The second derivative and concavity

Calculus, grade 11 (5 units) Lesson 3 of 26Learning

The second derivative and concavity

The second derivative

The second derivative of $\displaystyle f$ is the derivative of its derivative – the rate of change of the slope:

\displaystyle f''(x)

\displaystyle =

\displaystyle \left(f'(x)\right)'

For example, for $\displaystyle {f(x)}={x^3-5x}$ :

$\displaystyle {f'(x)}={3x^2-5}$ ;
$\displaystyle {f''(x)}={6x}$ .

The sign of

\displaystyle f''

and concavity

$\displaystyle f''$ is the derivative of $\displaystyle f'$ , so its sign says whether $\displaystyle f'$ rises or falls – and that is exactly the concavity of $\displaystyle f$ :

\displaystyle f''(x)>0

– $\displaystyle f'$ rises, and $\displaystyle f$ is concave up ( $\displaystyle \cup$ ).

\displaystyle f''(x)<0

– $\displaystyle f'$ falls, and $\displaystyle f$ is concave down ( $\displaystyle \cap$ ).

To find the intervals of concavity, solve the inequalities $\displaystyle f''(x)>0$ and $\displaystyle f''(x)<0$ , just as you find where a function rises and falls from $\displaystyle f'$ .

Example:

\displaystyle x^3-4x^2-3x+7

The example below finds the intervals of concavity of $\displaystyle {f(x)}={x^3-4x^2-3x+7}$ :

$\displaystyle {f'(x)}={3x^2-8x-3}$ .
$\displaystyle {f''(x)}={6x-8}$ .
$\displaystyle f''(x)>0$ where $\displaystyle x>\frac{4}{3}$ , and $\displaystyle f''(x)<0$ where $\displaystyle x<\frac{4}{3}$ .

\displaystyle f(x)=x^{3}-4x^{2}-3x+7

$\displaystyle f$ is concave down where $\displaystyle x<\frac{4}{3}$ , and concave up where $\displaystyle x>\frac{4}{3}$ .
The maximum point, $\displaystyle {x}={-\frac{1}{3}}$ , lies where the function is concave down, and the minimum point, $\displaystyle {x}={3}$ , where it is concave up.

Example:

\displaystyle x^4-6x^2

The example below finds the intervals of concavity of $\displaystyle {f(x)}={x^4-6x^2}$ :

$\displaystyle {f'(x)}={4x^3-12x}$ .
$\displaystyle {f''(x)}={12x^2-12}={12(x-1)(x+1)}$ .

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

\displaystyle f(x)=x^{4}-6x^{2}

$\displaystyle f$ is concave up where $\displaystyle x<-1$ and where $\displaystyle x>1$ , and concave down where $\displaystyle -1<x<1$ .
The graph changes the way it bends twice: at $\displaystyle {x}={-1}$ and at $\displaystyle {x}={1}$ .

Exercise 1

Find the second derivative of each function.

$\displaystyle f(x)=x^{3}-6x^{2}+5$

/ – 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

$\displaystyle f(x)=x^{4}+2x^{3}$

/ – 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

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

/ – 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

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

/ – 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

Exercise 2

The function is $\displaystyle {f(x)}={x^4+2x^3-12x^2}$ .

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

$\displaystyle f(x)=x^{4}+2x^{3}-12x^{2}$

/ – 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

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

Write the intervals of concavity of $\displaystyle f$ : where is it concave up, and where concave down?

Exercise 3

The function is $\displaystyle {f(x)}={\frac{1}{x}}$ .
Its second derivative is $\displaystyle {f''(x)}={\frac{2}{x^3}}$ .

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

Write the intervals of concavity of $\displaystyle f$ .
The concavity changes at $\displaystyle {x}={0}$ . What is special about that point?

Exercise 4

The function is $\displaystyle {f(x)}={x^3+ax^2}$ , where $\displaystyle a$ is a parameter.
The function is concave down where $\displaystyle x<2$ , and concave up where $\displaystyle x>2$ .

Find $\displaystyle f''(x)$ , in terms of $\displaystyle a$ .

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

/ – 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

The sign of $\displaystyle f''$ changes at $\displaystyle {x}={2}$ , so $\displaystyle {f''(2)}={0}$ .
Solve the equation and find $\displaystyle a$ .

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