<0$	$\displaystyle 0<x<\sqrt{3}$	$\displaystyle x>\sqrt{3}$
$\displaystyle f''(x)$	−	+	−	+
$\displaystyle f(x)$	$\displaystyle \cap$	$\displaystyle \cup$	$\displaystyle \cap$	$\displaystyle \cup$

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

\displaystyle A=(-1.732,-0.433)

\displaystyle O=(0,0)

\displaystyle B=(1.732,0.433)

Three inflection points: $\displaystyle \left(-\sqrt{3},-\frac{\sqrt{3}}{4}\right)$ , $\displaystyle (0,0)$ and $\displaystyle \left(\sqrt{3},\frac{\sqrt{3}}{4}\right)$ .
Far to the right the graph is concave up while it approaches $\displaystyle {y}={0}$ from above: it falls more and more slowly.

Example: concavity across an asymptote

The concavity of a rational function may change across a vertical asymptote, but there is no inflection point there: an inflection point is a point of the graph. The example below shows it for $\displaystyle {f(x)}={\frac{1}{x^2-1}}$ .

The domain: $\displaystyle x\neq\pm1$ ; the vertical asymptotes $\displaystyle {x}={-1}$ , $\displaystyle {x}={1}$ , and the horizontal asymptote $\displaystyle {y}={0}$ .
The function is even.
$\displaystyle {f'(x)}={\frac{-2x}{(x^2-1)^2}}$ : rising where $\displaystyle x<0$ (but $\displaystyle x\neq-1$ ), falling where $\displaystyle x>0$ (but $\displaystyle x\neq1$ ); $\displaystyle (0,-1)$ – a maximum point.
$\displaystyle {f''(x)}={\frac{6x^2+2}{(x^2-1)^3}}$ : the numerator is always positive, so $\displaystyle f''$ has the sign of $\displaystyle (x^2-1)^3$ , that is of $\displaystyle x^2-1$ .

$\displaystyle x$	$\displaystyle x<-1$	$\displaystyle -1<x<1$	$\displaystyle x>1$
$\displaystyle f''(x)$	+	−	+
$\displaystyle f(x)$	$\displaystyle \cup$	$\displaystyle \cap$	$\displaystyle \cup$

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

\displaystyle x=-1

\displaystyle x=1

\displaystyle (0,-1)

The concavity changes at $\displaystyle {x}={\pm1}$ , but $\displaystyle f$ is not defined there: no inflection points.

Exercise 1

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

Find $\displaystyle f'(x)$ .

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

/ – 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

Find $\displaystyle f''(x)$ .

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

/ – 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

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

Mark on the graph the extremum point and the inflection points.

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

Write where $\displaystyle f$ is concave up and where it is concave down, and the coordinates of its inflection points.

Exercise 2

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

Find $\displaystyle f'(x)$ .

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

/ – 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

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

Find $\displaystyle f''(x)$ .

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

/ – 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

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

Mark the extremum points on the graph.

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

\displaystyle x=1

Write the type of each extremum point, and where $\displaystyle f$ is concave up and concave down.
Explain why $\displaystyle f$ has no inflection point although its concavity changes.

Exercise 3

Organic waste is buried in a lake at the time $\displaystyle {t}={0}$ .
The share of oxygen in the lake, $\displaystyle t$ weeks later, is $\displaystyle {p(t)}={\frac{t^2-t+1}{t^2+1}}$ , for $\displaystyle t\geq0$ .
$\displaystyle {p(t)}={1}$ is the normal level.

Find $\displaystyle p'(t)$ .

$\displaystyle p(t)=\frac{t^{2}-t+1}{t^{2}+1}$

/ – 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

Solve the equation $\displaystyle {p'(t)}={0}$ .

When is the share of oxygen at its lowest? Work out the lowest share.

$\displaystyle p(t)=\frac{t^{2}-t+1}{t^{2}+1}$

/ – 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

The oxygen grows fastest where $\displaystyle p'$ is greatest, that is where $\displaystyle {p''(t)}={0}$ and its sign changes from + to −.
Find $\displaystyle p''(t)$ .

$\displaystyle p(t)=\frac{t^{2}-t+1}{t^{2}+1}$

/ – 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

Solve the equation $\displaystyle {p''(t)}={0}$ .

After how many weeks does the oxygen grow fastest? Explain which solutions fit the story.

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