Practice: the second derivative

Calculus, grade 11 (5 units) Lesson 8 of 26Practice

Practice: the second derivative

Find the second derivative of each function.

$\displaystyle f(x)=x^{4}-2x^{3}+5x$

/ – 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

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

/ – 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

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

/ – 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

The function is $\displaystyle {f(x)}={x^3+3x^2-9x}$ .
Its second derivative is $\displaystyle {f''(x)}={6x+6}$ .

Solve the inequality $\displaystyle f''(x)>0$ : where is $\displaystyle f$ concave up?

Mark the inflection point on the graph.

\displaystyle f(x)=x^{3}+3x^{2}-9x

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

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

$\displaystyle {f''(x)}={2x-2}$ .
Work out $\displaystyle f''$ at the two points you found.

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

/ – 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

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

/ – 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 points on the graph.

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

Write the type of each extremum point, from the sign of $\displaystyle f''$ .

The function is $\displaystyle {f(x)}={x^4-6x^3+12x^2}$ .

$\displaystyle f(x)=x^{4}-6x^{3}+12x^{2}$

/ – 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 equation $\displaystyle {f''(x)}={0}$ .

Mark the inflection points on the graph.

\displaystyle f(x)=x^{4}-6x^{3}+12x^{2}

The function is $\displaystyle {f(x)}={x^4-4x^3+6x^2}$ .

$\displaystyle f(x)=x^{4}-4x^{3}+6x^{2}$

/ – 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 equation $\displaystyle {f''(x)}={0}$ .

Is the point you found an inflection point? Give your reason.

The board shows the graph of the derivative $\displaystyle {f'(x)}={x(x-2)^2}$ of a function $\displaystyle f$ .

Mark on the graph the points where it meets the $\displaystyle x$ -axis, and its extremum points.

\displaystyle y=x(x-2)^{2}

Which points of $\displaystyle f$ do the points you marked belong to?
Write the extremum points of $\displaystyle f$ and their types, and the $\displaystyle x$ -coordinates of the inflection points of $\displaystyle f$ .

A body moves along a straight line by $\displaystyle {s(t)}={t^3-9t^2+24t}$ , with $\displaystyle t\geq0$ (metres, seconds).

Find the acceleration $\displaystyle {a(t)}={s''(t)}$ .

$\displaystyle s(t)=t^{3}-9t^{2}+24t$

/ – 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

When is the acceleration 0?

Work out the speed at that moment.
At that moment the speed is at its smallest: why?

$\displaystyle s(t)=t^{3}-9t^{2}+24t$

/ – 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 family of functions is $\displaystyle {f(x)}={x^4+ax^3}$ , where $\displaystyle a$ is a parameter.

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

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

/ – 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

For which value of $\displaystyle a$ does the function have an inflection point at $\displaystyle {x}={1}$ ?

The board shows the graph of $\displaystyle {f(x)}={x^4-2x^2}$ and two graphs whose formulas are hidden: of $\displaystyle f'$ and of $\displaystyle f''$ .
Write on the board the formula of each hidden graph.

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

\displaystyle y=\,?

\displaystyle y=\,?

For each claim choose: always, sometimes or never – and prove it.
A function of degree two has an inflection point.
A function of degree three has an inflection point.
A function of degree four has an inflection point.

The function $\displaystyle f$ can be differentiated twice, and it has a minimum point at $\displaystyle x_0$ .
Explain why $\displaystyle f''(x_0)$ is not negative.

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