Review: calculus

Calculus, grade 11 (4 units) Lesson 20 of 20Practice

Review: calculus

$\displaystyle {g(x)}={\frac{1}{x^2}}$ , $\displaystyle {f(x)}={\frac{1}{(x-2)^2}}$ .

Write the equations of the asymptotes of $\displaystyle g$ parallel to the axes on the board.
On which intervals does $\displaystyle g$ rise, and on which does it fall?

\displaystyle g(x)=\frac{1}{x^{2}}

The graph of $\displaystyle f$ is the graph of $\displaystyle g$ shifted. Write $\displaystyle f$ and the equations of its asymptotes on the board.

A tangent is drawn to the graph of $\displaystyle f$ at the point where it meets the $\displaystyle y$ -axis.
Work out the slope of the tangent, and write the equation of the tangent on the board.

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

/ – 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

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

\displaystyle x=2

Work out the area bounded by the graph of $\displaystyle f$ , the tangent and the line $\displaystyle {x}={-1}$ .

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

\displaystyle t(x)=\frac{x}{4}+\frac{1}{4}

\displaystyle x=-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

$\displaystyle {h(x)}={\frac{1}{(x-2)^2}+a}$ .

What values does the function $\displaystyle f$ take?
For which values of $\displaystyle a$ are all the values of $\displaystyle h$ greater than 1?
A tangent is drawn to the graph of $\displaystyle h$ at the point where it meets the $\displaystyle y$ -axis. Is the area bounded by the graph of $\displaystyle h$ , the tangent and the line $\displaystyle {x}={-1}$ larger than, smaller than or equal to the area you worked out in part 4? Explain.

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

Find the domain of $\displaystyle f$ and its vertical asymptote.

Find $\displaystyle f'(x)$ .

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

/ – 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

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

Mark the extremum points of $\displaystyle f$ on the board, and write their kind and the intervals where $\displaystyle f$ rises and falls.

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

\displaystyle x=-1

Solve the inequality.

$\displaystyle {f(x)}={\frac{ax+1}{x-2}}$ .
The line $\displaystyle {y}={3}$ is a horizontal asymptote of $\displaystyle f$ .

Find $\displaystyle a$ .

Substitute the $\displaystyle a$ you found, and find $\displaystyle f'(x)$ .
Does $\displaystyle f$ rise or fall? Explain.

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

/ – 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

A rectangle is inscribed in a semicircle: two of its vertices are on the $\displaystyle x$ -axis, and the other two on the graph of $\displaystyle {y}={\sqrt{9-x^2}}$ .
The upper right vertex is $\displaystyle \left(x,\sqrt{9-x^2}\right)$ , $\displaystyle 0<x<3$ , and the area of the rectangle is $\displaystyle {S(x)}={2x\sqrt{9-x^2}}$ .

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 largest 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

$\displaystyle {f'(x)}={\frac{2}{(x+1)^2}}$ , and the graph of $\displaystyle f$ passes through the point $\displaystyle B(0,1)$ .

Find $\displaystyle f(x)$ , and write it on the board. Work out $\displaystyle C$ in the card under the board.

\displaystyle B=(0,1)

\displaystyle x=-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

Work out the area bounded by the graph of $\displaystyle f$ , the line $\displaystyle {y}={3}$ , the $\displaystyle y$ -axis and the line $\displaystyle {x}={1}$ .

/ – 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

Formulas: calculus, grade 11 (4 units)

Open as a page

Transformations of the graph of $f$

A horizontal shift: $\displaystyle p$ units to the right ( $\displaystyle p>0$ ) or to the left ( $\displaystyle p<0$ )
$\displaystyle y=f(x-p)$
A vertical shift: $\displaystyle k$ units up ( $\displaystyle k>0$ ) or down ( $\displaystyle k<0$ )
$\displaystyle y=f(x)+k$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A vertical stretch ( $\displaystyle a>1$ ) or squeeze ( $\displaystyle 0<a<1$ )
$\displaystyle y=a\cdot f(x)$
Absolute value: the part below the $\displaystyle x$ -axis is reflected above it
$\displaystyle y=|f(x)|$

The rational function

The horizontal asymptote when the numerator and the denominator have the same degree – $\displaystyle a$ and $\displaystyle b$ the coefficients of the highest power
$\displaystyle y=\frac{a}{b}$
The horizontal asymptote when the numerator’s degree is lower than the denominator’s
$\displaystyle y=0$
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
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$

Derivatives and the tangent

The derivative of a power
$\displaystyle \left(x^{n}\right)'=nx^{n-1}$
The derivative of a product
$\displaystyle \left(u\cdot v\right)'$ $\displaystyle =$ $\displaystyle u'\cdot v+u\cdot v'$
The derivative of a quotient
$\displaystyle \left(\frac{u}{v}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}$
The derivative of $\displaystyle \frac{c}{x}$
$\displaystyle \left(\frac{c}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{c}{x^{2}}$
The derivative of a root – $\displaystyle u$ a function of $\displaystyle x$
$\displaystyle \left(\sqrt{u}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'}{2\sqrt{u}}$
The equation of the tangent at the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The slope of the tangent
$\displaystyle m$ $\displaystyle =$ $\displaystyle f'(x_1)$

Investigating a function with its derivative

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A possible extremum point – check the sign of $\displaystyle f'$ on each side of it
$\displaystyle f'(x)=0$

Integrals and areas

The integral of a power
$\displaystyle \int x^{n}\,dx$ $\displaystyle =$ $\displaystyle \frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
The integral of $\displaystyle \frac{c}{(ax+b)^{2}}$
$\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx$ $\displaystyle =$ $\displaystyle -\frac{c}{a(ax+b)}+C$
The definite integral – $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx$ $\displaystyle =$ $\displaystyle F(b)-F(a)$
The area between two graphs, from $\displaystyle {x}={a}$ to $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$