Practice: extremum problems

Calculus, grade 11 (5 units) Lesson 20 of 26Practice

Practice: extremum problems

1

The function $\displaystyle {f(x)}={\frac{x}{x^2+1}}$ is defined on the interval $\displaystyle 0\leq x\leq3$ .

a

Find $\displaystyle f'(x)$ .

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

/ – 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

b

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

c

Work out the values of the function at the point in the interval and at both ends of the interval.
Write the greatest and the least value of $\displaystyle f$ on the interval.

At the point you found

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

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/ – 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(0)

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

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/ – 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(3)

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

/ – 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 greatest value

/ – 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 least value

/ – 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

2

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

a

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x\sqrt{8-x^{2}}$

/ – 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

b

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

c

Mark on the graph the extremum points inside the domain.

\displaystyle f(x)=x\sqrt{8-x^{2}}

d

What is the domain of $\displaystyle f$ ?
What are the greatest and the least value of $\displaystyle f$ on the interval $\displaystyle 0\leq x\leq\sqrt{8}$ ?

The greatest value

/ – 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 least value

/ – 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

3

Two positive numbers $\displaystyle x$ and $\displaystyle y$ satisfy $\displaystyle {xy}={16}$ .
You are looking for the least value of $\displaystyle x^2+y^2$ .
With $\displaystyle {y}={\frac{16}{x}}$ this is $\displaystyle {g(x)}={x^2+\frac{256}{x^2}}$ .

a

Find $\displaystyle g'(x)$ .

$\displaystyle g(x)=x^{2}+\frac{256}{x^{2}}$

/ – 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

b

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

c

Work out the least value of $\displaystyle x^2+y^2$ .

$\displaystyle g(x)=x^{2}+\frac{256}{x^{2}}$

/ – 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

4

You are looking for the point of the graph of $\displaystyle {y}={\sqrt{x+3}}$ nearest to the point $\displaystyle A(2,0)$ .

a

Show that the distance from $\displaystyle A$ to the point $\displaystyle (x,\sqrt{x+3})$ of the graph is $\displaystyle {d(x)}={\sqrt{x^2-3x+7}}$ .
Explain why the distance is least where the expression $\displaystyle x^2-3x+7$ is least.

b

Differentiate the expression $\displaystyle x^2-3x+7$ , and solve the equation where the derivative is 0.

c

Work out the shortest distance.

$\displaystyle d(x)=\sqrt{x^{2}-3x+7}$

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5

A rectangular plot of 200 m² is fenced next to a long wall: the wall is one side, and the other three sides are fenced.
$\displaystyle x$ – the length of each of the two sides perpendicular to the wall, in metres.

a

Show that the length of the fence is $\displaystyle {L(x)}={2x+\frac{200}{x}}$ , and find $\displaystyle L'(x)$ .

$\displaystyle L(x)=2x+\frac{200}{x}$

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b

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

c

Work out the least length of the fence.

$\displaystyle L(x)=2x+\frac{200}{x}$

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6

A cylindrical cup, open at the top, holds $\displaystyle 27\pi$ cm³.
$\displaystyle r$ – the radius of the base, in cm.

a

Show that the surface area of the cup (one base and the curved surface) is $\displaystyle {S(r)}={\pi r^2+\frac{54\pi}{r}}$ , and find $\displaystyle S'(r)$ .

$\displaystyle S(r)=\pi r^{2}+\frac{54\pi}{r}$

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b

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

c

Work out the least surface area.
What is the height of this cup?

$\displaystyle S(r)=\pi r^{2}+\frac{54\pi}{r}$

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7

A rectangle is inscribed in a semicircle of radius 4: one side of it lies on the diameter.
$\displaystyle x$ – the distance from the centre to the side edge of the rectangle.
The area of the rectangle is $\displaystyle {S(x)}={2x\sqrt{16-x^2}}$ .

a

Find $\displaystyle S'(x)$ .

$\displaystyle S(x)=2x\sqrt{16-x^{2}}$

/ – 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

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b

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

c

Work out the greatest area.

$\displaystyle S(x)=2x\sqrt{16-x^{2}}$

/ – 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

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8

House $\displaystyle A$ is 3 km from a straight river, and house $\displaystyle B$ is 5 km from it, on the same side.
Along the river, the points nearest to the two houses are 6 km apart.
You walk from $\displaystyle A$ to a point $\displaystyle P$ on the bank, fill a bucket, and go on to $\displaystyle B$ .
$\displaystyle x$ – the distance along the river from the point nearest to $\displaystyle A$ to $\displaystyle P$ .

a

Show that the length of the way is $\displaystyle {L(x)}={\sqrt{x^2+9}+\sqrt{(6-x)^2+25}}$ , on the interval $\displaystyle 0\leq x\leq6$ .

b

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

c

Work out the length of the shortest way.
Check that the values at the ends of the interval are greater.

$\displaystyle L(x)=\sqrt{x^{2}+9}+\sqrt{(6-x)^{2}+25}$

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9

The function $\displaystyle {f(x)}={\frac{x^2+4}{x}}$ is defined on the interval $\displaystyle 1\leq x\leq4$ .
Its derivative is $\displaystyle {f'(x)}={1-\frac{4}{x^2}}$ .

a

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

b

Work out the values of the function at the point in the interval and at both ends of the interval.

At the point you found

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

/ – 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

\displaystyle f(1)

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

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\displaystyle f(4)

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

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c

What is the least value of $\displaystyle f$ on the interval, and what is the greatest?
Where does the function take its greatest value?

The least value

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The greatest value

/ – 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

Formulas: calculus, grade 11 (5 units)

Rules of differentiation and the tangent line

The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
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 quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
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 tangent line at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle y-y_1=f'(x_1)(x-x_1)$

The second derivative

The second derivative – the derivative of the derivative
$\displaystyle f''(x)$ $\displaystyle =$ $\displaystyle \left(f'(x)\right)'$
The function is concave up ( $\displaystyle \cup$ )
$\displaystyle f''(x)>0$
The function is concave down ( $\displaystyle \cap$ )
$\displaystyle f''(x)<0$
A possible inflection point – check that the sign of $\displaystyle f''$ changes there
$\displaystyle f''(x_0)=0$
A minimum point
$\displaystyle f'(x_0)=0,\ f''(x_0)>0$
A maximum point
$\displaystyle f'(x_0)=0,\ f''(x_0)<0$
Acceleration – the derivative of the speed
$\displaystyle a(t)=v'(t)=s''(t)$

Asymptotes

A vertical asymptote – $\displaystyle f(x)$ grows or falls without end as $\displaystyle x$ approaches $\displaystyle a$
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle f(x)$ approaches $\displaystyle b$ as $\displaystyle x\to\infty$ or as $\displaystyle x\to-\infty$
$\displaystyle y=b$

Roots

The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Radians and periods

Half a turn in radians
$\displaystyle \pi$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The length of an arc of $\displaystyle \alpha$ radians in a circle of radius $\displaystyle R$
$\displaystyle l$ $\displaystyle =$ $\displaystyle \alpha R$
The period of $\displaystyle \sin x$ and of $\displaystyle \cos x$ is $\displaystyle 2\pi$
$\displaystyle \sin(x+2\pi)$ $\displaystyle =$ $\displaystyle \sin x$
The period of $\displaystyle \tan x$ is $\displaystyle \pi$
$\displaystyle \tan(x+\pi)$ $\displaystyle =$ $\displaystyle \tan x$
The limit behind the derivative of $\displaystyle \sin x$ ( $\displaystyle x$ in radians)
$\displaystyle \lim_{x\to0}\frac{\sin x}{x}=1$

Trigonometric identities

The Pythagorean identity
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =$ $\displaystyle 1$
Tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$
Sine – an odd function
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
Cosine – an even function
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
The sine of the supplement to $\displaystyle \pi$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The cosine of the complement to $\displaystyle \frac{\pi}{2}$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$
The sine of a double angle
$\displaystyle \sin2\alpha$ $\displaystyle =$ $\displaystyle 2\sin\alpha\cdot\cos\alpha$
The cosine of a double angle
$\displaystyle \cos2\alpha=\cos^2\alpha-\sin^2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha$
The difference of two sines
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =$ $\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}$
The sine of a sum and of a difference
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta$

Derivatives of the trigonometric functions

The derivative of sine ( $\displaystyle x$ in radians)
$\displaystyle (\sin x)'$ $\displaystyle =$ $\displaystyle \cos x$
The derivative of cosine
$\displaystyle (\cos x)'$ $\displaystyle =$ $\displaystyle -\sin x$
The derivative of tangent
$\displaystyle (\tan x)'$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2x}$