Maclaurin Series - IB Math AA Question Bank

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Maclaurin Series

Practice exam-style IB Math AA questions for Maclaurin Series, aligned with the syllabus and grouped by topic.

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Question 1

HL • Paper 1

Easy

Non Calculator

HL • Paper 1

Easy

Non Calculator

The function $f$ is defined by $f(x)=e^{-2x}$ .

Find the Maclaurin series for $f(x)$ up to and including the term in $x^3$ .

[2]

Hence find a polynomial approximation to $e^{-\frac{1}{5}}$ , giving your answer as a single fraction.

[2]

Question 2

HL • Paper 2

Easy

Calculator Permitted

HL • Paper 2

Easy

Calculator Permitted

Consider the function $g(x)=\sqrt{1+3x}$ .

Find the Maclaurin series for $g(x)$ up to and including the term in $x^3$ .

[3]

Hence find $\displaystyle \lim_{x\to 0}\frac{\sqrt{1+3x}-1-\frac{3}{2}x}{x^2}$ .

[2]

Question 3

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the function $f(x)=\sqrt{1-3x}$ .

Find the Maclaurin series for $f(x)$ up to and including the term in $x^3$ .

[4]

State the interval of convergence for this series.

[1]

Question 4

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=e^x\sin(2x)$ .

Find the Maclaurin series for $f(x)$ up to and including the term in $x^4$ .

[5]

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Question 5

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the function $f(x)=\arctan(3x^2)$ .

Determine the Maclaurin series for $f(x)$ up to and including the term in $x^6$ .

[3]

State the set of values of $x$ for which this series converges.

[1]

Question 6

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let

I(x)=\int_0^x \frac{\ln(1+t)}{t}\,\mathrm{d}t

Using the Maclaurin series for $\ln(1+t)$ , find the Maclaurin series for $\dfrac{\ln(1+t)}{t}$ up to and including the term in $t^3$ .

[2]

Hence find the Maclaurin series for $I(x)$ up to and including the term in $x^4$ .

[3]

Question 7

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The geometric series is

\frac{1}{1+x}=1-x+x^2-x^3+x^4-\cdots

By differentiating term by term, find the Maclaurin series for $\dfrac{1}{(1+x)^2}$ up to and including the term in $x^4$ .

[4]

State the interval of convergence of this series.

[1]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Find

\lim_{x\to 0}\frac{\ln(1+2x)-\sin(2x)}{x^2}

[4]

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Question 9

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $f(x)=e^{2x}\ln(1+x)$ .

Find the Maclaurin series for $f(x)$ up to and including the term in $x^4$ .

[4]

Use your series to estimate $\displaystyle \int_0^{0.2} e^{2x}\ln(1+x)\,\mathrm{d}x$ .

[2]

Question 10

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $h(x)=\dfrac{\arctan x}{1+x^2}$ .

Find the Maclaurin series for $h(x)$ up to and including the term in $x^5$ .

[4]

Use your series to estimate $h(0.4)$ .

[2]

Question 11

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

I(x)=\int_0^x \frac{1-\cos(t^2)}{t^2}\,\mathrm{d}t

where the integrand is defined by its limiting value at $t=0$ .

Find the Maclaurin series for $I(x)$ up to and including the term in $x^{11}$ .

[4]

Use your series to estimate $I(1)$ .

[1]

Question 12

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

f(x)=\frac{1}{e^x+\cos x}

Determine the Maclaurin series for $f(x)$ up to and including the term in $x^3$ .

[4]

Hence find $f'''(0)$ .

[1]

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Question 13

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A function $y$ satisfies the differential equation

y'=y+x

with $y(0)=2$ .

Find $y'(0)$ , $y''(0)$ , $y'''(0)$ and $y^{(4)}(0)$ .

[3]

Hence find the Maclaurin series for $y$ up to and including the term in $x^4$ .

[2]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let

h(x)=e^x(1+ax)^{-\frac{1}{2}}

where $a$ is a real constant. The coefficient of $x^2$ in the Maclaurin series for $h(x)$ is $1$ .

Find the possible values of $a$ .

[6]

Question 15

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=e^x\cos x$ .

Find the Maclaurin series for $f(x)$ up to and including the term in $x^4$ .

[5]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let

f(x)=\frac{\ln(1+x)}{1-x}

Find the Maclaurin series for $f(x)$ up to and including the term in $x^3$ .

[4]

Hence find

\lim_{x\to 0}\frac{f(x)-x-\dfrac{x^2}{2}}{x^3}

[2]

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Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The Maclaurin series for $\tan x$ may be found using the series for $\sin x$ and $\cos x$ .

Show that $\tan x=x+\dfrac{x^3}{3}+\dfrac{2x^5}{15}+\cdots$ .

[4]

Using this series, estimate the smallest positive value of $x$ for which $\tan(2x)-2x=0.100$ .

[2]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A function $y$ satisfies the differential equation

y''+xy'+y=0

with $y(0)=1$ and $y'(0)=0$ .

Find $y''(0)$ and $y'''(0)$ .

[2]

Find the Maclaurin series for $y$ up to and including the term in $x^6$ .

[4]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $L(x)=\ln(\cos x)$ .

Find the Maclaurin series for $L(x)$ up to and including the term in $x^6$ .

[4]

Hence find $\displaystyle \lim_{x\to 0}\frac{\ln(\cos x)+\frac{x^2}{2}+\frac{x^4}{12}}{x^6}$ .

[2]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let $p(x)=\arctan(e^x-1)$ .

Find the Maclaurin series for $p(x)$ up to and including the term in $x^4$ .

[4]

Using this series, estimate the smaller positive solution (the solution close to $x=0$ ) of $\arctan(e^x-1)=0.500$ .

[2]

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Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the third-degree Maclaurin polynomial for $\ln(1-2x)$ .

Write down this polynomial and state the interval of convergence of the corresponding infinite series.

[3]

Use a GDC to determine the largest positive value of $x$ for which $|\ln(1-2x)-P_3(x)|=0.005$ .

[2]

Question 22

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=(1+x)^{-\frac12}\ln(1+2x)$ , where $x$ is close to $0$ .

I.

Find the Maclaurin series for $(1+x)^{-\frac12}$ up to and including the term in $x^3$ .

[2]

II.

Find the Maclaurin series for $\ln(1+2x)$ up to and including the term in $x^3$ .

[2]

Hence find the Maclaurin series for $f(x)$ up to and including the term in $x^3$ .

[3]

Hence determine $\lim_{x\to 0}\frac{f(x)-2x+3x^2}{x^3}$

[2]

Question 23

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $G$ is defined by $G(x)=\int_0^x \frac{\arctan t}{t}\,\mathrm{d}t$ where the integrand is defined by its limiting value at $t=0$ .

I.

Starting from $\dfrac{1}{1+t^2}$ , obtain the Maclaurin series for $\arctan t$ up to and including the term in $t^7$ .

[3]

II.

State the interval of convergence of the series for $\arctan t$ .

[1]

Hence find the Maclaurin series for $G(x)$ up to and including the term in $x^7$ .

[3]

Using the first four non-zero terms of this series, find an approximate value for $G(1)$ as a single fraction.

[2]

Question 24

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

For $x\ne0$ , let $f(x)=\dfrac{\sin x}{x}$ , and define $f(0)=1$ .

I.

Find the Maclaurin series for $f(x)$ up to and including the term in $x^6$ .

[2]

II.

Hence find the Maclaurin series for $\dfrac{1}{f(x)}$ up to and including the term in $x^4$ . If you did not obtain the series in part (a)(i), use $f(x)=1-\dfrac{x^2}{6}+\dfrac{x^4}{120}+\cdots$ .

[3]

Hence determine $\lim_{x\to0}\frac{\dfrac{x}{\sin x}-1-\dfrac{x^2}{6}}{x^4}$

[3]

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Question 25

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $F$ is defined by $F(x)=\int_0^x (1+t^2)^{-\frac32}\,\mathrm{d}t$

I.

Find the Maclaurin series for $(1+t^2)^{-\frac32}$ up to and including the term in $t^6$ .

[3]

II.

Hence find the Maclaurin series for $F(x)$ up to and including the term in $x^7$ .

[2]

I.

Show that $F(x)=\dfrac{x}{\sqrt{1+x^2}}$ .

[2]

II.

Hence determine $\lim_{x\to0}\frac{F(x)-x+\frac{x^3}{2}-\frac{3x^5}{8}}{x^7}$

[2]

Question 26

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=e^{x^2}\cos x$

I.

Write down the Maclaurin series for $e^{x^2}$ and for $\cos x$ up to and including the term in $x^6$ .

[2]

II.

Hence find the Maclaurin series for $f(x)$ up to and including the term in $x^6$ .

[2]

Using the series from part (a), approximate $\int_0^1 \frac{f(x)-1}{x^2}\,\mathrm{d}x$

[3]

Justify why every odd derivative of $f$ at $0$ is zero.

[2]

Question 27

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A function $y$ satisfies

y''+y=x

with $y(0)=1$ and $y'(0)=0$ .

By writing $y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+\cdots$ , determine the Maclaurin series for $y$ up to and including the term in $x^5$ .

[6]

Question 28

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The polynomial $P_5(x)$ is the Maclaurin polynomial for $\sin x$ up to and including the term in $x^5$ .

I.

Write down $P_5(x)$ .

[1]

II.

State why the error function $E(x)=\sin x-P_5(x)$ is an odd function.

[2]

Use a GDC to determine the largest positive value of $x$ for which

|\sin x-P_5(x)|=0.001

[2]

Hence state the interval, symmetric about zero, on which $P_5(x)$ approximates $\sin x$ with absolute error less than $0.001$ .

[2]

Explain why the Maclaurin series for $\sin x$ itself converges for all real $x$ , but the polynomial $P_5(x)$ does not give a uniformly accurate approximation for all real $x$ .

[2]

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Question 29

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A function $y$ satisfies

y'=x+y^2

with $y(0)=1$ .

Determine the Maclaurin series for $y$ up to and including the term in $x^4$ .

[5]

Use this series to estimate $y(0.2)$ .

[1]

Question 30

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

For $x\ne 0$ , define $F(x)=(1+x)^{1/x}$ , and define $F(0)=e$ by its removable continuous extension. Show that the Maclaurin series for $F(x)$ , up to and including the term in $x^3$ , is $e\left(1-\dfrac{x}{2}+\dfrac{11x^2}{24}-\dfrac{7x^3}{16}+\cdots\right)$ .

Show that the Maclaurin series for $F(x)$ , up to and including the term in $x^3$ , is $e\left(1-\dfrac{x}{2}+\dfrac{11x^2}{24}-\dfrac{7x^3}{16}+\cdots\right)$ .

[5]

Hence find $\displaystyle \lim_{x\to 0}\frac{(1+x)^{1/x}-e}{x}$ .

[1]

Question 31

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

The value of $\pi$ can be approximated using $\pi=6\arcsin\left(\frac12\right)$ . In this question, a Maclaurin series for $\arcsin x$ is obtained by integration.

I.

Use the binomial expansion to show that $\frac{1}{\sqrt{1-t^2}}=1+\frac{t^2}{2}+\frac{3t^4}{8}+\frac{5t^6}{16}+\cdots$

[3]

II.

Hence find the Maclaurin series for $\arcsin x$ up to and including the term in $x^7$ .

[2]

Use the series from part (a) to estimate $\pi$ .

[1]

The next term in the series for $\arcsin x$ is $\frac{35x^9}{1152}$ . You may use that, for the terms of this series evaluated at $x=\frac12$ , the ratio of each term to the preceding term is less than $\frac14$ . Use this to find an upper bound for the error in your estimate in part (b), and state whether the estimate is an underestimate or an overestimate.

[2]

Question 32

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A function $y$ satisfies the differential equation $y''+2xy'+2y=0$ with $y(0)=1$ and $y'(0)=0$ .

I.

Find $y''(0)$ and $y'''(0)$ .

[3]

II.

Find $y^{(4)}(0)$ and $y^{(5)}(0)$ . If you did not obtain $y''(0)=-2$ and $y'''(0)=0$ , use these values in this part.

[3]

Hence find the Maclaurin series for $y$ up to and including the term in $x^5$ .

[3]

Using this series, approximate $\displaystyle\int_0^1 y\,\mathrm{d}x$ .

[2]

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Question 33

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For a real constant $a$ , define $f_a(x)=e^{ax}\cos x$

I.

Find the Maclaurin series for $f_a(x)$ up to and including the term in $x^4$ .

[4]

II.

Given that the coefficient of $x^3$ is zero and $a>0$ , find $a$ .

[2]

Hence determine $\lim_{x\to0}\frac{e^{\sqrt3 x}\cos x-1-\sqrt3 x-x^2}{x^4}$

[3]

Question 34

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A function $y$ satisfies the differential equation $y''=xy$ with $y(0)=0$ and $y'(0)=1$ . Suppose $y=a_0+a_1x+a_2x^2+a_3x^3+\cdots$

I.

Write down $a_0$ , $a_1$ and $a_2$ .

[2]

II.

By substituting the series into the differential equation, show that $(n+2)(n+1)a_{n+2}=a_{n-1}\quad \text{for } n\ge1$

[2]

Hence find the Maclaurin series for $y$ up to and including the term in $x^7$ .

[3]

Using this series, find an approximate value for $y(1)$ as a single fraction.

[2]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let

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

I.

Write down the Maclaurin series for $\sin x$ up to and including the term in $x^5$ .

[1]

II.

Using the Maclaurin series for $\ln(1+u)$ , find the Maclaurin series for $f(x)$ up to and including the term in $x^5$ .

[5]

Hence determine $\lim_{x\to0}\frac{\ln(1+\sin x)-x+\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{12}}{x^5}$

[2]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $h(x)=\arctan(\sin x)$

I.

Write down the Maclaurin series for $\arctan u$ up to and including the term in $u^5$ .

[1]

II.

Find the Maclaurin series for $h(x)$ up to and including the term in $x^5$ .

[4]

I.

Hence find $h'(0)$ , $h'''(0)$ and $h^{(5)}(0)$ .

[3]

II.

Determine $\lim_{x\to0}\frac{h(x)-x+\frac{x^3}{2}}{x^5}$

[2]

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Question 37

HL • Paper 1

Hard

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HL • Paper 1

Hard

Non Calculator

A function $y$ satisfies $y'=(1+x)y+x^2$ with $y(0)=1$ .

I.

Find $y'(0)$ , $y''(0)$ , $y'''(0)$ and $y^{(4)}(0)$ .

[5]

II.

Hence find the Maclaurin series for $y$ up to and including the term in $x^4$ . If you did not obtain the derivative values in part (a)(i), use $y'(0)=1$ , $y''(0)=2$ , $y'''(0)=6$ and $y^{(4)}(0)=12$ .

[2]

Using this series, approximate $\displaystyle\int_0^1 y\,\mathrm{d}x$ .

[2]

Question 38

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $L(x)=\ln\left(\frac{1+x}{1-x}\right),\qquad -1<x<1$

I.

Using the Maclaurin series for $\ln(1+x)$ and $\ln(1-x)$ , find the Maclaurin series for $L(x)$ up to and including the term in $x^5$ .

[3]

II.

State the interval of convergence of the series for $L(x)$ .

[1]

Let $M(x)=\sqrt{1-x^2}\,L(x)$ . Find the Maclaurin series for $M(x)$ up to and including the term in $x^5$ .

[4]

Hence determine $\lim_{x\to0}\frac{M(x)-2x+\frac{x^3}{3}}{x^5}$

[2]

Question 39

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined by

f(x)=e^{-x^2}\\cos(3x), \quad x\in\mathbb{R}

The polynomial $P_6(x)$ is the Maclaurin polynomial for $f$ up to and including the term in $x^6$ .

I.

Write down the Maclaurin series for $e^{-x^2}$ and for $\cos(3x)$ up to and including terms in $x^6$ .

[2]

II.

Hence determine $P_6(x)$ .

[3]

Use $P_6(x)$ to find a polynomial approximation for

I(a)=\int_0^a e^{-x^2}\cos(3x)\,\mathrm{d}x

[2]

Using the approximation from part (b), determine the two positive values of $a$ for which $I(a)=0.240$ .

[3]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

g(x)=\ln(1+\sin x)

I.

Write down the Maclaurin series for $\sin x$ up to and including the term in $x^5$ .

[1]

II.

Hence find the Maclaurin series for $g(x)$ up to and including the term in $x^5$ .

[4]

Using your series, estimate the smallest positive solution of $\ln(1+\sin x)=0.250$ .

[2]

Hence determine the exact value of

\lim_{x\to 0}\frac{g(x)-x+\dfrac{x^2}{2}-\dfrac{x^3}{6}+\dfrac{x^4}{12}}{x^5}

[3]

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Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

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A function $y$ satisfies the differential equation

y''+xy'+2y=0

with $y(0)=1$ and $y'(0)=0$ .

I.

Find $y''(0)$ and $y'''(0)$ .

[2]

II.

By writing $y=a_0+a_1x+a_2x^2+ \cdots$ , find a recurrence relation for the coefficients.

[3]

Hence determine the Maclaurin series for $y$ up to and including the term in $x^6$ .

[3]

Using the polynomial found in part (b), estimate the first positive value of $x$ for which $y=0$ .

[2]

Question 42

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Consider

F(x)=\frac{1}{1+x+x^2}

and

J(x)=\int_0^x F(t)\,\mathrm{d}t

I.

By writing $F(x)=a_0+a_1x+a_2x^2+ \cdots$ , find a recurrence relation for the coefficients.

[3]

II.

Hence write the Maclaurin series for $F(x)$ up to and including the term in $x^6$ .

[2]

Use the series in part (a) to find the Maclaurin series for $J(x)$ up to and including the term in $x^7$ .

[2]

Show that

J(x)=\frac{2}{\sqrt{3}}\left(\arctan\left(\frac{2x+1}{\sqrt{3}}\right)-\frac{\pi}{6}\right)

[2]

Determine, using a GDC, the positive solution of $J(x)=0.500$ . Then find the value given by solving the polynomial approximation from part (b).

[2]

Question 43

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The inverse sine function may be written as

\arcsin x=\int_0^x \frac{1}{\sqrt{1-t^2}}\,\mathrm{d}t, \quad |x|<1

I.

Use the binomial expansion to find the Maclaurin series for $(1-x^2)^{-\frac12}$ up to and including the term in $x^6$ .

[3]

II.

Hence find the Maclaurin series for $\arcsin x$ up to and including the term in $x^7$ .

[2]

Use the series in part (a) with $x=\dfrac12$ to obtain an approximation for $\pi$ .

[2]

Calculate the absolute error in the approximation from part (b).

[1]

Using a GDC and the series generated by integrating the binomial expansion, determine the least odd power $x^{2n+1}$ that must be included so that the resulting approximation to $\pi$ has absolute error less than $10^{-6}$ .

[2]

Question 44

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let

H(x)=e^{x^2}\ln(1+2x)

I.

Write down the Maclaurin series for $e^{x^2}$ and $\ln(1+2x)$ up to the terms needed to find $H(x)$ as far as $x^5$ .

[2]

II.

Hence determine the Maclaurin series for $H(x)$ up to and including the term in $x^5$ .

[3]

Use your series to estimate

\int_0^{0.25} e^{x^2}\ln(1+2x)\,\mathrm{d}x

[2]

Use a GDC to evaluate the integral in part (b) directly and hence find the absolute error in the estimate.

[2]

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Question 45

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A function $y$ satisfies

y'=1+xy

with $y(0)=1$ .

I.

Let $y=a_0+a_1x+a_2x^2+ \cdots$ . Find a recurrence relation for the coefficients.

[3]

II.

Hence find the Maclaurin series for $y$ up to and including the term in $x^6$ .

[2]

Using the polynomial found in part (a), estimate the positive value of $x$ for which $y=2$ .

[2]

The exact solution may be written in the form

y=e^{x^2/2}\left(1+\int_0^x e^{-t^2/2}\,\mathrm{d}t\right)

Verify this expression and use a GDC to find the positive value of $x$ for which the exact solution has value $2$ .

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A damped oscillation is modelled by $f(x)=e^{-x}\cos(2x)$ for $0\le x\le 0.5$ . A fourth-degree Maclaurin polynomial $P_4(x)$ is to be used as a simple approximation to $f(x)$ near $x=0$ .

I.

Write down the Maclaurin series for $e^{-x}$ and $\cos(2x)$ up to and including the term in $x^4$ .

[2]

II.

Hence find $P_4(x)$ for $f(x)$ .

[3]

I.

Use $P_4$ to estimate $f(0.30)$ .

[2]

II.

Using a GDC, determine the largest value of $k$ in $0\le k\le 0.5$ such that $|f(x)-P_4(x)|\le 0.005$ for all $0\le x\le k$ . If you did not obtain $P_4(x)$ , use $P_4(x)=1-x-\frac{3x^2}{2}+\frac{11x^3}{6}-\frac{7x^4}{24}$ .

[3]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $0\le x\le 0.5$ , define $A(x)=2\arctan x-\arctan(2x)$ . This function represents the small difference between two angle measurements.

I.

Write down the first four non-zero terms of the Maclaurin series for $\rctan x$ and for $\rctan(2x)$ .

[2]

II.

Hence find the first three non-zero terms in the Maclaurin series for $A(x)$ .

[3]

Use the series from part (a) to estimate $A(0.4)$ .

[1]

Justify that $A(x)>0$ for $0<x<0.5$ .

[2]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A function $y$ is defined by the differential equation $y'=1+xy$ with $y(0)=0$ .

I.

By writing $y=\sum_{n=0}^{\infty}a_nx^n$ , find a recurrence relation for the coefficients.

[3]

II.

Find the Maclaurin series for $y$ up to and including the term in $x^7$ .

[2]

Use your series to estimate $y(0.6)$ .

[1]

Deduce a general expression for the non-zero coefficients of the series.

[2]

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Question 49

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The curve $y=\sec x$ is to be approximated near $x=0$ by an even polynomial $S_6(x)=1+ax^2+bx^4+cx^6$ .

Expression	Role
$y=\sec x$	Curve near $x=0$
$S_6(x)=1+ax^2+bx^4+cx^6$	Even polynomial approximation

I.

Use the identity $\cos x\sec x=1$ to find $a$ , $b$ and $c$ .

[4]

Use $S_6$ to estimate $\int_0^{0.4}\sec x\,\mathrm{d}x$ .

[2]

Given that the next non-zero term in the Maclaurin series for $\sec x$ is positive, state whether your estimate in part (b) is an underestimate or an overestimate. Justify your answer by considering the sign of the entire omitted tail, not only the next term.

[2]

Question 50

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For a real parameter $a$ , let $H_a(x)=e^{ax}\ln(1+2x)$ A value of $a$ is chosen so that the quadratic term in the Maclaurin series of $H_a$ is zero.

I.

Find the coefficient of $x^2$ in the Maclaurin series for $H_a(x)$ .

[3]

II.

Determine the required value of $a$ , and find the Maclaurin series for $H_a(x)$ up to and including the term in $x^4$ .

[2]

Hence find $\lim_{x\to0}\frac{e^x\ln(1+2x)-2x}{x^3}$

[1]

Using the polynomial found in part (a), estimate the small positive solution (the solution closest to $x=0$ ) of $e^x\ln(1+2x)=0.25$ . If you did not obtain a polynomial, use $2x+\frac{5x^3}{3}-2x^4$ .

[2]

Question 51

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For small positive $x$ , the difference $D(x)=\sin x-\arctan x$ may be approximated by a polynomial.

I.

Find the Maclaurin series for $D(x)$ up to and including the term in $x^5$ .

[3]

Find the value of $a$ for which $\lim_{x\to0}\frac{\sin x-\arctan x-ax^3}{x^5}$ is finite and non-zero, and find the value of the limit.

[3]

Using the polynomial from part (a), estimate the smallest positive solution of $\sin x-\arctan x=0.020$ .

[2]

Question 52

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $n\in\mathbb{Z}^{+}$ , define

S_n=4\sum_{r=0}^{n}(-1)^r\left(\frac{(1/2)^{2r+1}}{2r+1}+\frac{(1/3)^{2r+1}}{2r+1}\right)

I.

Show that $\arctan\dfrac{1}{2}+\arctan\dfrac{1}{3}=\dfrac{\pi}{4}$ .

[2]

II.

Explain why $S_n$ is an approximation to $\pi$ .

[2]

Calculate $S_2$ and the absolute error $|S_2-\pi|$ .

[3]

Using the alternating series error estimate, show that

|S_n-\pi|\le \frac{4}{2n+3}\left(\frac{1}{2^{2n+3}}+\frac{1}{3^{2n+3}}\right)

[2]

Determine the least value of $n$ for which this error bound is less than $10^{-6}$ .

[2]

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Question 53

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For sufficiently small $x$ , let $w$ be defined implicitly by

w e^w=x

Assume that

w=a_1x+a_2x^2+a_3x^3+a_4x^4+\cdots

I.

Write the Maclaurin series for $e^w$ up to and including the term in $w^4$ .

[1]

II.

By substituting the assumed series into $we^w=x$ , show that

w=x-x^2+\frac{3x^3}{2}-\frac{8x^4}{3}+\cdots

[5]

Use this series to estimate the solution of $we^w=0.300$ .

[2]

Use a GDC to find the actual solution of $we^w=0.300$ and hence find the percentage error in the estimate from part (b).

[2]

Question 54

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A rational approximation to $e^x$ is given by

R(x)=\frac{1+ax}{1+bx+cx^2}

where $a$ , $b$ and $c$ are constants. The coefficients of $x$ , $x^2$ and $x^3$ in the Maclaurin series for $R(x)$ are to match those of $e^x$ .

I.

Find the Maclaurin series for $(1+bx+cx^2)^{-1}$ up to and including the term in $x^3$ .

[3]

II.

Hence show that $a=\dfrac13$ , $b=-\dfrac23$ and $c=\dfrac16$ .

[3]

Use $R(1)$ to approximate $e$ .

[2]

Compare the percentage error of this approximation with that of the third-degree Maclaurin polynomial for $e^x$ at $x=1$ .

[2]

Question 55

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A function $y$ is represented by a Maclaurin series and satisfies

xy''+y'+xy=0

with $y(0)=1$ . Assume that $y'(0)=0$ .

I.

Let $y=a_0+a_1x+a_2x^2+ \cdots$ . Show that $a_1=0$ and that, for $n\ge 1$ ,

(n+1)^2a_{n+1}+a_{n-1}=0

[3]

II.

Hence find the Maclaurin series for $y$ up to and including the term in $x^8$ .

[3]

Using this polynomial approximation, estimate the first positive zero of $y$ .

[2]

Explain why the approximation in part (b) should be treated with caution.

[2]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A function $y$ satisfies the differential equation $y''+xy'-3y=0$ with $y(0)=1$ and $y'(0)=2$ . Let $y=\sum_{n=0}^{\infty}a_nx^n$

I.

Show that $a_{n+2}=\frac{3-n}{(n+2)(n+1)}a_n$ for $n\ge0$ .

[3]

II.

Hence find the Maclaurin series for $y$ up to and including the term in $x^6$ .

[2]

Use your series to estimate $y(0.4)$ .

[1]

Prove that all odd-power coefficients after $a_3$ are zero.

[3]

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Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $x\ne0$ , define $F(x)=\frac{\ln(1+x^2)}{x^2}$ and define $F(0)$ by continuity. The integral $I=\int_0^{0.8}F(x)\,\mathrm{d}x$ is to be estimated using a Maclaurin series.

I.

Find the Maclaurin series for $F(x)$ .

[2]

II.

Use the first three non-zero terms to estimate $I$ .

[2]

The integrated series is alternating for $0\le x\le0.8$ . Determine the minimum number of terms of the series for $F(x)$ that must be integrated so that the alternating-series error bound for $I$ is less than $10^{-4}$ .

[4]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A function $y$ satisfies $y''=xy$ with $y(0)=1$ and $y'(0)=-1$ . Let $y=\sum_{n=0}^{\infty}a_nx^n$

I.

Find a recurrence relation for the coefficients $a_n$ .

[3]

II.

Find the Maclaurin series for $y$ up to and including the term in $x^6$ .

[2]

Use your series to estimate $y(1)$ .

[1]

Prove that $a_{3k+2}=0$ for all integers $k\ge0$ .

[2]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The function $D(x)=e^{-x^2}\int_0^x e^{t^2}\,\mathrm{d}t$ occurs in a model of diffusion. Its Maclaurin series may be found using products of series.

I.

Find the Maclaurin series for $\int_0^x e^{t^2}\,\mathrm{d}t$ up to and including the term in $x^7$ .

[2]

II.

Hence find the Maclaurin series for $D(x)$ up to and including the term in $x^7$ .

[3]

Use the series from part (a) to estimate $D(0.5)$ .

[1]

Show that $D$ satisfies the differential equation $D'=1-2xD$ , and use this equation to verify the coefficient of $x^7$ in your series.

[3]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $x\ne0$ , define $L(x)=\ln\left(\frac{\sin x}{x}\right)$ with $L(0)$ defined by continuity. This function is used to study the central peak of a wave pattern.

I.

Show that $\frac{\sin x}{x}=1-\frac{x^2}{6}+\frac{x^4}{120}-\frac{x^6}{5040}+\cdots$

[2]

II.

Hence find the Maclaurin series for $L(x)$ up to and including the term in $x^6$ .

[3]

Use your answer to part (a) to estimate the positive value of $x$ for which $\frac{\sin x}{x}=0.95$ .

[2]

Explain why the Maclaurin series for $L(x)$ contains only even powers of $x$ .

[1]

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Kinematics

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