5.11 MacLaurin Series

Question 1b

Marks: 2

b)

Hence approximate the value of

e^{2 x}

when

x = 1

.

Key Concepts

Question 1c

Marks: 3

c)

(i)

Compare the approximation found in part (b) to the exact value of

e^{2 x}

when

x = 1 .

(ii)

Explain how the accuracy of the Maclaurin series approximation could be improved.

Key Concepts

Question 1d

Marks: 2

d)

Use the general Maclaurin series formula to show that the general term of the Maclaurin series for

e^{2 x}

is

$\frac{{(2 x)}^{n}}{n!}$

Key Concepts

Question 2a

Marks: 3

a)

Use substitution into the Maclaurin series for

\sin x

$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots$

to find the first four terms of the Maclaurin series for

\sin (\frac{x}{2})

.

Key Concepts

Question 2b

Marks: 3

b)

Hence approximate the value of

\sin \frac{π}{2}

and compare this approximation to the exact value.

Key Concepts

Question 2c

Marks: 2

c)

Without performing any additional calculations, explain whether the answer to part (a) would be expected to give an approximation of

\sin \frac{π}{4}

that is more accurate or less accurate than its approximation for

\sin \frac{π}{2}

.

Key Concepts

Question 3a

Marks: 4

The Maclaurin series for $e^{x}$ and $\sin x$ are

$e^{x} = 1 + x + \frac{x^{2}}{2!} + \dots and \sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots$

a)

Find the Maclaurin series for

e^{x} s i n x

up to and including the term in

x^{4}

.

Key Concepts

Differentiating & Integrating Maclaurin Series

Question 3b

Marks: 3

b)

Use the Maclaurin series for

\sin x

, along with the fact that

\frac{d}{d x} (\sin x) = \cos x

, to find the first four terms of the Maclaurin series for

\cos x

.

Key Concepts

Question 4a

Marks: 4

a)

Use the general Maclaurin series formula to find the first four terms of the Maclaurin series for

\frac{1}{1 + x}

.

Key Concepts

Question 4b

Marks: 3

b)

Confirm that the answer to part (a) matches the first four terms of the binomial theorem expansion of

​ \frac{1}{1 + x}

.

Key Concepts

Differentiating & Integrating Maclaurin Series

Question 4c

Marks: 2

The Maclaurin series for $\ln (1 + x)$ is

$\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \dots$

c)

Differentiate the Maclaurin series for

\ln (1 + x)

up to its fourth term and compare this to the answer from part (a). Give an explanation for any similarities that are found.

Key Concepts

Question 5a

Marks: 3

a)

Use the Maclaurin series for

\sin x

and

\cos x

to find a Maclaurin series approximation for

2 \sin x \cos x

up until the term in

x^{4} .

Key Concepts

Question 5b

Marks: 3

The double angle identity for sine tells us that

$\sin 2 x = 2 \sin x \cos x$

b)

Use substitution into the Maclaurin series for

\sin x

to find a Maclaurin series approximation for

\sin 2 x

up until the term in

x^{4}

, and confirm that this matches the answer to part (a).

Key Concepts

Question 6a

Marks: 4

a)

Use the Binomial theorem to find a Maclaurin series for the function

f

defined by

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

Give the series up to and including the term in

x^{6}

.

Key Concepts

Question 6b

Marks: 2

b)

State any limitations on the validity of the series expansion found in part (a).

Key Concepts

Question 6c

Marks: 4

c)

Use the answer to part (a) to estimate the value of

\sqrt{0.5}

, and compare the accuracy of that estimated value to the actual value of

\sqrt{0.5}

.

Key Concepts

Maclaurin Series from Differential Equations
Implicit Differentiation

Question 7a

Marks: 4

Consider the differential equation

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

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

a)

(i)

Show that

y'' = 4 y y^{'} + 1

.

(ii)

Use an equivalent method to find expressions for

y^{'''}

,

y^{(4)}

and

y^{(5)}

. Each should be given in terms of

y

and of lower-order derivatives of

y

.

Key Concepts

Question 7b

Marks: 3

b)

Using the boundary condition above, calculate the values of

y^{'} (0)

,

y^{''} (0)

,

y^{'''} (0)

,

y^{(4)} (0)

and

y^{(5)} (0)

.

Key Concepts

Maclaurin Series from Differential Equations

Question 7c

Marks: 4

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

c)

Using the answers to part (b), find the first six terms of the Maclaurin series for

f (x)

.

Key Concepts

Maclaurin Series from Differential Equations

Question 7d

Marks: 2

d)

Hence approximate the value of to 4 d.p. when

x = 0.1

.

Key Concepts

Maclaurin Series from Differential Equations
Implicit Differentiation

Question 8a

Marks: 5

Consider the differential equation

$y^{'} = 2 x y^{2}$

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

a)

(i)

Find

y^{''}

.

(ii)

Hence show that

y^{'''} = 8 y y^{'} + 4 x {(y^{'})}^{2} + 4 x y y^{''}

and

y^{(4)} = 12 {(y^{'})}^{2} + 12 y y^{''} + 12 x y^{'} y^{''} + 4 x y y^{'''}

Key Concepts

Question 8b

Marks: 4

b)

Use the results from part (a) along with the given initial condition to find a Maclaurin series to approximate the solution of the differential equation, giving the approximation up to the term in

x^{4}

.

Key Concepts

Separation of Variables

Question 8c

Marks: 4

c)

Use separation of variables to show that the exact solution of the differential equation with the given initial condition is

$y = \frac{1}{1 - x^{2}}$

Key Concepts

Separation of Variables

Question 8d

Marks: 3

d)

Use the binomial theorem to find an approximation for

​ \frac{1}{1 - x^{2}}

up to the term in

x^{4}

, and verify that it matches the answer to part (b).

Key Concepts