Investigating a root function

Calculus, grade 11 (4 units) Lesson 12 of 20Learning

Investigating a root function

The steps of an investigation

Investigating a root function takes the same steps as investigating a rational function, and one more – the ends of the domain:

The domain: the expression under the root is greater than or equal to 0.
The derivative, as one fraction.
The points where $\displaystyle {f'(x)}={0}$ , inside the domain.
The sign of $\displaystyle f'$ between these points: $\displaystyle f'>0$ – rising, $\displaystyle f'<0$ – falling.
The extremum points: inside – where $\displaystyle f'$ changes sign; and at the end of the domain.
The points on the axes, and then a sketch.

An extremum at the end of the domain

At the end of the domain the graph starts or stops. If it falls towards the end, the end is lower than every point near it – a minimum point at the end. If it rises towards the end – a maximum point at the end.

The derivative at the end need not be 0, and usually it is not defined there: the root in its denominator is 0.
So the ends are checked on their own: put them into the function, and see whether it rises or falls next to them.

Example

$\displaystyle {f(x)}={x\sqrt{6-x}}$ .
In the investigation below we find the domain, the extremum points and where the function rises and falls:

The domain: $\displaystyle 6-x\geq0$ , that is $\displaystyle x\leq6$ .
The derivative: $\displaystyle {f'(x)}={\sqrt{6-x}-\frac{x}{2\sqrt{6-x}}}={\frac{12-3x}{2\sqrt{6-x}}}$ .
$\displaystyle {f'(x)}={0}$ when $\displaystyle {12-3x}={0}$ , that is $\displaystyle {x}={4}$ ; $\displaystyle {f(4)}={4\sqrt{2}}$ .
The sign of $\displaystyle f'$ : the denominator is positive, so the sign is that of $\displaystyle 12-3x$ .

$\displaystyle x$	$\displaystyle x<4$	$\displaystyle 4$	$\displaystyle 4<x<6$	$\displaystyle 6$
$\displaystyle f'(x)$	$\displaystyle +$	$\displaystyle 0$	$\displaystyle -$	not defined
$\displaystyle f(x)$	rises	maximum	falls	minimum at the end

$\displaystyle (4,4\sqrt{2})$ – a maximum point; $\displaystyle (6,0)$ – a minimum point at the end of the domain.
The points on the axes: $\displaystyle (0,0)$ and $\displaystyle (6,0)$ .

\displaystyle f(x)=x\sqrt{6-x}

\displaystyle M=(4,5.66)

\displaystyle B=(6,0)

When $\displaystyle x$ is very small, $\displaystyle x$ is a large negative number and $\displaystyle \sqrt{6-x}$ grows, so $\displaystyle f$ falls without end to the left. $\displaystyle f$ has no value greater than $\displaystyle 4\sqrt{2}$ , but it has no least value.

Exercise 1

$\displaystyle {f(x)}={(x-3)\sqrt{x}}$ , and its domain is $\displaystyle x\geq0$ .

Differentiate $\displaystyle f$ , and write the derivative as one fraction.

$\displaystyle f(x)=(x-3)\sqrt{x}$

/ – 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 $\displaystyle {f'(x)}={0}$ .

On the board, mark the extremum point of $\displaystyle f$ inside the domain and the points where its graph meets the $\displaystyle x$ -axis.

\displaystyle f(x)=(x-3)\sqrt{x}

What kind of extremum point is the one inside the domain? Which extremum point is at the end of the domain, and of which kind? Write where $\displaystyle f$ rises and where it falls.

Exercise 2

$\displaystyle {f(x)}={\sqrt{x^2-4}}$ .
The domain: $\displaystyle x\leq-2$ or $\displaystyle x\geq2$ .

Differentiate $\displaystyle f$ .

$\displaystyle f(x)=\sqrt{x^{2}-4}$

/ – 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 $\displaystyle {f'(x)}={0}$ . Careful: a solution outside the domain is no solution.

On the board, mark the points where the graph meets the $\displaystyle x$ -axis.

\displaystyle f(x)=\sqrt{x^{2}-4}

Where does $\displaystyle f$ rise, and where does it fall? What are its extremum points?

Exercise 3

$\displaystyle {f(x)}={x+2\sqrt{1-x}}$ , and its domain is $\displaystyle x\leq1$ .

Differentiate $\displaystyle f$ .

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

/ – 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

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

On the board, mark the extremum point inside the domain.

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

Work out $\displaystyle f(1)$ . What kind of point is $\displaystyle (1,f(1))$ ? Does $\displaystyle f$ have a least value?

\displaystyle f(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

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