Question 1a

Marks: 6

(a)

Find the first three non-zero terms of the Maclaurin series for

\tan x

in ascending powers of

x

.

Key Concepts

Maclaurin Series of Standard Functions

Question 1b

Marks: 2

(b)

Confirm that the result from part (a) gives the same type of function – either even or odd – as

\tan x

.

Key Concepts

Odd & Even Functions

Question 1c

Marks: 4

(c)

Hence approximate the value of

\tan 1

(i)

by substituting the value

x = 1

(ii)

by substituting another positive value of

x

.

Key Concepts

Maclaurin Series of Standard Functions

Question 1d

Marks: 4

(d)

(i)

Compare the approximations found in part (c) to the exact value of

\tan 1

.

(ii)

Explain briefly the reason for the difference in accuracy between the two approximations.

Key Concepts

Maclaurin Series of Standard Functions

Question 2a

Marks: 4

(a)

Find the first four non-zero terms of the Maclaurin series for

e^{- 2 x}

in ascending powers of

x

.

Key Concepts

Maclaurin Series of Composites & Products

Question 2b

Marks: 3

(b)

Hence approximate the value of

\sqrt{e}

and compare this approximation to the exact value.

Key Concepts

Maclaurin Series of Standard Functions

Question 2c

Marks: 1

(c)

Explain how the accuracy of the Maclaurin series approximation in part (b) could be improved.

Key Concepts

Maclaurin Series of Standard Functions

Question 3a

Marks: 5

(a)

Find the Maclaurin series for

e^{x} (\sin 3 x + \cos \sqrt{x})

in ascending powers of

x

, up to and including the term in

x^{3}

.

Key Concepts

Maclaurin Series of Composites & Products

Question 3b

Marks: 4

(b)

Hence find the first three non-zero terms, in ascending powers of

x

, of the Maclaurin series for

e^{x} (2 \sin 3 x + 6 \cos 3 x + 2 \cos \sqrt{x} - \frac{\sin \sqrt{x}}{\sqrt{x}})

Key Concepts

Differentiating & Integrating Maclaurin Series

Question 4a

Marks: 5

Consider the function $f$ defined by $f (x) = e^{3 x} \cos 2 x$ .

(a)

Show that

f'' (x) = p f (x) + p f' (x)

, where

p

and

q

are constants to be determined.

Key Concepts

Second Order Derivatives

Question 4b

Marks: 3

(b)

Hence find the Maclaurin series for

f (x)

in ascending powers of

x

, up to and including the term in

x^{5}

.

Key Concepts

Maclaurin Series of Standard Functions

Question 4c

Marks: 7

(c)

Show that

\int f (x) d x = \frac{e^{3 x}}{13} (2 \sin 2 x + 3 \cos 2 x) + c

.

Key Concepts

Repeated Integration by Parts

Question 4d

Marks: 4

(d)

Hence find the first seven terms, in ascending powers of

x

, of the Maclaurin series for

e^{3 x} (2 \sin 2 x + 3 \cos 2 x)

.

Key Concepts

Differentiating & Integrating Maclaurin Series

Question 5a

Marks: 4

(a)

Find the Maclaurin series for

e^{\frac{1}{2} x^{2}}

in ascending powers of

x

, up to and including the term in

x^{8}

.

Key Concepts

Maclaurin Series of Composites & Products

Question 5b

Marks: 3

The probability density function for the random variable $X ~ N (0, 1)$ is

$f (x) = \frac{1}{\sqrt{2 π}} e^{- \frac{1}{2} x^{2}}$

(b)

Use the result of part (a) to find an approximation for the probability

P (0 \leq X \leq 1)

.

Key Concepts

Calculating Probabilities using PDF
Differentiating & Integrating Maclaurin Series

Question 5c

Marks: 3

(c)

Determine the percentage error of your approximation from part (b).

Question 6

Marks: 9

Consider the function $f$ defined by

$f (x) = \frac{1}{\sqrt{1 - 2 x^{2}}}$

By first determining the Maclaurin series of $f (x)$ in ascending powers of $x$ , up to and including the term in $x^{6}$ , show that

$\sin \frac{π}{4} \approx 0.70710675$

Be sure to justify that the Maclaurin series is valid for the value of $x$ used to produce your approximation.

Key Concepts

Binomial Theorem: Fractional & Negative Indices

Question 7a

Marks: 5

Consider the differential equation

$y' = \cos x + x y^{2}$

together with the initial condition $y (0) = 1$ .

(a)

Find expressions for

y'', y''', y^{(4)}

and

y^{(5)}

. Each should be given in terms of

x

and

y

and of lower-order derivatives of

y

.

Key Concepts

Maclaurin Series from Differential Equations
Implicit Differentiation

Question 7b

Marks: 7

Let $f (x)$ be the solution to the differential equation above with the given boundary condition, so that $y = f (x)$ .

(b)

Find the first six terms in ascending powers of

x

of the Maclaurin series for

f (x)

.

Key Concepts

Maclaurin Series from Differential Equations

Question 7c

Marks: 2

(c)

Hence find an approximation for the value of

y

when

x = 0.1

.

Key Concepts

Maclaurin Series of Standard Functions

Question 8a

Marks: 9

Consider the differential equation

$y' = \frac{y}{x + 1} + 1, x > - 1$

with the initial condition $y (0) = - 1$ .

(a)

By first finding expressions for

y'', y''',

and

y^{(4)}

in terms of

x, y

and lower-order derivatives of

y

, find a Maclaurin series for the solution to the differential equation with the given boundary condition, in ascending powers of

x

up to and including the term in

x^{4}

.

Key Concepts

Maclaurin Series from Differential Equations
Implicit Differentiation

Question 8b

Marks: 5

(b)

Solve the differential equation with the given boundary condition analytically to find an exact solution in the form

y = f (x)

.

Key Concepts

Integrating Factor

Question 8c

Marks: 3

(c)

Find the first four non-zero terms of the Maclaurin series for the answer to part (b), and confirm that they match those in the answer to part (a).

Key Concepts

Maclaurin Series of Composites & Products

DP IB Maths: AA HL

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5.11 MacLaurin Series

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