Review: algebra and calculus

Algebra and calculus, grade 10 (4 units) Lesson 20 of 20Practice

Review: algebra and calculus

Solve the equation.

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

What is the extremum point of $\displaystyle f(x)$ , and what kind is it?
What is the extremum point of $\displaystyle {h(x)}={f(x-2)}$ ?

The board also shows the graph of $\displaystyle {k(x)}={-f(x)+3}$ .
Write $\displaystyle k(x)$ as a polynomial on the graph board.

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

\displaystyle k(x)=\,?

What is the extremum point of $\displaystyle k(x)$ , and what kind is it?
Give a reason without differentiating.

The function is $\displaystyle {f(x)}={8(2x-1)^3}$ .

Find the points where the graph meets the axes.

Find $\displaystyle f'(x)$ .

/ – 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

Find where the function increases and where it decreases, if anywhere.
Has the function extremum points?
Give a reason.

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

Find $\displaystyle f'(x)$ .

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

/ – 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 on the graph the points where it meets the $\displaystyle x$ -axis.
What are the extremum points of the function, and what kind are they?

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

The function is $\displaystyle {g(x)}={|f(x)|}$ .
What are the extremum points of $\displaystyle g(x)$ , and what kind are they?

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

Find the domain and the points where the graph meets the axes.

Find $\displaystyle f'(x)$ , over a common denominator.

/ – 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 on the graph the extremum point inside the domain.
The point where $\displaystyle {x}={-1}$ is marked on the graph: write on the graph board the equation of the tangent there.

\displaystyle f(x)=x\sqrt{3-x}

\displaystyle (-1,-2)

Where does the function increase, and where does it decrease?
What kind is the extremum point inside the domain, and what is the extremum point at the end of the domain?

The board shows the graph of the derivative $\displaystyle {y}={f'(x)}$ of a function $\displaystyle f(x)$ .
The derivative equals 0 only when $\displaystyle {x}={-2}$ and when $\displaystyle {x}={1}$ .

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

Where does $\displaystyle f(x)$ increase, and where does it decrease?

At the points where $\displaystyle {x}={-2}$ and $\displaystyle {x}={1}$ the tangent to the graph of $\displaystyle f(x)$ is parallel to the $\displaystyle x$ -axis.
Which of them is an extremum point, and what kind?
Give a reason.

The function is $\displaystyle {f(x)}={x^3+ax^2+b}$ , where $\displaystyle a$ and $\displaystyle b$ are parameters.
The point $\displaystyle (2,-1)$ is an extremum point of the function.

Find $\displaystyle f'(x)$ .

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

/ – 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 an equation and solve it: what is $\displaystyle a$ ?

Write an equation and solve it: what is $\displaystyle b$ ?

What kind of extremum point is $\displaystyle (2,-1)$ ?
Give a reason.

The board shows the graph of $\displaystyle {f(x)}={-x^2+6x}$ .
A rectangle has two vertices on the $\displaystyle x$ -axis and two vertices on the graph, and it is symmetric about the line $\displaystyle {x}={3}$ .
$\displaystyle x$ – the distance from the line $\displaystyle {x}={3}$ to each upright side of the rectangle.

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

Show that the area of the rectangle is $\displaystyle {S(x)}={18x-2x^3}$ .
What is the domain of $\displaystyle x$ ?

Find $\displaystyle S'(x)$ .

/ – 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

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

What is the greatest area of the rectangle?

/ – 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

Formulas: algebra and calculus, grade 10 (4 units)

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Algebraic technique

The square of a sum
$\displaystyle (a+b)^2$ $\displaystyle =$ $\displaystyle a^2+2ab+b^2$
The square of a difference
$\displaystyle (a-b)^2$ $\displaystyle =$ $\displaystyle a^2-2ab+b^2$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
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 roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

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

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
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 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 composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

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
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

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
$\displaystyle f'(x)=0$