Extremum problems with square root and rational functions

Calculus, grade 10 (5 units) Lesson 24 of 26Learning

Extremum problems with square root and rational functions

The steps of a solution

An extremum problem with a root or a fraction is solved in the same steps as one with a polynomial:

Choose a variable $\displaystyle x$ , and write the quantity to make smallest or largest as a function of it.
Fix the domain of $\displaystyle x$ from the problem: a positive length, a point between two places.
Differentiate, and solve $\displaystyle {f'(x)}={0}$ . A solution outside the domain is rejected.
Make sure it is a minimum or a maximum: in an open domain – a table of signs of $\displaystyle f'$ ; in a closed domain – compare with the values at its ends too.

In today’s problems the function holds a fraction such as $\displaystyle \frac{72}{x}$ or a root such as $\displaystyle \sqrt{x^2+9}$ .

Example: the least perimeter

A rectangle has an area of $\displaystyle 36\text{ cm}^2$ .
What is the least perimeter it can have?

The solution below takes one side as the variable and writes the perimeter with it:

One side is $\displaystyle x$ , the other $\displaystyle \frac{36}{x}$ ; the domain is $\displaystyle x>0$ .
$\displaystyle {P(x)}={2x+\frac{72}{x}}$ .
$\displaystyle {P'(x)}={2-\frac{72}{x^2}}={0}$ , so $\displaystyle {x^2}={36}$ and $\displaystyle {x}={6}$ ( $\displaystyle {x}={-6}$ is outside the domain).

$\displaystyle x$	$\displaystyle 0<x<6$	$\displaystyle 6$	$\displaystyle x>6$
$\displaystyle P'(x)$	−	0	+
$\displaystyle P(x)$	falls	minimum	rises

The rectangle is a $\displaystyle 6\times6$ square, and the least perimeter is $\displaystyle {P(6)}={12+12}={24}$ cm.

Example: the shortest distance from a point to a graph

Find the point on the graph of $\displaystyle {y}={\sqrt{x}}$ closest to the point $\displaystyle A(3,0)$ .

The solution below writes the distance with the $\displaystyle x$ -coordinate of a point on the graph:

A point on the graph is $\displaystyle (x,\sqrt{x})$ , and $\displaystyle x\geq0$ .
By the distance formula: $\displaystyle {d(x)}={\sqrt{(x-3)^2+(\sqrt{x}-0)^2}}={\sqrt{x^2-5x+9}}$ .
$\displaystyle {d'(x)}={\frac{2x-5}{2\sqrt{x^2-5x+9}}}$ . The denominator is positive, so $\displaystyle {d'(x)}={0}$ where $\displaystyle {2x-5}={0}$ , that is $\displaystyle {x}={2.5}$ .
To the left of 2.5 the derivative is negative, to the right positive: a minimum.

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

\displaystyle A=(3,0)

\displaystyle P=(2.5,1.5811)

The closest point is $\displaystyle (2.5,\sqrt{2.5})$ , and the shortest distance is $\displaystyle {d(2.5)}={\sqrt{2.75}}={\frac{\sqrt{11}}{2}\approx1.66}$ .
The sign of $\displaystyle d'$ is the sign of $\displaystyle 2x-5$ – the derivative of the expression under the root. So you can also find the minimum of $\displaystyle x^2-5x+9$ : where the expression under the root is smallest, the root is smallest too.

Exercise 1

A box open at the top is built on a square base.
The volume of the box is $\displaystyle 32\text{ dm}^3$ .
$\displaystyle x$ – the side of the base, in decimetres.

Write the height of the box with $\displaystyle x$ .
Show that the area of the base and the four sides is $\displaystyle {S(x)}={x^2+\frac{128}{x}}$ .

Find $\displaystyle S'(x)$ .

$\displaystyle S(x)=x^{2}+\frac{128}{x}$

/ – 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

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

Work out the least area.

$\displaystyle S(x)=x^{2}+\frac{128}{x}$

/ – 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

What are the measurements of the box with the least area?
Show by a table of signs that it is a minimum.

base side

/ – 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

height

/ – 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

Exercise 2

A factory makes $\displaystyle x$ products a day.
The total cost of making them is $\displaystyle 0.5x^2+20x+800$ ₪.
The average cost per product is the total cost divided by $\displaystyle x$ : $\displaystyle {A(x)}={\frac{1}{2}x+20+\frac{800}{x}}$ .

Find $\displaystyle A'(x)$ .

$\displaystyle A(x)=\frac{1}{2}x+20+\frac{800}{x}$

/ – 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 $\displaystyle {A'(x)}={0}$ .

Work out the average cost per product for the number you found.

$\displaystyle A(x)=\frac{1}{2}x+20+\frac{800}{x}$

/ – 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

How many products a day should the factory make for the least average cost per product?
Which solution of the equation did you reject, and why?

/ – 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

Exercise 3

A swimmer is on an island $\displaystyle B$ , 3 km from the point $\displaystyle C$ on the shore.
The village $\displaystyle A$ is on the shore, 10 km from $\displaystyle C$ .
The swimmer swims at 3 km/h, and runs along the shore at 5 km/h.
He swims to the point $\displaystyle P$ on the shore, between $\displaystyle C$ and $\displaystyle A$ , and runs from there to the village.
$\displaystyle x$ – the distance $\displaystyle CP$ , in kilometres.

Show that the time, in hours, is $\displaystyle {T(x)}={\frac{\sqrt{x^2+9}}{3}+\frac{10-x}{5}}$ .
What is the domain of $\displaystyle x$ ?

Find $\displaystyle T'(x)$ .

$\displaystyle T(x)=\frac{\sqrt{x^{2}+9}}{3}+\frac{10-x}{5}$

/ – 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 $\displaystyle {T'(x)}={0}$ .

The domain is closed: work out the time at the point you found and at the two ends of the domain.

\displaystyle T(0)

$\displaystyle T(x)=\frac{\sqrt{x^{2}+9}}{3}+\frac{10-x}{5}$

/ – 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

extremum

$\displaystyle T(x)=\frac{\sqrt{x^{2}+9}}{3}+\frac{10-x}{5}$

/ – 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 T(10)

$\displaystyle T(x)=\frac{\sqrt{x^{2}+9}}{3}+\frac{10-x}{5}$

/ – 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

To which point on the shore should the swimmer swim?
What is the shortest time?

\displaystyle CP

/ – 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

time

/ – 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

Exercise 4

The point is $\displaystyle A(4,0)$ , and the graph is that of $\displaystyle {y}={\sqrt{x}}$ .
The distance from $\displaystyle A$ to the point $\displaystyle (x,\sqrt{x})$ on the graph is $\displaystyle {d(x)}={\sqrt{x^2-7x+16}}$ .

Find $\displaystyle d'(x)$ .

$\displaystyle d(x)=\sqrt{x^{2}-7x+16}$

/ – 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 $\displaystyle {d'(x)}={0}$ .

Work out the shortest distance.

$\displaystyle d(x)=\sqrt{x^{2}-7x+16}$

/ – 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 the point on the graph closest to $\displaystyle A$ .
Show that $\displaystyle x^2-7x+16$ reaches its minimum at the same $\displaystyle x$ .

Exercise 5

The function $\displaystyle {f(x)}={x+\frac{9}{x}}$ is defined in the domain $\displaystyle 1\leq x\leq6$ .
Its derivative is $\displaystyle {f'(x)}={1-\frac{9}{x^2}}$ .

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

Work out the values of the function at the ends of the domain and at the point you found inside it.

\displaystyle f(1)

$\displaystyle f(x)=x+\frac{9}{x}$

/ – 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

extremum

$\displaystyle f(x)=x+\frac{9}{x}$

/ – 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

\displaystyle f(6)

$\displaystyle f(x)=x+\frac{9}{x}$

/ – 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

What is the least value of $\displaystyle f$ in the domain, and what is the greatest?
Which of them is an extremum at the end?

least

/ – 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

greatest

/ – 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 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
$\displaystyle y=b$