Subjects

IB Math

IB Math AA SL

Question Bank

IB Math AA HL

Question Bank

IB Biology

Coming soon

IB Chemistry

Coming soon

IB Physics

Coming soon

IB Psychology

Coming soon

IB Economics

Coming soon

Educators

Coming soon

Students

Coming soon

Analysis and Approaches SL — Questionbank

All Concepts - Calculus

All questions in Topic 5 Calculus

Question 27

NO CALCULATOR

Spicy

[Maximum mark: 15]

A ball is attached to the end of a spring, which is suspended from a ceiling.

The height, $h$ metres, of the ball above the ground at time $t$ seconds after being released can be modelled by the function $h(t) = 0.4cos(\pi t) + 1.8$ where $t \geq 0$ .


(a) Find the height of the ball above the ground when it is released.	[2]
(b) Find the minimum height of the ball above the ground.	[2]
(c) Show that the ball takes 2 seconds to return to its initial height above the ground for the first time.	[2]
(d) For the first 2 seconds of its motion, determine the amount of time that the ball is less than $1.8 + 0.2\sqrt{2}$ metres above the ground	[5]
(e) Find the rate of change of the ball's height above the ground when $t = \frac{1}{3}$ . Give your answer in the form $p\pi\sqrt{q}{\text{ }ms}^{- 1}$ where $p \in \mathbb{Q}$ and $q \in \mathbb{Z}^{+}$ .	[4]

Question 28

NO CALCULATOR

Medium

[Maximum mark: 7]

Consider the function defined by $f(x) = x^{2} - 10x$ . The graph of $f$ passes through the point $A(4, - 24)$ .


(a) Find the gradient of the tangent to the graph of f at the point A .	[2]
(b) Hence, write down the gradient of the normal to the graph of f at Point A .	[1]
(c) Write down the equation of the normal to the graph of f at Point A. The normal to the graph of f at point A intersects the graph of f again at a second point B .	[1]
(d) Find the coordinates of B.	[3]

Question 29

NO CALCULATOR

Medium

[Maximum mark: 13]

The temperature of a cup of tea, $t$ minutes after it is poured, can be modelled by $H(t) = 21 + 75e^{- 0.08t},t \geq 0$ . The temperature is measured in degrees Celsius ( $\ ^{\circ}C$ ).


(a.i) Find the initial temperature of the tea.	[1]
(a.ii) Find the temperature of the tea three minutes after it is poured.	[1]
(b) Write down the value of HI(3).	[2]
(c) Interpret the meaning of your answer to part (b) in the given context.	[2]
(d) After k minutes, the tea will be below 67^∘C and cool enough to drink. Find the least possible value of k, where k ∈ ℤ⁺. As the tea cools, H(t) approaches the temperature of the room, which is constant.	[3]
(e) Find the temperature of the room.	[2]
(f) Find the limit of H_′(t) as t approaches infinity.	[2]

Question 30

NO CALCULATOR

Mild

[Maximum mark: 6]

Consider the vectors $\mathbf{u} = \mathbf{i} + \mathbf{j}$ and $\mathbf{v} = \left( cos\frac{1}{n} \right)\mathbf{i} + \left( sin\frac{1}{n} \right)\mathbf{j}$ , where $n \in \mathbb{Z}^{+}$ .
Let $\theta$ be the angle between $\mathbf{u}$ and $\mathbf{v}$ .


(a) Find an expression for $cos\theta$ in terms of $n$ .	[3]
(b) Find the exact value of the limit approached by $\theta$ as $n \rightarrow \infty$ .	[3]

Question 31

NO CALCULATOR

Spicy

[Maximum mark: 21]

The population, $P$ , of a particular species of marsupial on a small remote island can be modelled by the logistic differential equation

\frac{dP}{\text{ }dt} = kP\left( 1 - \frac{P}{N} \right)

where $t$ is the time measured in years and $k,N$ are positive constants.
The constant $N$ represents the maximum population of this species of marsupial that the island can sustain indefinitely.


(a) In the context of the population model, interpret the meaning of $\frac{dP}{\text{ }dt}$	[1]
(b) Show that $\frac{d^{2}P}{\text{ }dt^{2}} = k^{2}P\left( 1 - \frac{P}{N} \right)\left( 1 - \frac{2P}{N} \right)$ .	[4]
(c) Hence show that the population of marsupials will increase at its maximum rate when $P = \frac{N}{2}$ . Justify your answer.	[5]
(d) Hence determine the maximum value of $\frac{dP}{\text{ }dt}$ in terms of k and N. Let P₀ be the initial population of marsupials.	[3]
(e) By solving the logistic differential equation, show that its solution can be expressed in the form $kt = ln\frac{P}{P_{0}}\left( \frac{N - P_{0}}{N - P} \right)$	[7]
(f) After 10 years, the population of marsupials is 3P₀. It is known that N = 4P₀. Find the value of k for this population model	[2]

Question 32

NO CALCULATOR

Spicy

[Maximum mark: 15]

A scientist conducted a nine-week experiment on two plants, $A$ and $B$ , of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant $A$ was given fertilizer regularly, while Plant $B$ was not.

The scientist found that the height of Plant $A,h_{A}\text{ }cm$ , at time $t$ weeks can be modelled by the function $h_{A}(t) = sin(2t + 6) + 9t + 27$ , where $0 \leq t \leq 9$ .

The scientist found that the height of Plant $B,h_{B}\text{ }cm$ , at time $t$ weeks can be modelled by the function $h_{B}(t) = 8t + 32$ , where $0 \leq t \leq 9$ .

Use the scientist's models to find the initial height of


(a.i) Plant $B$ .	[1]
(a.ii) Plant $A$ correct to three significant figures.	[2]
(b) Find the values of $t$ when $h_{A}(t) = h_{B}(t)$ .	[3]
(c) For $t > 6$ , prove that Plant $A$ was always taller than Plant $B$ .	[3]
(d) For $0 \leq t \leq 9$ , find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$ .	[6]

Question 33

NO CALCULATOR

Spicy

[Maximum mark: 8]

Consider the function $f(x) = \frac{1}{3}x^{3} + \frac{3}{4}x^{2} - x - 1$ .

(a) Write down the y-intercept of the graph of y = f(x).

[1]

(b) Sketch the graph of y = f(x) for −3 ≤ x ≤ 3 and −4 ≤ y ≤ 12.

The function has one local maximum at x = p and one local minimum at x = q.

[4]

(c) Determine the range of f(x) for p ≤ x ≤ q.

[3]

Question 34

NO CALCULATOR

Mild

[Maximum mark: 4]

Let $l$ be the tangent to the curve $y = xe^{2x}$ at the point $\left( 1,e^{2} \right)$ .

Find the coordinates of the point where $l$ meets the $x$ -axis.

Question 35

NO CALCULATOR

Spicy

[Maximum mark: 17]

The derivative of a function $f$ is given by $f'(x) = \frac{2x + 2}{x^{2} + 2x + 2}$ , for $x\mathbb{\in R}$ .


(a.i) Show that $x^{2} + 2x + 2 > 0$ for all values of $x$ .	[2]
(a.ii) Hence, find the values of $x$ for which $f$ is increasing.	[1]
(b.i) Write down the value of $x$ for which $f'(x) = 0$ .	[1]
(b.ii) Show that $f''(x) = \frac{- 2x^{2} - 4x}{\left( x^{2} + 2x + 2 \right)^{2}}$ .	[4]
(b.iii) Hence, justify that the value of $x$ found in part (b)(i) corresponds to a local minimum point on the graph of $f$ .	[2]
(c) Find an expression for $f(x)$ .	[4]
(d) Find the equation of the normal to the graph of $f$ at $(2,\ 3 + ln\, 10)$ .	[3]

Question 36

CALCULATOR

Spicy

[Maximum mark: 16]

Consider the differential equation $\frac{dy}{dx} - y\, cosec\, 2x = \sqrt{\tan\, x}$ , where $0 < x < \frac{\pi}{2}$ and $y = \frac{\pi}{4}$ at $x = \frac{\pi}{4}$ .


(a) Use Euler’s method with step length $\frac{\pi}{12}$ to find an approximate value of y when $x = \frac{5\pi}{12}$ . Give your answer correct to three significant figures.	[3]
(b) Show that $\frac{d}{dx}\left( \frac{1}{2}\,\ln\,\left( \cot\, x \right) \right) = - cosec\, 2x$ .	[4]
(c) Show that $\sqrt{\cot\, x}$ is an integrating factor for this differential equation.	[4]
(d) Hence, by solving the differential equation, show that $y = x\sqrt{\tan\, x}$ .	[5]