C UNIVERSITY OF SURREY

[Pages:3]MAT1030/3/ Semr1 11/12 (0 handouts)

c

UNIVERSITY OF SURREY

Faculty of Engineering and Physical Sciences Department of Mathematics

Undergraduate Programmes in Mathematical Studies

Module MAT1030 -- 15 Credits

Calculus

Level 1 Examination Time allowed: Two hours

Semester 1, 2011/12

Answer ALL questions. All working must be shown. Please note that some questions carry more marks than others.

Where appropriate the mark carried by an individual part of a question is indicated in square brackets [ ].

Approved calculators are allowed. Additional material: None

c Please note that this exam paper is copyright of the University of Surrey and may not be reproduced, republished or redistributed without written permission.

MAT1030/3/ Semr1 11/12 (0 handouts)

?2?

Question 1

(a) Write down the definitions of sinh x and cosh x in terms of exponentials. From these

definitions show that

(i) ex = cosh x + sinh x, (ii) cosh2 x - sinh2 x = 1, (iii) sinh 2x = 2 sinh x cosh x.

[6]

(b)

Given

that sinh x =

12 5

,

find

cosh x,

sinh 2x

and

ex.

[4]

Question 2

eix - e-ix

eix + e-ix

Assuming that sin x =

and cos x =

where i = -1, show that

2i

2

sin4

x

=

3 8

-

1 2

cos 2x

+

1 8

cos 4x

[7]

Question 3 Assuming that the Maclaurin expansion for cos x is

x2 x4 x6 cos x = 1 - + - + ? ? ? ,

2! 4! 6!

deduce expansions, with the coefficients simplified as far as possible, for (i) cos 2x, (ii) cos 4x

and (iii) sin4 x, in each case up to and including the x6 term. For part (iii) you may use the

result of Question 2.

[9]

Question 4 Evaluate the following limits using L'Hopital's rule, with a substitution where necessary:

x2 - 49 (a) lim

cos 5x - cos 3x (b) lim

(c) lim x - x2 ln 1 + x

[10]

x7 x - 7

x0

x2

x

x

Question 5 Differentiate each of the following, simplifying your answers

x3 (a)

(b) cos 3x2

(c) x sin-1 x

[7]

1 + x2

Question 6

Find the sum to infinity of the geometric series xn+1.

n=1

By differentiating your answer show that, for -1 < x < 1,

(n

+

1)xn

=

2x - x2 (1 - x)2

n=1

(n + 1)

and hence evaluate

.

[7]

2n

n=1

Question 7 Evaluate the integrals

x

3x + 7

(a) x cos x dx

(b)

dx

(c)

dx

[11]

4 - x2

x2 + 3x - 4

MAT1030/3/ Semr1 11/12 (0 handouts)

?3?

Question 8 By writing the integrand as cosn-1 x cos x and then integrating by parts show that, if n is

an integer with n 2, then

/2

cosn x dx

=

n

-

1

/2

cosn-2 x dx.

[5]

0

n0

/2

Find

cos9 x dx.

0

[4]

Question 9 Solve the following differential equations, expressing the solution in the form y = f (x):

(i) dy = -2y3x dx

[3]

dy y2 + 1

(ii) =

subject to y(0) = 2

dx y

[4]

dy

(iii) (x + 1) + 4y = 1 dx

[4]

Question 10 Solve the differential equations:

d2y

dy

(i)

dx2

-

10 dx

+

26y

=

0

[4]

(ii)

d2y dx2

-

dy 4

dx

+

3y

=

2e-4x

subject

to

y(0)

=

1

and

y

(0)

=

2

[8]

Question 11 1

Show that the transformation v = transforms the differential equation y2

dy + y = 3exy3 dx

into

dv - 2v = -6ex.

[4]

dx

Hence solve the original differential equation, expressing the answer in the form y = f (x). [3]

END OF PAPER

INTERNAL EXAMINER: S.A. Gourley

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