IB Mathematics AA HL 5.2 Calculus Question Bank

Practise IB Mathematics AA HL 5.2 by solving advanced integration, differential-equation, series, numerical and multivariable-calculus problems.

Exam points

HL syllabus concepts and methods
HL extended reasoning

Covered syllabus points

AHL 5.12 (HL)—Continuity, differentiability and first principles AHL 5.13 (HL)—L'Hopital's rule and limits AHL 5.14 (HL)—Implicit differentiation and related rates AHL 5.15 (HL)—Further derivatives and integrals AHL 5.16 (HL)—Advanced integration techniques AHL 5.17 (HL)—Areas and volumes of revolution AHL 5.18 (HL)—Differential equations AHL 5.19 (HL)—Maclaurin series

[Maximum number: 6]

The general term of a sequence $\left\{a_{n}\right\}$ is given by the formula $a_{n}=\frac{\mathrm{e}^{n}+2^{n}}{2 \mathrm{e}^{n}}, n \in \mathbb{Z}^{+}$ .

(a)

Show that the sequence $\left\{a_{n}\right\}$ is convergent and find the limit L.

[ 2 ]

$\quad L=\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \frac{1}{2}+\frac{1}{2}\left(\frac{2}{\mathrm{e}}\right)^{n}=\frac{1}{2}+\frac{1}{2} \times 0=\frac{1}{2}$

(b)

Find the smallest value of $N \in \mathbb{Z}^{+}$ such that $\left|a_{n}-L\right|<0.001$ , for all $n \geq N$ .

[ 4 ]

$\quad\left|a_{n}-\frac{1}{2}\right|=\left|\frac{1}{2}+\frac{1}{2}\left(\frac{2}{\mathrm{e}}\right)^{n}-\frac{1}{2}\right|=\left|\frac{1}{2}\left(\frac{2}{\mathrm{e}}\right)^{n}\right|<\frac{1}{1000}$

EITHER
$\Rightarrow\left(\frac{\mathrm{e}}{2}\right)^{n}>500$ $\Rightarrow n>20.25 \ldots$

OR
$\Rightarrow\left(\frac{2}{\mathrm{e}}\right)^{n}<500$ $\Rightarrow n>20.25 \ldots$

Note: A1 for correct inequality; A1 for correct value.

THEN
therefore N=21

[Maximum number: 6]

In this question you will investigate series of the form

\sum_{i=1}^{n} \boldsymbol{i}^{q}=1^{q}+2^{q}+3^{q}+\ldots+n^{q} \text { where } n, q \in \mathbb{Z}^{+}

and use various methods to find polynomials, in terms of n, for such series.
When q=1, the above series is arithmetic.

(a)

Show that $\lim _{x \rightarrow 1} f_{1}(x)$ is in indeterminate form.

[ 1 ]

$\quad \lim _{x \rightarrow 1} f_{1}(x)=\frac{n-(n+1)+1}{0}\left(=\frac{0}{0}\right)$

Note: Only award R1 for sufficient simplification of the numerator, for example, as shown above.

Do not award R1 if $\lim _{x \rightarrow 1}$ is not referred to or stated.

(b)

Hence, by applying l'Hôpital's rule, show that $\lim _{x \rightarrow 1} f_{1}(x)=\frac{1}{2} n(n+1)$ .

[ 5 ]

attempts to differentiate both the numerator and the denominator M1

\lim _{x \rightarrow 1} \frac{n(n+2) x^{n+1}-(n+1)^{2} x^{n}+1}{2(x-1)}

Note: Award A1 for $\left(\left(\lim _{x \rightarrow 1}\right) \frac{n(n+2) x^{n+1}-n(n+1) x^{n}-(n+1) x^{n}+1}{2(x-1)}\right)$ . This form can be used in subsequent work.
(l'Hôpital's rule applies again since)

\lim _{x \rightarrow 1} \frac{n(n+2) x^{n+1}-(n+1)^{2} x^{n}+1}{2(x-1)}=\frac{0}{0}

Note: Do not award R1 if $\lim _{x \rightarrow 1}$ is not referred to or stated.
Subsequent marks are independent of this R mark.
attempts to differentiate both the numerator and the denominator

\begin{aligned} & \lim _{x \rightarrow 1} \frac{n(n+2)(n+1) x^{n}-n(n+1)^{2} x^{n-1}}{2} \\ & =\frac{n(n+2)(n+1)-n(n+1)^{2}}{2}\left(=\frac{n^{3}+3 n^{2}+2 n-\left(n^{3}+2 n^{2}+n\right)}{2}\right) \\ & =\frac{n(n+1)((n+2)-(n+1))}{2}\left(=\frac{n^{2}+n}{2}\right) \\ & =\frac{1}{2} n(n+1) \end{aligned}

[Maximum number: 6]

Two boats A and B travel due north.
Initially, boat B is positioned 50 metres due east of boat A .
The distances travelled by boat A and boat B , after t seconds, are x metres and y metres respectively. The angle $\theta$ is the radian measure of the bearing of boat B from boat A . This information is shown on the following diagram.

Figure for Question 5.2 Calculus - AHL content question 3 — IB Maths AA HL

Find the speed of boat A at time T.

attempt to differentiate with respect to t
$\frac{\mathrm{d} y}{\mathrm{~d} t}=\frac{\mathrm{d} x}{\mathrm{~d} t}-50(\operatorname{cosec} \theta)^{2} \frac{\mathrm{~d} \theta}{\mathrm{~d} t}$
attempt to set speed of B equal to double the speed of A
$2 \frac{\mathrm{~d} x}{\mathrm{~d} t}=\frac{\mathrm{d} x}{\mathrm{~d} t}-50(\operatorname{cosec} \theta)^{2} \frac{\mathrm{~d} \theta}{\mathrm{~d} t}$ $\frac{\mathrm{d} x}{\mathrm{~d} t}=-50(\operatorname{cosec} \theta)^{2} \frac{\mathrm{~d} \theta}{\mathrm{~d} t}$ $\theta=\arctan 5(=1.373 \ldots=78.69 \ldots)$ OR $\operatorname{cosec}^{2} \theta=1+\cot ^{2} \theta=1+\left(\frac{1}{5}\right)^{2}=\frac{26}{25}$

Note: This A1 can be awarded independently of previous marks.
$\frac{\mathrm{d} x}{\mathrm{~d} t}=-50\left(\frac{26}{25}\right) \times-0.1$
So the speed of boat A is $5.2\left(\mathrm{~ms}^{-1}\right)$

Note: Accept 5.20 from the use of inexact values.

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Covered syllabus points

5.2 Calculus - AHL content question 1

Question (a)

Question (b)

5.2 Calculus - AHL content question 2

Question (a)

Question (b)

5.2 Calculus - AHL content question 3