Investigating functions with square roots

Calculus, grade 11 (5 units) Lesson 15 of 26Learning

Investigating functions with square roots

The steps with a root

A function with a root is investigated by the same steps as a rational function, with three things to watch:

the domain comes first, and every table of signs starts and ends at its ends;
an end of the domain is an extremum point – an extremum at the end – by what the function does beside it;
the asymptotes may differ at $\displaystyle \infty$ and at $\displaystyle -\infty$ , by $\displaystyle {\sqrt{x^2}}={|x|}$ .

Example:

\displaystyle x\sqrt{4-x^2}

The example below investigates $\displaystyle {f(x)}={x\sqrt{4-x^2}}$ in full.

The domain: $\displaystyle 4-x^2\geq0$ , that is $\displaystyle -2\leq x\leq2$ .
$\displaystyle {f(-x)}={-x\sqrt{4-x^2}}={-f(x)}$ : the function is odd.
The zeros: $\displaystyle {x}={0}$ and $\displaystyle {x}={\pm2}$ .
$\displaystyle {f'(x)}={\frac{4-2x^2}{\sqrt{4-x^2}}}$ (lesson 14): 0 at $\displaystyle {x}={\pm\sqrt{2}}$ .

$\displaystyle x$	$\displaystyle -2$	$\displaystyle -2<x<-\sqrt{2}$	$\displaystyle -\sqrt{2}$	$\displaystyle -\sqrt{2}<x<\sqrt{2}$	$\displaystyle \sqrt{2}$	$\displaystyle \sqrt{2}<x<2$	$\displaystyle 2$
$\displaystyle f'(x)$		−	0	+	0	−
$\displaystyle f(x)$	maximum at the end	falls	minimum	rises	maximum	falls	minimum at the end

The second derivative, by the quotient rule on $\displaystyle f'$ :

f''(x)=\frac{x(2x^2-12)}{(4-x^2)\sqrt{4-x^2}}

In the domain $\displaystyle 2x^2-12<0$ and the denominator is positive, so $\displaystyle f''$ has the sign of $\displaystyle -x$ .
$\displaystyle f$ is concave up where $\displaystyle -2<x<0$ and concave down where $\displaystyle 0<x<2$ : $\displaystyle (0,0)$ is an inflection point.
$\displaystyle {f'(0)}={2}$ : the tangent at the inflection point is slanted, not level.

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

\displaystyle A=(1.414,2)

\displaystyle B=(-1.414,-2)

\displaystyle (0,0)

$\displaystyle (\sqrt{2},2)$ – a maximum point, $\displaystyle (-\sqrt{2},-2)$ – a minimum point.
$\displaystyle (-2,0)$ – a maximum at the end, $\displaystyle (2,0)$ – a minimum at the end.

Example:

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

The example below investigates $\displaystyle {f(x)}={\frac{x}{\sqrt{x^2+1}}}$ , whose asymptotes $\displaystyle {y}={1}$ and $\displaystyle {y}={-1}$ you found in lesson 13.

The domain: every $\displaystyle x$ ; the function is odd.
$\displaystyle {f'(x)}={\frac{\sqrt{x^2+1}-x\cdot\frac{x}{\sqrt{x^2+1}}}{x^2+1}}={\frac{1}{(x^2+1)\sqrt{x^2+1}}>0}$ : $\displaystyle f$ rises throughout, and has no extremum points.
$\displaystyle {f''(x)}={\frac{-3x}{(x^2+1)^2\sqrt{x^2+1}}}$ : positive where $\displaystyle x<0$ , negative where $\displaystyle x>0$ .
$\displaystyle (0,0)$ is an inflection point: the graph rises steepest there, with the slope $\displaystyle {f'(0)}={1}$ .

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

\displaystyle y=1

\displaystyle y=-1

\displaystyle (0,0)

Exercise 1

The function is $\displaystyle {f(x)}={\sqrt{x}-\frac{x}{4}}$ .

Find $\displaystyle f'(x)$ .

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

/ – 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

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

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

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

/ – 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

Mark on the graph the extremum point inside the domain and the zeros.

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

Write all the extremum points of $\displaystyle f$ and their types, the end included.
Where is $\displaystyle f$ concave up, and where concave down? Does it have an inflection point?

Exercise 2

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

Find $\displaystyle f'(x)$ , over one denominator.

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

/ – 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 $\displaystyle f''(x)$ .

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

/ – 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 on the graph the zeros and the extremum point inside the domain.

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

Write all the extremum points and their types, and the concavity of $\displaystyle f$ .

Exercise 3

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

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

Work out the value of the function at the point you found.

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

/ – 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

Mark the extremum point inside the domain.

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

Write the domain, the asymptote, all the extremum points and their types.
What is the tangent at $\displaystyle (0,0)$ ?

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