Course review

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

Course review

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

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

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

/ – 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 inequality $\displaystyle f''(x)<0$ : where is $\displaystyle f$ concave down?

Mark the inflection points on the graph.

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

The function is $\displaystyle {f(x)}={x^3-3x}$ , and its derivative is $\displaystyle {f'(x)}={3x^2-3}$ .

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

Work out $\displaystyle f''$ at each of the points you found.

\displaystyle x<0

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

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\displaystyle x>0

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

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Decide by the second derivative the kind of each extremum point.
Why can this not decide the kind of the point of $\displaystyle {g(x)}={x^3}$ where $\displaystyle {x}={0}$ ?

The board shows the graph of the derivative $\displaystyle f'$ of a function $\displaystyle f$ : $\displaystyle {f'(x)}={x^2-4x+3}$ .
Mark on the graph of $\displaystyle f'$ the points where $\displaystyle {f'(x)}={0}$ and the extremum point of $\displaystyle f'$ .

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

Write, from the graph of $\displaystyle f'$ :

where $\displaystyle f$ rises and where it falls;
the $\displaystyle x$ -coordinates of the extremum points of $\displaystyle f$ and their kind;
the $\displaystyle x$ -coordinate of the inflection point of $\displaystyle f$ , and its intervals of concavity.

The

\displaystyle x

of the inflection point

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

The function is defined for every $\displaystyle x$ . Write on the board the equation of its horizontal asymptote.

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

Find $\displaystyle f'(x)$ .

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

/ – 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

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

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

/ – 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 $\displaystyle f''(x)>0$ : where is $\displaystyle f$ concave up?

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

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

\displaystyle y=1

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

$\displaystyle f$ has two different horizontal asymptotes: one as $\displaystyle x\to\infty$ and one as $\displaystyle x\to-\infty$ .
Write the equations of both on the board.

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

Find $\displaystyle f'(x)$ .

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

/ – 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

Explain why $\displaystyle f$ rises throughout its domain.
Show that $\displaystyle f$ is odd, and explain how this fits the two asymptotes.

The function is $\displaystyle {f(x)}={\frac{4}{x^2+1}}$ , with the point $\displaystyle (1,2)$ on its graph.

Work out the slope of the tangent at the point.

$\displaystyle f(x)=\frac{4}{x^{2}+1}$

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Write the equation of the tangent on the board.

\displaystyle f(x)=\frac{4}{x^{2}+1}

A rectangle is inscribed in a semicircle of radius $\displaystyle \sqrt{8}$ : one side of it is on the diameter.
$\displaystyle x$ – the distance from the centre to the side edge of the rectangle.
The area of the rectangle is $\displaystyle {S(x)}={2x\sqrt{8-x^2}}$ , on the interval $\displaystyle 0<x<\sqrt{8}$ .

Find $\displaystyle S'(x)$ .

$\displaystyle S(x)=2x\sqrt{8-x^{2}}$

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

Work out the greatest area.

$\displaystyle S(x)=2x\sqrt{8-x^{2}}$

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The function $\displaystyle {f(x)}={x+2\cos x}$ is defined on the interval $\displaystyle 0\leq x\leq2\pi$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x+2\cos x$

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

$\displaystyle {f''(x)}={-2\cos x}$ . Solve the equation $\displaystyle {f''(x)}={0}$ in the interval.

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

\displaystyle f(x)=x+2\cos x

Differentiate.

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