Applications of calculus

Use differentiation and integration to sketch curves, solve optimisation problems, and analyse rates of change and motion.

Calculus · MAV-12-06

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Open Math

Applications of calculus

Calculus · MAV-12-06

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Curve sketching

Q1Straightforward

A function

f(x)

has

f'(x) = 0

x = 2

and

f''(2) = -5

. What can you conclude about

x = 2

A.Local minimum
B.Local maximum
C.Point of inflection
D.Cannot be determined

Q2Straightforward

Find the

x

-coordinate of the stationary point of

f(x) = x^2 - 6x + 11

Q3Straightforward

For

f(x) = x^3 - 3x^2 + 4

, find the value of

f''(x)

x = 1

Q4Straightforward

A curve has

f''(x) > 0

on the interval

(1, 4)

. What does this mean?

A.The curve is decreasing on $(1, 4)$
B.The curve is concave up on $(1, 4)$
C.The curve is concave down on $(1, 4)$
D.The curve has a stationary point on $(1, 4)$

Q5Moderate

For

f(x) = x^3 - 3x^2 + 4

, find the

x

-coordinate of the point of inflection.

Q6Moderate

Find the

y

-value at the local minimum of

f(x) = 2x^3 - 9x^2 + 12x - 1

Q7Moderate

For

f(x) = 2x^3 - 9x^2 + 12x - 1

, what is the

y

-value at the local maximum?

Q8Moderate

The graph of

f'(x)

is shown to cross the

x

-axis from negative to positive at

x = -1

, and from positive to negative at

x = 3

. Which statement is correct about

f(x)

A. $f(x)$ has a local minimum at $x = -1$ and a local maximum at $x = 3$
B. $f(x)$ has a local maximum at $x = -1$ and a local minimum at $x = 3$
C. $f(x)$ has points of inflection at $x = -1$ and $x = 3$
D. $f(x)$ is concave up at both $x = -1$ and $x = 3$

Q9Moderate

Consider

h(x) = x^3 - 3x

. On which interval is

h(x)

decreasing?

A. $(-\infty, -1)$
B. $(-1, 1)$
C. $(1, \infty)$
D. $(-\infty, 0)$

Q10Moderate

For

f(x) = x^3 - 6x^2 + 9x + 1

, find the

x

-value of the point of inflection.

Q11Challenging

The function

f(x) = x^3 - 6x^2 + 9x + 1

has two stationary points. Find the sum of their

x

-coordinates.

Q12Challenging

For

f(x) = e^x(x - 2)

, find the

x

-coordinate of the stationary point. Give your answer as an integer.

Q13Challenging

For

f(x) = \ln x - x

(defined for

x > 0

), find the

x

-coordinate of the stationary point.

Optimisation

Q14Straightforward

A rectangle has a perimeter of 40 cm. If its length is

x

cm, its width is

(20 - x)

cm. What value of

x

maximises the area?

Q15Straightforward

The daily profit (in dollars) from selling

x

items is

P(x) = -2x^2 + 80x - 200

. How many items per day maximise profit?

Q16Straightforward

A ball is thrown so that its height above ground (in metres) is

h(t) = 20t - 5t^2

after

t

seconds. What is the maximum height reached, in metres?

Q17Straightforward

The number of bacteria in a culture after

t

hours is

N(t) = -t^2 + 8t + 20

for

0 \leq t \leq 10

. At what time (in hours) is the population at its maximum?

Q18Moderate

A farmer has 120 m of fencing to enclose a rectangular paddock against a straight wall (the wall forms one side and needs no fencing). What length (in metres) should each side perpendicular to the wall be, to maximise the enclosed area?

Q19Moderate

A piece of wire 60 cm long is bent into a rectangle. What is the maximum possible area, in cm²?

Q20Moderate

Two positive numbers sum to 12. Find the maximum value of their product.

Q21Moderate

A closed cylindrical can (open top) has volume

125\pi

cm³. Its surface area is

S = \pi r^2 + 2\pi r h

. Using the constraint

\pi r^2 h = 125\pi

(so

h = \dfrac{125}{r^2}

), find the radius

r

(in cm) that minimises the surface area.

Q22Moderate

The revenue from selling

x

units is

R(x) = 500x - 2x^2

dollars and the cost is

C(x) = 100x + 3000

dollars. Find the number of units

x

that maximises profit.

Q23Challenging

A rectangular piece of cardboard measures 16 cm × 10 cm. Equal squares of side

x

cm are cut from each corner, and the sides are folded up to make an open box. Find the value of

x

(in cm) that maximises the volume.

Q24Challenging

A closed rectangular box has a square base of side

x

cm and height

h

cm. Its total surface area is

600

cm². Given

h = \dfrac{600 - 2x^2}{4x}

, find the value of

x

(in cm) that maximises the volume

V = x^2 h

Q25Challenging

A rectangle is inscribed in a semicircle of radius 4. The base of the rectangle lies along the diameter. If half the base has length

x

, the height is

\sqrt{16 - x^2}

. Find the maximum area of the rectangle.

Rates of change and motion

Q26Straightforward

A particle moves along a line so that its displacement at time

t

seconds is

x(t) = t^2 - 4t + 3

metres. What is the velocity at

t = 1

A. $-2$ m/s
B. $0$ m/s
C. $2$ m/s
D. $4$ m/s

Q27Straightforward

A particle has displacement

x(t) = t^2 - 4t + 3

metres at time

t

seconds. At what time (in seconds) is the particle momentarily at rest?

Q28Straightforward

A particle has velocity

v(t) = 3t^2 - 12

m/s. Find the acceleration at

t = 2

seconds, in m/s².

Q29Straightforward

A particle moves so that its velocity at time

t

v(t) = 2t - 6

m/s. When is the particle moving in the positive direction?

A. $t < 3$
B. $t > 3$
C. $t = 3$
D.Always

Q30Moderate

A particle's displacement is

x(t) = t^3 - 6t^2 + 9t

metres, where

t \geq 0

. Find the displacement (in metres) at the first time the particle is at rest.

Q31Moderate

A particle's velocity is

v(t) = 6 - 2t

m/s. It starts at position

x = 3

m at

t = 0

. Find its displacement (in metres) at

t = 4

Q32Moderate

A water tank is being drained. The volume of water (in litres) at time

t

minutes is

V(t) = 500 - 20t - t^2

for

0 \leq t \leq 10

. Find the rate at which water is draining (in litres per minute) at

t = 3

Q33Moderate

A particle's acceleration is

a(t) = 6t - 4

m/s². Its initial velocity is

v(0) = 5

m/s. Find the velocity (in m/s) at

t = 3

Q34Moderate

The population of a town grows at a rate of

\dfrac{dP}{dt} = 200 + 30t

people per year. If the population is 5000 at

t = 0

, find the population after 4 years.

Q35Challenging

A particle moves in a straight line with displacement

x(t) = t^3 - 6t^2 + 9t

metres. Find the total distance (in metres) travelled in the first 3 seconds.

Q36Challenging

A particle moves so that its velocity is

v(t) = t^3 - 6t^2 + 9t

m/s (

t \geq 0

). Find the acceleration (in m/s²) at

t = 2

Q37Challenging

Oil leaks from a tanker at a rate of

R(t) = 50e^{-0.1t}

litres per hour, where

t

is hours after the leak starts. How many litres leak in the first 10 hours? Give your answer to the nearest whole number.

(Use

e^{-1} \approx 0.3679

Worked solutions and answers at openmath.au/year-12/advanced/applications-of-calculus