Calculus — Free IB Maths AA HL Practice Questions

1FoundationSAQ-SDefinition and calculation of limits5 marksPaper 1~8 min

The function

g

is defined as

g(x) = \begin{cases} 3x + 2, & x < 1 \\ k, & x = 1 \\ 2x + 3, & x > 1 \end{cases}

where

k

is a constant.

(a)

Find

\lim_{x \to 1^-} g(x)

. [1 mark]

(b)

Find

\lim_{x \to 1^+} g(x)

. [1 mark]

(c)

Determine the value of

k

for which

g

is continuous at

x = 1

. Justify your answer. [2 marks]

(d)

For a different value

k = 7

, identify which condition(s) of continuity fail at

x = 1

and explain why. [1 mark]

Solutions

2MasterySAQ-SDefinition and calculation of limits5 marksPaper 1~8 min

Consider the function

f(x) = \dfrac{x^2 - 4}{x - 2}

, for

x \neq 2

.

(a)

Show that

\displaystyle\lim_{x \to 2} f(x) = 4

. [3 marks]

(b)

Hence find the value of

\displaystyle\lim_{x \to 2} \frac{[f(x)]^2 - 16}{x - 2}

. [2 marks]

Solutions

3ChallengeSAQ-LDefinition and calculation of limits8 marksPaper 1~12 min

A function is defined by

g(x) = \begin{cases} \dfrac{x^2 - 9}{x-3}, & x \neq 3 \\[6pt] k, & x = 3 \end{cases}

where

k

is a constant.

(a)

Determine

\lim_{x \to 3} g(x)

by simplifying the expression for

x \neq 3

. [2 marks]

(b)

Determine the value of

k

for which

g

is continuous at

x = 3

. [1 mark]

(c)

For

k = 0

, classify the discontinuity of

g

at

x = 3

as removable or non-removable. Justify your answer using the formal definition of continuity. [5 marks]

Solutions

4FoundationSAQ-SContinuity of functions at a points5 marksPaper 1~8 min

A function

f

is defined by

f(x) = \begin{cases} x^2 + k, & x < 2 \\ 5, & x = 2 \\ 2x + 1, & x > 2 \end{cases}

where

k

is a constant.

(a)

State three conditions required for

f

to be continuous at

x = 2

. [3 marks]

(b)

Find the value of

k

such that

f

is continuous at

x = 2

. [2 marks]

Solutions

5FoundationSAQ-SDefinition of a derivative (rate of change)8 marksPaper 1~12 min

The height of a tree,

h

metres,

t

years after it was planted is modelled by

h(t) = 1.2 + 0.8\ln(t+1), \quad t \geq 0.

(a)

Using the limit definition of the derivative, show that

h'(t) = \frac{0.8}{t+1}.

[3 marks]

(b)

Find the instaneous rate of growth of the tree when

t = 4

, and interpret this value in context. [2 marks]

(c)

The owner claims the tree's rate of growth will eventually fall below

0.05

metres per year. Determine the value of

t

at which this occurs and comment on whether the model remains realistic for large

t

. [3 marks]

Solutions

6MasterySAQ-SDefinition of a derivative (rate of change)5 marksPaper 1~8 min

A particle moves along a straight line such that its displacement

s

metres from a fixed point

O

at time

t

seconds is given by

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

, for

t \geq 0

.

(a)

Use the definition of the derivative as a limit to show that

v(t) = s'(t) = 3t^2 - 12t + 9

. [3 marks]

(b)

Hence find the values of

t

at which the particle is instaneously at rest. [2 marks]

Solutions

7ChallengeSAQ-LDefinition of a derivative (rate of change)10 marksPaper 1~15 min

The graph of

y = f(x)

for

0 \leq x \leq 4

is shown below. The function

f

is continuous and differentiable on

(0, 4)

. Let

g(x) = x\,f(x)

.

(a)

Using the geometric interpretation of the derivative, state the value of

f'(1)

, justifying your answer. [2 marks]

(b)

Determine the value of

g'(1)

. [3 marks]

(c)

(i) Show that

g'(x) = f(x) + x\,f'(x)

. Hence write

g'(3)

in terms of

f(3)

and

f'(3)

. [2]
(ii) Given that

f'(3) = -2

, find the value of

g'(3)

. Hence determine a value of

x

in the interval

(0, 4)

at which

g'(x) = 0

, other than

x = 1

, justifying your answer. [3 marks]

Solutions

8FoundationSAQ-SDefinition of a derivative (rate of change)5 marksPaper 1~8 min

The graph of

y = f(x)

passes through the point

P(2,\,5)

. The tangent to the graph at

P

has equation

y = 3x - 1

.

(a)

Write down the value of

f(2)

. [1 mark]

(b)

Find the value of

f'(2)

. [2 marks]

(c)

A student claims that

f''(2)

can be determined from the information given. Determine whether the student is correct, justifying your answer. [2 marks]

Solutions

9FoundationSAQ-SIndefinite integrals and their propertiess5 marksPaper 1~8 min

A company designs a logo consisting of a curved line. The gradient of the curve at any point

(x, y)

is given by

\dfrac{dy}{dx} = 6x^2 - 4x + 5

. The curve passes through the point

(1, 8)

.

(a)

Find an expression for

y

in terms of

x

. [3 marks]

(b)

The designer claims the curve has no local maximum or minimum points. Justify whether this claim is correct. [2 marks]

Solutions

10MasterySAQ-SIndefinite integrals and their propertiess5 marksPaper 1~8 min

A function

f

is defined for

x > 0

such that

f'(x) = \dfrac{2}{x} + 3x^2

.

(a)

Find

f(x)

, giving your answer in the form

f(x) = 2\ln x + x^3 + c

, where

c \in \mathbb{R}

. [2 marks]

(b)

Given that the graph of

f

passes through the point

(1, 5)

, find the value of

c

. [2 marks]

(c)

The graph of

f

has exactly one stationary point for

x > 0

. Determine the nature of this stationary point, justifying your answer. [1 mark]

Solutions

11ChallengeSAQ-LIndefinite integrals and their propertiess8 marksPaper 1~12 min

The region bounded by the curve

y = \dfrac{4}{\sqrt{x^2+4}}

, the

x

-axis, and the lines

x = 0

and

x = 2

is rotated

360°

about the

x

-axis to form a solid of revolution.

(a)

Write down a definite integral expression for the volume

V

of this solid. [1 mark]

(b)

Hence find the exact value of

V

. [5 marks]

(c)

The upper bound is extended to

x = a

. Determine whether the volume converges to a finite limit as

a \to \infty

, stating the limit if it exists. [2 marks]

Solutions

12FoundationSAQ-SIndefinite integrals and their propertiess5 marksPaper 1~8 min

A particle moves along a straight line such that its velocity

v\,\text{m s}^{-1}

at time

t

seconds is given by

v(t) = 3t^2 + 2\cos t

. At time

t = 0

, the displacement of the particle from the origin is

4\,\text{m}

.

(a)

Find an expression for the displacement

s(t)

. [3 marks]

(b)

Determine whether the particle is moving towards or away from the origin at time

t = \pi

seconds. Justify your answer. [2 marks]

Solutions

13FoundationSAQ-SSolving first-order differential equationss5 marksPaper 1~8 min

A population of bacteria,

P

(measured in thousands), grows according to the differential equation

\frac{\mathrm{d}P}{\mathrm{d}t} = kP

where

t

is the time in hours and

k

is a positive constant.

(a)

By separating variables, show that the differential equation can be written as

\int \frac{1}{P}\,\mathrm{d}P = \int k\,\mathrm{d}t.

[1 mark]

(b)

Hence find the general solution of the differential equation, expressing

P

in terms of

t

and an arbitrary constant

A

. [2 marks]

(c)

Given that

P = 2

when

t = 0

, and

P = 6

when

t = 3

, find the exact value of

k

. [2 marks]

Solutions

14MasterySAQ-SSolving first-order differential equationss19 marksPaper 1~29 min

A population of bacteria grows according to the differential equation

\frac{dP}{dt} = kP(1000 - P)

where

P

is the population size at time

t

days and

k

is a positive constant.

(a)

Show that

\dfrac{1}{P(1000-P)} = \dfrac{1}{1000}\left(\dfrac{1}{P} + \dfrac{1}{1000-P}\right)

. [2 marks]

(b)

Hence, given that

P(0) = 100

, solve the differential equation to show that

P = \frac{1000\,e^{1000kt}}{9 + e^{1000kt}}.

[3 marks]

(c)

Deduce the long-run behaviour of

P

as

t \to \infty

and explain, with reference to the differential equation, why this value is expected. [2] (Paper 1 extension — no calculator) Total: [7] > (If restricted to 5 marks, omit (c) and award (a) [2], (b) .) [3 marks]

Solutions

15ChallengeSAQ-LSolving first-order differential equationss10 marksPaper 1~15 min

A tank initially contains

100

litres of pure water. A salt solution of concentration

0.5\,\text{kg}\,\text{L}^{-1}

flows into the tank at

3\,\text{L}\,\text{min}^{-1}

, and the well-mixed solution flows out at the same rate. Let

S(t)\,\text{kg}

be the amount of salt in the tank at time

t

minutes.

(a)

Show that

S(t)

satisfies the differential equation

\frac{dS}{dt} = 1.5 - \frac{3S}{100}.

[2 marks]

(b)

Determine the general solution of this differential equation, giving

S

as a function of

t

. [3 marks]

(c)

State the particular solution satisfying the initial condition

S(0) = 0

. [1 mark]

(d)

Determine the time, to the nearest minute, for the salt in the tank to reach

40\,\text{kg}

, and comment on whether the tank's salt content could ever reach

50\,\text{kg}

. [4 marks]

Solutions

16FoundationSAQ-SSolving first-order differential equationss5 marksPaper 1~8 min

Consider the differential equation

\frac{dy}{dx} = \frac{x}{y^2}, \quad y > 0.

(a)

Show that this differential equation is separable by writing it in the form

f(y)\,dy = g(x)\,dx

. [1 mark]

(b)

Find the general solution of the differential equation, expressing

y

in terms of

x

. [3 marks]

(c)

Given that

y = 2

when

x = 0

, determine whether

y

is increasing or decreasing at the point where

x = -3

. [1 mark]

Solutions

17MasterySAQ-SDefinition and calculation of limits6 marksPaper 2~9 min

The below shows the graph of

y = h(x)

for

-3 \leq x \leq 5

. The graph has vertical asymptotes at

x = -2

and

x = 3

.

(a)

State the value of

\displaystyle\lim_{x \to -2^{-}} h(x)

. [1 mark]

(b)

State the value of

\displaystyle\lim_{x \to +\infty} h(x)

. [1 mark]

(c)

Determine whether

\displaystyle\lim_{x \to 1} h(x)

exists. If it does, state its value. [2 marks]

(d)

Given that

h(1) = 4

, determine whether

h

is continuous at

x = 1

. Justify your answer. [2 marks]

Solutions

18MasterySAQ-SContinuity of functions at a points6 marksPaper 2~9 min

A function

f

is defined as

f(x) = \begin{cases} \dfrac{x^2 - kx - 3}{x - 3}, & x \neq 3 \\ 4, & x = 3 \end{cases}

where

k \in \mathbb{R}

.

(a)

Find the value of

k

such that

f

is continuous at

x = 3

. [3 marks]

(b)

For the value of

k

found in part (a), the function value at

x = 3

is changed to

f(3) = c

, where

c \in \mathbb{R}

. (i) State the value of

c

for which

f

is continuous at

x = 3

, and identify the type of discontinuity that occurs for all other values of

c

. [2 marks]
(ii) Explain why no real value of

c

can produce a discontinuity of a different type at

x = 3

. [1 mark]

Solutions

19MasterySAQ-SDefinition and calculation of limits6 marksPaper 2~9 min

The function

f

is defined by

f(x) = \dfrac{\sqrt{x^2 + 3x + 2} - \sqrt{6}}{x^2 - 1}

, for

x \in \mathbb{R}

,

x \neq \pm 1

.

(a)

Calculate

\displaystyle\lim_{x \to 1} f(x)

. [3 marks]

(b)

By evaluating the one-sided limits

\displaystyle\lim_{x \to -1^-} f(x)

and

\displaystyle\lim_{x \to -1^+} f(x)

, determine whether

\displaystyle\lim_{x \to -1} f(x)

exists. [3 marks]

Solutions

20MasterySAQ-SDefinition and calculation of limits6 marksPaper 2~9 min

A function

g

is defined by

g(x) = \dfrac{|x-2|}{x^2 - 5x + 6}

, for

x \in \mathbb{R}

,

x \neq 2,\, 3

.

(a)

Show that

\displaystyle\lim_{x \to 2} g(x)

does not exist. [3 marks]

(b)

Write down the equation of the vertical asymptote of

g

at

x = 3

, and determine whether

g(x) \to +\infty

or

g(x) \to -\infty

as

x \to 3^-

and as

x \to 3^+

. Hence state whether

\displaystyle\lim_{x \to 3} g(x)

exists, justifying your answer. [3 marks]

Solutions

21MasterySAQ-SRules of differentiation (power, product, quotient, chain rule)s6 marksPaper 2~9 min

A curve is defined by

y = \dfrac{2x-1}{x^2+3}

for

x \in \mathbb{R}

.

(a)

Find

\dfrac{dy}{dx}

. [3 marks]

(b)

Determine the

x

-coordinates of the points on the curve where the tangent is horizontal. [3 marks]

Solutions

22MasterySAQ-SRules of differentiation (power, product, quotient, chain rule)s6 marksPaper 2~9 min

The function

f

is defined by

f(x) = (x^2+1)\ln x

for

x > 0

.

(a)

Find

f'(x)

. [3 marks]

(b)

Hence find the exact coordinates of the local minimum of

f

, and justify that is a minimum. [3 marks]

Solutions

23MasterySAQ-SRules of differentiation (power, product, quotient, chain rule)s6 marksPaper 2~9 min

A particle moves along a straight line such that its displacement,

s

metres, from a fixed point

O

at time

t

seconds is given by

s(t) = (t^2 + 1)\ln(t^2 + 1), \quad t \geq 0.

(a)

Show that the velocity of the particle is

v(t) = 2t\bigl(\ln(t^2+1)+1\bigr)

. [3 marks]

(b)

Determine the value of

t

for which

v(t) = 0

, justifying that no other solution exists for

t \geq 0

. [2 marks]

(c)

Hence determine whether the particle ever returns to

O

for

t > 0

. Justify your answer. [1 mark]

Solutions

24MasterySAQ-SRules of differentiation (power, product, quotient, chain rule)s8 marksPaper 2~12 min

Consider the function

f(x) = \dfrac{x^2}{e^x}

, for

x \in \mathbb{R}

.

(a)

Find

f'(x)

. [3 marks]

(b)

Determine the

x

-coordinates of the stationary points of

f

. [2 marks]

(c)

Justify, using

f''(x)

, the nature of each stationary point found in part (b). [3 marks]

Solutions

25MasterySAQ-SIndefinite integrals and their propertiess7 marksPaper 2~11 min

A particle moves along the

x

-axis. Its velocity at time

t

seconds is given by

v(t) = t^2 \cos(2t)\,\text{m s}^{-1}

for

t \geq 0

.

(a)

Show that

\displaystyle\int t^2 \cos(2t)\,dt = \frac{1}{2}t^2\sin(2t) + \frac{1}{2}t\cos(2t) - \frac{1}{4}\sin(2t) + C

. [3 marks]

(b)

Hence, find the displacement of the particle from

t = 0

to

t = \dfrac{\pi}{2}

, giving your answer in exact form. [2 marks]

(c)

Determine the total distance travelled by the particle from

t = 0

to

t = \dfrac{\pi}{2}

. Give your answer correct to 3 significant figures. [2 marks]

Solutions

26MasterySAQ-SIndefinite integrals and their propertiess6 marksPaper 2~9 min

The graph of

y = \ln(3x)

for

x > 0

is shown below. The shaded region is bounded by the curve, the

x

-axis, and the line

x = 4

.

(a)

State the

x

-intercept of the curve. [1 mark]

(b)

Show that

\displaystyle\int \ln(3x)\,dx = x\ln(3x) - x + C

. [3 marks]

(c)

Hence, calculate the area of the shaded region. Give your answer in the form

a\ln b + c

, where

a,\,b,\,c \in \mathbb{Z}

. [2 marks]

Solutions

27MasterySAQ-SIndefinite integrals and their propertiess6 marksPaper 2~9 min

A curve is such that its derivative is given by

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

, for

x > 0

.

(a)

Show that

\displaystyle\int \frac{2x^2 + 3}{\sqrt{x}}\, dx = \frac{4}{5}x^{5/2} + 6x^{1/2} + c

. [3 marks]

(b)

Given that the curve passes through the point

(4,\, 20)

, determine the equation of the curve in the form

y = f(x)

. [2 marks]

(c)

The region

R

is bounded by the curve, the

x

-axis, and the lines

x = 1

and

x = 4

. Determine whether

R

lies entirely above the

x

-axis, justifying your answer. [1 mark]

Solutions

28MasterySAQ-SIndefinite integrals and their propertiess8 marksPaper 2~12 min

Consider the function

f(x) = \dfrac{x^3 - 2x^2 + 4}{x^2}

, for

x \neq 0

.

(a)

Express

f(x)

in the form

ax + b + cx^{-2}

, where

a

,

b

,

c

are constants to be stated. [2 marks]

(b)

Find

\displaystyle\int f(x)\,dx

. [3 marks]

(c)

The curve

y = f(x)

has a local minimum at

x = p

. Determine the value of

p

and justify that is a minimum. [3 marks]

Solutions

29MasterySAQ-SSolving first-order differential equationss7 marksPaper 2~11 min

A chemical reaction is modelled by the differential equation

\dfrac{dC}{dt} = -kC^2

, where

C(t)

is the concentration of a reactant in

\text{mol\,L}^{-1}

at time

t

seconds, and

k

is a positive constant.

(a)

Show that

C(t) = \dfrac{1}{kt + \dfrac{1}{C_0}}

, where

C_0 = C(0)

. [3 marks]

(b)

Given that

C(0) = 2.0\,\text{mol\,L}^{-1}

and

C(10) = 0.50\,\text{mol\,L}^{-1}

, calculate the value of

k

. [2 marks]

(c)

Hence calculate the time, in seconds, at which the concentration reaches

0.10\,\text{mol\,L}^{-1}

. Give your answer correct to 3 significant figures. [2 marks]

Solutions

30MasterySAQ-SSolving first-order differential equationss8 marksPaper 2~12 min

A metal rod cools in a room held at constant temperature

20^\circ\text{C}

. Newton's Law of Cooling gives

\frac{dT}{dt} = -\lambda(T - 20)

where

T(t)

is the rod's temperature in

^\circ\text{C}

at time

t

minutes, and

\lambda > 0

.

(a)

Show that

T(t) = 20 + Ae^{-\lambda t}

, where

A

is a constant. [3 marks]

(b)

The rod's initial temperature is

100^\circ\text{C}

. After

5

minutes its temperature is

60^\circ\text{C}

. Calculate the value of

\lambda

. [2 marks]

(c)

A second identical rod also starts at

100^\circ\text{C}

in the same room, but its cooling constant is

2\lambda

. Determine the time at which the second rod's temperature is exactly half that of the first rod. Give your answer in minutes correct to 3 significant figures. [3 marks]

Solutions

31MasterySAQ-SSolving first-order differential equationss6 marksPaper 2~9 min

A pharmaceutical company is developing a new drug. The rate of change of the concentration

C(t)

(in

\text{mg}\,\text{L}^{-1}

) of the drug in a patient's bloodstream

t

hours after administration is modelled by the differential equation

\frac{dC}{dt} = k(6 - C),

where

k

is a positive constant. Initially, the concentration of the drug is

0\,\text{mg}\,\text{L}^{-1}

.

(a)

Solve the differential equation to find an expression for

C(t)

in terms of

k

and

t

. [4 marks]

(b)

After 2 hours, the concentration is measured to be

3\,\text{mg}\,\text{L}^{-1}

. Determine the value of

k

, correct to 3 significant figures. [2 marks]

Solutions

32MasterySAQ-SSolving first-order differential equationss8 marksPaper 2~12 min

The population

P(t)

of a species of bird on an island, measured in thousands, is modelled by the differential equation

\frac{dP}{dt} = 0.02P(10 - P), \quad t \geq 0,

where

t

is measured in years. Initially, the population is

1

thousand birds.

(a)

By solving the differential equation, show that

P(t) = \frac{10}{1 + 9e^{-0.2t}}.

[4 marks]

(b)

Determine the population after

5

years, correct to 2 significant figures. [1 mark]

(c)

The island's ecosystem can sustain a maximum of

8

thousand birds. Determine the value of

t

at which this limit is first reached, and hence comment on the suitability of the model for large

t

. [3 marks]

Solutions

33ChallengeLAQDefinition and calculation of limits10 marksPaper 3~15 min

A function

h

is defined by

h(x) = \dfrac{\sqrt{x^2 + 2x + 5} - (x+1)}{x}

for

x \neq 0

.

(a)

Show that

h(x)

can be written in the form

h(x) = \sqrt{1 + \dfrac{2}{x} + \dfrac{5}{x^2}} - 1 - \dfrac{1}{x}

for

x > 0

, and hence prove that

\displaystyle\lim_{x \to \infty} h(x) = 0

. [4 marks]

(b)

Show that for

x < 0

,

h(x) = -\sqrt{1 + \frac{2}{x} + \frac{5}{x^2}} - 1 - \frac{1}{x}.

Hence determine

\displaystyle\lim_{x \to -\infty} h(x)

. [3 marks]

(c)

The graph of

y = h(x)

has two distinct horizontal asymptotes. State both asymptotes and explain why the asymptote as

x \to -\infty

differs from the asymptote as

x \to \infty

, with reference to the role of

\sqrt{x^2} = |x|

in the simplification of

h(x)

. [3 marks]

Solutions

34ChallengeLAQContinuity of functions at a points10 marksPaper 3~15 min

Consider the function

f: \mathbb{R} \to \mathbb{R}

defined by

f(x) = \begin{cases} \dfrac{\sin(ax)}{x} & \text{if } x < 0 \\[6pt] \ln 4 & \text{if } x = 0 \\[6pt] \dfrac{4^x - 1}{x} & \text{if } x > 0 \end{cases}

where

a

is a real constant. The following results may be used without proof:

\lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1, \qquad \lim_{x \to 0} \frac{b^x - 1}{x} = \ln b \text{ for } b > 0

(a)

Prove that for

f

to be continuous at

x = 0

, the constant

a

must satisfy

a = \ln 4

. [6 marks]

(b)

Using

a = \ln 4

and the definition

f'(0) = \displaystyle\lim_{h \to 0} \frac{f(h) - f(0)}{h}

, determine whether

f

is differentiable at

x = 0

. Justify your answer by evaluating the left-hand right-hand derivatives separately. [4 marks]

Solutions

35ChallengeLAQDefinition of a derivative (rate of change)10 marksPaper 3~15 min

Consider the function

f(x) = \sqrt{x+1}

defined for

x \geq -1

.

(a)

Prove, using the limit definition of the derivative, that

f'(x) = \dfrac{1}{2\sqrt{x+1}}

for

x > -1

. [5 marks]

(b)

Determine the equation of the tangent line to

y = f(x)

at the point where

x = 3

. [2 marks]

(c)

Use the tangent line found in part (b) to write down a linear approximation for

f(3 + h)

for small

h

. Hence calculate the approximate value of

f(3.4)

given by this approximation, and show that the true value of

f(3.4)

is less than this approximation. Justify why the approximation always overestimates

f(x)

near

x = 3

. [3 marks]

Solutions

36ChallengeLAQDefinition of a derivative (rate of change)15 marksPaper 3~23 min

A particle moves along a straight line. Its position,

s

metres, at time

t

seconds is given by

s(t) = t^3 - 6t^2 + 9t + 2, \quad t \geq 0.

(a)

Using the limit definition of the derivative,

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h},

prove that the velocity of the particle is

v(t) = 3t^2 - 12t + 9

. [5 marks]

(b)

Calculate

v

and

v

. Explain the geometric interpretation of the sign of each velocity value in terms of the graph of

s(t)

. [3 marks]

(c)

The acceleration of the particle is

a(t) = 6t - 12

. (i) Determine the time interval(s) during which the particle is speeding up for

0 \leq t \leq 4

. [3 marks]
(ii) Hence calculate the total distance travelled by the particle over

0 \leq t \leq 4

. [4 marks]

Solutions

37ChallengeLAQIndefinite integrals and their propertiess10 marksPaper 3~15 min

Consider the function

f

defined by

f(x) = \dfrac{2x^3 - 5x^2 + 5x - 3}{x^2 - 2x + 1}

, for

x \neq 1

.

(a)

Prove that

\displaystyle\int f(x)\,dx = x^2 - x + \ln|x-1| + \frac{1}{x-1} + C

, where

C

is an arbitrary constant. [6 marks]

(b)

Hence, evaluate

\displaystyle\int_2^4 f(x)\,dx

, justifying why the Fundamental Theorem of Calculus applies. [4 marks]

Solutions

38ChallengeLAQDefinite integrals and the area under a curves10 marksPaper 3~15 min

A function

f

is defined for

x \geq 0

by

f(x) = \ln(1+x)

.

(a)

Prove that, for

n \in \mathbb{N}

,

\int_0^1 x^n f(x)\,dx = \frac{1}{n+1}\int_0^1 \frac{1-x^{n+1}}{1+x}\,dx.

[5 marks]

(b)

(i) Using the result from part (a) and the booklet identity for the sum of a geometric series, show that

\int_0^1 \frac{\ln(1+x)}{x}\,dx = \int_0^1 \frac{1}{1+x}\sum_{n=0}^{\infty}\frac{x^n}{n+1}\,dx.

[3 marks]
(ii) Hence show that

\int_0^1 \frac{\ln(1+x)}{x}\,dx = \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^2},

and state its exact value. [2 marks]

Solutions

39ChallengeLAQSolving first-order differential equationss10 marksPaper 3~15 min

A population of a bird species on an isolated island has size

P

(in hundreds) at time

t

(in years), satisfying the logistic differential equation

\frac{dP}{dt} = \frac{1}{5}P\!\left(1 - \frac{P}{12}\right), \quad t \geq 0,

with initial condition

P(0) = 3

.

(a)

By separating variables and using partial fractions, show that

P(t) = \frac{12}{1 + 3e^{-t/5}}.

[5 marks]

(b)

Find the limiting value of

P(t)

as

t \to \infty

and state what this represents in context. [2 marks]

(c)

The differential equation has two equilibrium solutions. Justify, by analysing the sign of

\dfrac{dP}{dt}

in each region, that the equilibrium

P = 12

is stable and

P = 0

is unstable. Hence explain why the result in (b) holds for any initial condition

P(0) = P_0

where

0 < P_0 < 12

, but not for

P_0 = 0

. [3 marks]

Solutions

40ChallengeLAQSolving first-order differential equationss10 marksPaper 3~15 min

A tank initially contains

100

litres of brine with a salt concentration of

0.2\,\text{kg per litre}

. Brine with a concentration of

0.5\,\text{kg per litre}

flows into the tank at

4\,\text{litres per minute}

. The mixture is kept uniform by stirring and flows out at

4\,\text{litres per minute}

. Let

S(t)\,\text{kg}

be the amount of salt in the tank at time

t

minutes,

t \geq 0

.

(a)

Show that

S(t)

satisfies the differential equation

\frac{dS}{dt} = 2 - \frac{S}{25}.

[3 marks]

(b)

Show that the solution to the differential equation in (a), subject to the appropriate initial condition, is

S(t) = 50 - 30e^{-t/25}.

[4 marks]

(c)

Find the value of

t

at which the salt concentration in the tank equals

0.4\,\text{kg per litre}

. [1 mark]

(d)

The inflow concentration is now changed to

c\,\text{kg per litre}

, where

c > 0

, while all flow rates remain the same. Determine the range of values of

c

for which the salt concentration in the tank never exceeds

0.4\,\text{kg per litre}

at any time

t \geq 0

, justifying your answer with reference to the long-term behaviour of the solution. [2 marks]

Solutions

Calculus — Free Maths AA HL Practice Questions