Practice: the derivative of a polynomial and its uses

Calculus, grade 10 (5 units) Lesson 17 of 26Practice

Practice: the derivative of a polynomial and its uses

Work out the average rate of change of $\displaystyle {f(x)}={x^3}$ from $\displaystyle {x}={1}$ to $\displaystyle {x}={3}$ .

/ – 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

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

Simplify the difference quotient.

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What does the difference quotient approach as $\displaystyle h\to0$ ?
What is $\displaystyle f'(x)$ ?

Differentiate.

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A body moves in a straight line.
The distance it covers in metres after $\displaystyle t$ seconds: $\displaystyle {s(t)}={t^3-6t^2+15t}$ .

Find the speed of the body after $\displaystyle t$ seconds.

$\displaystyle s(t)=t^{3}-6t^{2}+15t$

/ – 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

What is the speed after 3 seconds?

$\displaystyle s(t)=t^{3}-6t^{2}+15t$

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

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

The function is $\displaystyle {f(x)}={x^2-6x+5}$ .
The tangent to the graph at a certain point is parallel to the line $\displaystyle {y}={2x+1}$ .

Write an equation for the $\displaystyle x$ -coordinate of the point, and solve it.

Write the equation of the tangent on the board.

\displaystyle f(x)=x^{2}-6x+5

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

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

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

Mark the extremum points on the graph.

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

Write the extremum points and their kind, and where the function rises and falls.

The function $\displaystyle {f(x)}={x^3-12x}$ is defined in the interval $\displaystyle 0\leq x\leq3$ .

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

Work out the values of the function at the point in the interval and at the right-hand end.

extremum

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

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\displaystyle f(3)

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

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What is the absolute maximum of $\displaystyle f$ in the interval, and what is the absolute minimum?

maximum

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minimum

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The function $\displaystyle {f(x)}={ax^3-3x}$ ( $\displaystyle a$ is a parameter) has an extremum point at $\displaystyle {x}={1}$ .
Write an equation for $\displaystyle a$ and solve it.

The board below shows the graph of the derivative $\displaystyle {f'(x)}={-2x+4}$ of a function $\displaystyle f$ .

Mark the point where the graph of the derivative meets the $\displaystyle x$ -axis.

\displaystyle y=-2x+4

Where does $\displaystyle f$ rise, and where does it fall?
What does $\displaystyle f$ have at $\displaystyle {x}={2}$ ?

The derivative of a function $\displaystyle f$ is $\displaystyle {f'(x)}={2x+1}$ .
The graph of $\displaystyle f$ passes through the point $\displaystyle (2,1)$ .

$\displaystyle {f(x)}={x^2+x+C}$ .
Substitute the point and find $\displaystyle C$ .

Write $\displaystyle f(x)$ on the board.

\displaystyle A=(2,1)

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

Find $\displaystyle f'(x)$ .

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

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

Mark the extremum points on the graph.

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

From a square sheet with a side of 30 cm, a square with a side of $\displaystyle x$ is cut from each corner, and it is folded into a box with no lid.

Write the volume $\displaystyle V(x)$ and the domain of $\displaystyle x$ .

Find $\displaystyle V'(x)$ .

$\displaystyle V(x)=x(30-2x)^{2}$

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

Work out the greatest volume.

$\displaystyle V(x)=x(30-2x)^{2}$

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The function $\displaystyle {f(x)}={x^3-3x^2}$ has a maximum $\displaystyle (0,0)$ and a minimum $\displaystyle (2,-4)$ .
How many solutions does the equation $\displaystyle {x^3-3x^2}={-2}$ have?
And the equation $\displaystyle {x^3-3x^2}={1}$ ?
Explain.

\displaystyle {}={-2}

/ – 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

\displaystyle {}={1}

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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
$\displaystyle y=b$