Review: calculus

Calculus, grade 10 (5 units) Lesson 26 of 26Practice

Review: calculus

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

Find the points where the graph meets the axes.
At which point does the graph cross the $\displaystyle x$ -axis, and at which does it only touch it?
Explain by the multiplicity of each root.

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

The board also shows the graph of $\displaystyle {g(x)}={f(-x)}$ .
Write $\displaystyle g(x)$ on the board.

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

\displaystyle g(x)=\,?

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

The average rate of change of $\displaystyle f$ from $\displaystyle {x}={2}$ to $\displaystyle {x}={2+h}$ is $\displaystyle \frac{f(2+h)-f(2)}{h}$ .
Simplify it.

/ – 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

Work out $\displaystyle f'(2)$ by the definition: what does your expression approach as $\displaystyle h\to0$ ?

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$\displaystyle {f(2)}={-2}$ .
Write on the board the equation of the tangent to the graph at the point $\displaystyle (2,-2)$ .

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

\displaystyle (2,-2)

The function is $\displaystyle {f(x)}={x^3-3x^2-9x+5}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x^{3}-3x^{2}-9x+5$

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

Mark the extremum points on the graph.
Write on the board the equation of the tangent to the graph at the point where $\displaystyle {x}={0}$ .

\displaystyle f(x)=x^{3}-3x^{2}-9x+5

Write where $\displaystyle f$ rises and where it falls.
For which values of $\displaystyle k$ does the equation $\displaystyle {f(x)}={k}$ have three solutions?
Explain by the extremum points.

Differentiate by the product rule and the chain rule, and factor the result.

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The function is $\displaystyle {f(x)}={\sqrt{8-2x}+x}$ .

Find the domain of $\displaystyle f$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\sqrt{8-2x}+x$

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

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

$\displaystyle f(x)=\sqrt{8-2x}+x$

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Write all the extremum points of $\displaystyle f$ , an extremum at the end too, and their kind.
Write where the function rises and where it falls.

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

Write on the board the equations of all the asymptotes of $\displaystyle f$ .

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

Find $\displaystyle f'(x)$ .

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

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Show that $\displaystyle f'(x)<0$ throughout the domain.
What follows about the extremum points of $\displaystyle f$ ?
Is $\displaystyle f$ even, odd or neither?

Write on the board the equation of the tangent to the graph at the origin.

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

\displaystyle x=-1

\displaystyle x=1

A farmer wants to fence a rectangular plot of 800 m² along a long wall.
No fence is needed along the wall, and the other three sides are fenced.
$\displaystyle x$ – the length of each of the two sides perpendicular to the wall, in metres.
What are the measurements of the plot with the shortest fence, and how long is the fence?

\displaystyle x

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other side

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fence

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From a square sheet of tin with a side of 12 cm, a square with a side of $\displaystyle x$ cm is cut at each corner, and the edges are folded up into an open box.

Show that the volume of the box is $\displaystyle {V(x)}={x(12-2x)^2}$ .
What is the domain of $\displaystyle x$ ?

Find $\displaystyle V'(x)$ .

$\displaystyle V(x)=x(12-2x)^{2}$

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Solve the equation $\displaystyle {V'(x)}={0}$ .
Which solution is outside the domain?

Work out the greatest volume.

$\displaystyle V(x)=x(12-2x)^{2}$

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Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$

The properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units right ( $\displaystyle p<0$ – left)
$\displaystyle y=f(x-p)$
A vertical stretch ( $\displaystyle |a|>1$ ) or squeeze ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$
A horizontal squeeze by $\displaystyle a$ ( $\displaystyle a>1$ ) or stretch ( $\displaystyle 0<a<1$ )
$\displaystyle y=f(ax)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
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 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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$