Recap: the calculus of grade 10

Calculus, grade 11 (5 units) Lesson 1 of 26Learning

Recap: the calculus of grade 10

The rules of differentiation

In grade 10 you learnt that the derivative $\displaystyle f'(x)$ is the rate of change of $\displaystyle f$ at a point – the slope of the tangent to the graph. These are the rules you learnt for working it out:

\displaystyle (x^n)'

\displaystyle =

\displaystyle nx^{n-1}

\displaystyle [f(x)\cdot g(x)]'

\displaystyle =

\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)

\displaystyle \left[\frac{f(x)}{g(x)}\right]'

\displaystyle =

\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}

\displaystyle [f(u(x))]'

\displaystyle =

\displaystyle f'(u)\cdot u'(x)

\displaystyle (\sqrt{x})'

\displaystyle =

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

And the tangent line to the graph at a point $\displaystyle (x_1,y_1)$ on it:

\displaystyle y-y_1

\displaystyle =

\displaystyle f'(x_1)(x-x_1)

The steps of an investigation

Investigating a function, as you learnt in grade 10, goes through these steps:

The domain: a denominator is not 0, an expression under a root is not negative.
The points on the axes: $\displaystyle {f(x)}={0}$ , and $\displaystyle f(0)$ if 0 is in the domain.
Asymptotes – vertical and horizontal.
The derivative: solve $\displaystyle {f'(x)}={0}$ , and build a table of signs of $\displaystyle f'$ .
Rising, falling and extremum points: $\displaystyle f'>0$ – rising, $\displaystyle f'<0$ – falling; the sign changes from + to − – a maximum, from − to + – a minimum.
A sketch of the graph.

Example:

\displaystyle x^3-3x^2

The example below investigates $\displaystyle {f(x)}={x^3-3x^2}$ :

$\displaystyle {f'(x)}={3x^2-6x}={3x(x-2)}$ , so $\displaystyle {f'(x)}={0}$ where $\displaystyle {x}={0}$ or $\displaystyle {x}={2}$ .

$\displaystyle x$	$\displaystyle x<0$	$\displaystyle 0$	$\displaystyle 0<x<2$	$\displaystyle 2$	$\displaystyle x>2$
$\displaystyle f'(x)$	+	0	−	0	+
$\displaystyle f(x)$	rises	maximum	falls	minimum	rises

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

\displaystyle M=(0,0)

\displaystyle N=(2,-4)

A maximum point: $\displaystyle (0,0)$ ; a minimum point: $\displaystyle (2,-4)$ .
The function rises where $\displaystyle x<0$ and where $\displaystyle x>2$ , and falls where $\displaystyle 0<x<2$ .

Square root and rational functions

At the end of grade 10 you met two families of functions, which you investigate in depth in this course:

A square root function, such as $\displaystyle \sqrt{f(x)}$ : defined where $\displaystyle f(x)\geq0$ ; an extremum point may be at an end of the domain.
A rational function, a quotient of polynomials: defined where the denominator is not 0.
A zero of the denominator that is not a zero of the numerator – a vertical asymptote; a zero of both – usually a removable discontinuity, a hole in the graph.
A horizontal asymptote – the value the function approaches as $\displaystyle x$ grows or falls without end.

Exercise 1

Differentiate the functions.

/ – 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

Exercise 2

The function is $\displaystyle {f(x)}={x^3-6x^2+9x}$ .
Its derivative is $\displaystyle {f'(x)}={3x^2-12x+9}$ .

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

Mark the extremum points on the graph.

\displaystyle f(x)=x^{3}-6x^{2}+9x

Write the type of each extremum point, and where $\displaystyle f$ rises and where it falls.

Exercise 3

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

Work out the slope of the tangent to the graph at the point $\displaystyle (2,2)$ .

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

/ – 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

Write the equation of the tangent on the board.

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

\displaystyle (2,2)

Exercise 4

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

Write on the board the equations of the two vertical asymptotes and of the horizontal asymptote.

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

Find $\displaystyle f'(x)$ .

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

/ – 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

Find the extremum point of $\displaystyle f$ and its type.

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