Practice: the derivative and investigating a polynomial

Algebra and calculus, grade 10 (4 units) Lesson 16 of 20Practice

Practice: the derivative and investigating a polynomial

The function is $\displaystyle {f(x)}={x^2-3x}$ .
Find the average rate of change of the function from $\displaystyle {x}={1}$ to $\displaystyle {x}={4}$ .

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

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

Simplify the slope of the chord from the point where $\displaystyle {x}={2}$ to the point where $\displaystyle {x}={2+h}$ .

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$\displaystyle h$ gets closer to 0.
What is $\displaystyle f'(2)$ ?

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Find the derivative.

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The function is $\displaystyle {f(x)}={x^3-2x^2+5}$ .
Find $\displaystyle f'(-1)$ .

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

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The function is $\displaystyle {f(x)}={x^2-4x+1}$ .
The point where $\displaystyle {x}={3}$ is marked on the graph.
Write on the graph board the equation of the tangent to the graph at this point.

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

\displaystyle (3,-2)

The function is $\displaystyle {f(x)}={x^3-3x}$ .
At some points of the graph the slope of the tangent is 9.
Write an equation and solve it, and find the coordinates of the points.

The function is $\displaystyle {f(x)}={x^3-12x+1}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x^{3}-12x+1$

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Write an inequality and solve it: where does the function increase?

Mark the extremum points on the graph.
Say what kind each one is.

\displaystyle f(x)=x^{3}-12x+1

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

Find $\displaystyle f'(x)$ .

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

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

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

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

How many solutions has the equation $\displaystyle {f(x)}={2}$ ?
Give a reason from the graph.

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The function $\displaystyle {f(x)}={x^2-4x}$ is defined on $\displaystyle -1\leq x\leq3$ .
Find the absolute maximum point and the absolute minimum point of the function on this domain.

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

\displaystyle y=x^{2}-4

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

What are the $\displaystyle x$ -coordinates of the extremum points of $\displaystyle f(x)$ ?
Say what kind each one is, and give a reason.

The function is $\displaystyle {f(x)}={x^3+ax^2}$ , where $\displaystyle a$ is a parameter.
The function has an extremum point where $\displaystyle {x}={2}$ .
Write an equation and solve it: what is $\displaystyle a$ ?

The function is $\displaystyle {f(x)}={ax^2+3x}$ , where $\displaystyle a$ is a parameter.
The slope of the tangent to the graph at the point where $\displaystyle {x}={1}$ is 7.

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

Write the equation of the tangent at this point.

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
$\displaystyle f'(x)=0$