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Further applications of calculus

Apply calculus to differential equations, exponential growth and decay models, related rates of change, and volumes of solids of revolution.

Calculus · ME-12-05

Sub-topics

Differential equations

Solve differential equations of the form

\frac{dy}{dx} = f(x)

by direct integration and separable equations of the form

\frac{dy}{dx} = g(x)h(y)

; interpret direction fields qualitatively; model and solve simple problems with first-order differential equations.

Exponential growth and decay

Solve and interpret exponential growth and decay models of the form

\frac{dN}{dt} = kN

\frac{dN}{dt} = k(N-P)

; apply Newton's law of cooling; use logarithms to find unknown times and rates.

Related rates and volumes of revolution

Apply the chain rule to problems involving related rates of change; calculate volumes of solids of revolution by integrating

\pi y^2

(rotation about the

x

\pi x^2

(rotation about the

y

Open Math

Further applications of calculus

Calculus · ME-12-05

Name:

Date:

Differential equations

Q1Straightforward

Which of the following is the general solution of

\dfrac{dy}{dx} = 3x^2

?

A. $y = x^3 + C$
B. $y = 6x + C$
C. $y = x^3$
D. $y = \dfrac{x^3}{3} + C$

Q2Straightforward

The differential equation

\dfrac{dy}{dx} = 2x + 1

has general solution

y = x^2 + x + C

. If

y = 5

when

x = 1

, find the value of

C

.

Q3Straightforward

A differential equation is called **separable** if it can be written in the form

\dfrac{dy}{dx} = g(x)h(y)

. Which of the following is separable?

A. $\dfrac{dy}{dx} = \dfrac{x}{y}$
B. $\dfrac{dy}{dx} = x + y$
C. $\dfrac{dy}{dx} = \sin(x+y)$
D. $\dfrac{dy}{dx} = x^2 + y^2$

Q4Straightforward

The differential equation

\dfrac{dy}{dx} = e^x

with initial condition

y(0) = 2

has solution

y = e^x + C

. Find the value of

C

.

Q5Moderate

Solve the separable equation

\dfrac{dy}{dx} = 2xy

with initial condition

y(0) = 3

. Find

y(1)

to 4 decimal places.

Q6Moderate

Solve

\dfrac{dy}{dx} = \dfrac{x}{y}

with

y(0) = 2

. Find

y(3)

to 4 decimal places.

Q7Moderate

A direction field (slope field) for

\dfrac{dy}{dx} = y

shows line segments with positive slope wherever

y > 0

and negative slope wherever

y < 0

. Which function family is the general solution?

A. $y = Ce^x$
B. $y = Cx$
C. $y = C\ln x$
D. $y = x^2 + C$

Q8Moderate

The rate of change of a quantity

P

satisfies

\dfrac{dP}{dt} = 5 - P

. Find the general solution

P(t)

. If

P(0) = 2

, what is

P(\ln 2)

to 2 decimal places?

Q9Moderate

Solve

\dfrac{dy}{dx} = \cos x

with

y(0) = 1

. Find

y\!\left(\dfrac{\pi}{2}\right)

.

Q10Challenging

Solve the separable equation

\dfrac{dy}{dx} = \dfrac{1-y^2}{y}

with

y(0) = 2

. Find

y(1)

to 4 decimal places.

Q11Challenging

A tank initially contains 100 L of pure water. Salt solution containing 2 g/L flows in at 5 L/min, and the well-mixed solution drains at 5 L/min. The amount of salt

Q

grams after

t

minutes satisfies

\dfrac{dQ}{dt} = 10 - \dfrac{Q}{20}

. If

Q(0) = 0

, find

Q(20)

to 2 decimal places.

Q12Challenging

The velocity

v

(m/s) of a particle satisfies

\dfrac{dv}{dt} = 4 - v

with

v(0) = 0

. Find the time

t

(in seconds, to 4 decimal places) at which

v = 2

.

Exponential growth and decay

Q13Straightforward

A population

N

grows according to

\dfrac{dN}{dt} = 0.4N

, with

N(0) = 250

. Find

N

when

t = 3

. Round your answer to the nearest whole number.

Q14Straightforward

A radioactive substance decays so that

\dfrac{dM}{dt} = -0.05M

, with

M(0) = 8000

g. Find

M

when

t = 20

. Round your answer to the nearest whole number.

Q15Straightforward

A quantity

Q

satisfies

\dfrac{dQ}{dt} = kQ

with

Q(0) = 50

and

Q(4) = 200

. Find

Q(6)

. Give an exact answer.

Q16Straightforward

A substance decays according to

M = M_0 e^{kt}

. The initial mass is 400 g and the half-life is 6 hours (i.e.,

M(6) = 200

). Find the value of

k

to 4 decimal places.

Q17Moderate

A metal rod is heated to 120°C and placed in a room at 20°C. It cools according to Newton's law of cooling:

T(t) = 20 + 100\,e^{-kt}

. After 10 minutes the temperature is 70°C. Find the temperature after 20 minutes (in °C).

Q18Moderate

A bacterial culture grows from 500 to 2000 cells in 4 hours. Using

N = N_0 e^{kt}

, find the number of cells after 7 hours. Round to the nearest whole number.

Q19Moderate

The half-life of carbon-14 is 5730 years. An ancient wooden artefact contains 30% of its original carbon-14. Find the age of the artefact to the nearest year. Use

k = -\dfrac{\ln 2}{5730}

.

Q20Moderate

A cup of tea cools from 90°C to 50°C in 8 minutes in a room at 20°C. Using Newton's law of cooling in the form

T(t) = 20 + 70\,e^{-kt}

, find

k

to 4 decimal places.

Q21Moderate

Show that the function

N = N_0 e^{kt}

satisfies the differential equation

\dfrac{dN}{dt} = kN

and the initial condition

N(0) = N_0

.

Q22Moderate

A population satisfies

\dfrac{dP}{dt} = k(P - 500)

. Initially

P = 2000

and after 5 years

P = 3200

. Find

P

to the nearest whole number when

t = 10

.

Q23Challenging

Carbon-14 dating. A sample of wood from an archaeological site contains 60% of the carbon-14 that would be present in living wood. The half-life of carbon-14 is 5730 years. Find the age of the sample to the nearest year.

Q24Challenging

Two populations

A

and

B

grow according to

P_A(t) = 1000\,e^{0.1t}

and

P_B(t) = 2000\,e^{0.05t}

, where

t

is in years. After how many full years will population

A

first exceed population

B

?

Q25Challenging

A patient is given a drug that is eliminated from the bloodstream at a rate proportional to the current concentration. The initial concentration is 80 mg/L and after 4 hours it is 50 mg/L. Find the concentration (in mg/L, to 2 decimal places) when

t = 10

hours.

Related rates and volumes of revolution

Q26Straightforward

A spherical balloon is being inflated so its volume increases at

20 \text{ cm}^3/\text{s}

. Find the rate at which the radius is increasing (in cm/s) when the radius is 5 cm. Give your answer to 4 decimal places. (Use

V = \tfrac{4}{3}\pi r^3

.)

Q27Straightforward

A 10 m ladder leans against a vertical wall. Its base slides outward along the floor at

2 \text{ m/s}

. Find the rate at which the top of the ladder is sliding down the wall (in m/s) when the base is 6 m from the wall. Give a positive value for the speed of descent.

Q28Moderate

A circular oil spill expands so that its area increases at

5 \text{ m}^2/\text{min}

. Find the rate at which the radius is increasing (in m/min) when the radius is 3 m. Give your answer to 4 decimal places.

Q29Moderate

A conical tank with its apex at the bottom has a height of 6 m and a base radius of 3 m. Water drains out at

2 \text{ m}^3/\text{min}

. How fast is the water level falling (in m/min) when the depth is 3 m? Give your answer to 4 decimal places.

Q30Straightforward

Find the volume of the solid formed by rotating

y = \sqrt{x}

about the

x

-axis from

x = 0

to

x = 4

. Give your answer to 2 decimal places.

Q31Straightforward

Find the volume of the cone formed by rotating

y = x

about the

x

-axis from

x = 0

to

x = 5

. Give your answer to 2 decimal places.

Q32Moderate

Find the volume of the solid formed by rotating

y = 2x - x^2

about the

x

-axis from

x = 0

to

x = 2

. Give your answer to 2 decimal places.

Q33Moderate

Find the volume of the solid formed by rotating

y = e^x

about the

x

-axis from

x = 0

to

x = 1

. Give your answer to 2 decimal places.

Q34Challenging

Find the volume of the solid formed by rotating the region between

y = \sqrt{x}

and

y = x

(for

0 \le x \le 1

) about the

x

-axis. Give your answer to 4 decimal places.

Q35Moderate

Find the volume of the solid formed by rotating

y = x^2 + 1

about the

x

-axis from

x = 0

to

x = 2

. Give your answer to 2 decimal places.

Q36Challenging

Find the volume of the solid formed by rotating

y = \sin x

about the

x

-axis from

x = 0

to

x = \pi

. Give your answer to 4 decimal places.

Q37Challenging

Use the volume of revolution formula to prove that the volume of a sphere of radius

r

is

V = \dfrac{4}{3}\pi r^3

. Start by rotating the semicircle

y = \sqrt{r^2 - x^2}

about the

x

-axis from

x = -r

to

x = r

.

Worked solutions and answers at openmath.au/year-12/extension-1/further-applications-of-calculus