Revision Questions - Department of Earth Sciences

Lara Kalnins
Mathematics for Materials and Earth Sciences
Revision Questions, Hilary Term 2010
I. Integrate the following:
1. Ù H7 x - 1L12 â x
3
2
x +2x +x+1
2. Ù
âx
x2 + 2 x
1
3. Ù
2 âx
1 + tan x
4. Ù sin x sin 3 x â x
5. Ù x4 ln x â x
1
6. à
âx
"############################
#
4 x2 + x + 1
7. Ù Isin3 x cos x + 4 sin2 x cos x - 7 sin x cos xM â x
8. Ù sin x ã2 x â x
1
9. Ù 1
âx
+ sin x
1
10. à !!!!!!!!!!!!!!!!

âx
x!
1+ã
"##################
#
a2 - x2
11. à
âx
x4
2
3x+6x
12. à
2
2 âx
Hx + 2L Hx - 1L
13. Ù ã x + 3 H2 x + 1L â x
14. Ù sin 3 x cos 2 x â x
3
x
15. Ù
âx
x2 + 3 x + 2
1
16. Ù 3
âx
x2 - 4 x + 6
17. Ù t2 sin t â t
1
18. à
ây
y
"################
#
y3 - 1
2
3 x - 8 x + 13
19. Ù
âx
x3 + x2 - 5 x + 3
Π2
20. Ù0 sin5 Θ cos5 Θ â Θ
21. Ù lnIx4 M â x
22. Ù 3 sin2 x cos x â x
2
x
23. à
âx
"###############
9 -x2
1
24. à
!!!!!!!!!!!!!! â x
Hx + 1L
x+6
25. Ù sin 4 x sin 2 x â x
26. Ù cos H3 x L ã x + 3 â x
4
27. Ù
âx
21. Ù lnIx4 M â x
2
Ù 3 sin x cos
222.
MMES_HTRev.nb
xâx
2
x
23. à
âx
"###############
9 -x2
1
24. à
!!!!!!!!!!!!!! â x
Hx + 1L
x+6
25. Ù sin 4 x sin 2 x â x
26. Ù cos H3 x L ã x + 3 â x
4
27. Ù 4
âx
cos x - 3 sin x
II. Divide the following polynomials:
3
2
3
2
4
3
x - 5 x + 10 x - 3
1. 3

3x+1
x - 9 x + 15
2. 2

2 x -5
x +3x +2x+1
3. 4

x2 + x + 2
III. Reduction Formulae
1. Find a reduction formula for
1
In = Ù0 xn ãx â x
2. Use the reduction formula from 1 to evaluate
1
1 6 x
Ù0 5 x ã â x
3. Find a reduction formula for
In = Ù tann x â x
4. Use the reduction formula from 3 to evaluate
3
Ù tan x â x
IV. Applications of Integration
1. Calculate the area between the curves y = x2 - 6 x + 8 and y = 2 x - 7
2. Calculate the area enclosed by y2 = x - 1and y = x - 3
3. Find the coordinates of the centre of mass of the region between y =
!!!!!
4. Compute the average value of y = x cos(x2 + ΠM on [0, Π ]
2
!!!!!!!!!!!!!!!
x + 4 and x = 0 on the interval 0 £ x £ 4
3

5. Find the arc length of x = 3 Hy - 1L 2 on [1, 4]
6. Find the arc length of y = H2 x + 5L32 + 3 on [0, 3]
-1
7. Find the area enclosed by the curves y = 3, y = x2 - 2 x + 4, and y =
x+5
2
8. Find the area enclosed by y = sin x and the x-axis on the interval [0, 2Π ]
9. Find the moment of inertia of a thin rod of mass M and length L rotated about its centre and perpendicular to its length. Then
find the moment of inertia if it is instead rotated about its end and perpendicular to its length.
10. Find the moment of inertia of a thin rectangular sheet of mass M and dimensions a and b rotated about a perpendicular axis
through its centre.
11. Find the centroid of a square pyramid of height h.
12. Using the result in 10, find the moment of inertia of a square pyramid of mass M, height h, and bottom side length s.
V. Complex Numbers
MMES_HTRev.nb 3
V. Complex Numbers
a
For each a and b find: 2 a, a + b, a* - b, a b, b , |a|, arg(b)
1. a = 2 + 3 ä, b = 1 + ä
!!!!!
2. a = -1 + 2 ä, b = 1 + 3 ä
3. a = 3 + 4 ä, b = 2 - 2 ä
4. a = 2 - ä, b = -3 + ä
Use de Moivre’s theorem to write the following as products of sinΘ and cosΘ
5. cos4Θ
6. sin3Θ
Use trigonometric formulae (see pg. 8 of the lecture notes) to reduce the following
1
1
Example: sin x cos3 x = 8 sin 4 x + 4 sin 2 x
7. sin2 x cos3 x
8. sin2 x sin2 2 x
Find all z such that:
9. z7 = 1
10. z3 = 27
11. z4 = 2 + 2 ä
!!!!!
12. z2 = 3 - ä
Show that:
ix
ie
1
13.
= 2

2ix
sin x
e
-1
iIe4 i x + 1M
14.
= cot 2 x
4ix
e
-1
1
15. sec 3 x Hcos 2 x - i sin 2 xL =

2ix
e
cos 3 x
Write the following so that you could evaluate it on a calculator with no complex number functions:
16. sinH3 + 2 äL
17. coshH7 - äL
Evaluate:
18. lnH2 + äL
19. lnH6 - 2 äL
20. lnH-13 - 13 äL
Evaluate:
21. H1 + 3 äL2 ä
22. H-6L1-ä
!!!!!!
3
23. 1
Integrate:
24. Ù I2 sin x + 3 ä x2 M â x
Π2
25. Ù-Π2 H2 ä cos x + cos 3 xL â x
Evaluate:
21. H1 + 3 äL2 ä
22. H-6L1-ä
4 MMES_HTRev.nb
!!!!!!
23. 1 3
Integrate:
24. Ù I2 sin x + 3 ä x2 M â x
Π2
25. Ù-Π2 H2 ä cos x + cos 3 xL â x
VI. Series and Limits
1. Find the first four terms of Taylor series for
!!!!!!!!!!!!!
1 + x about x = 0
sin x
2. Find the first five terms of the Taylor series for
about x = 0
x
3. Find the Taylor series up to and including OIx2 M for x2 lnHxL about x = ã
x sinx
4. Find the Taylor series up to and including order 4 of
2
1-x cosx
5. Find the first six terms of the Taylor series for x5 about x = 1
6. Repeat question 5 using the binomial theorem
7. Use the binomial theorem to expand H6 + xL4
1
8. Use the binomial theorem to find the Taylor series up to and including OIx3 M of 2 about x = 0
4+x
9. Use the binomial theorem to find the Taylor series up to and including OIx3 M of H1 + 2 xL32 about x = 0
Do the following converge?
n!
10. Ú¥
n=1 nn
n!
11. Ú¥
n=0 n
1000
2
-n
12. Ú¥
n=1 ã
2 H-1Ln xn+1
13. Ú¥

n=0
H2 nL!
n
14. Ú¥
n=0 H-2 xL
Evaluate the following:
4x+
"######################
#
1 + 4 x2
15. limx®¥

x
sin 2 t
16. limt®0

2
t +4t
cos 3 x -1
17. limx®0

2
5x
18.
x2 cos x
limx®0

x sin x
19. limx®-¥ x ã2 x
Use the leading term method to evaluate the following:
1 - ãx
20. limx®0
sin x
1 - ãx
21. limx®0+

2
1 - cos x
sin 3 x
22. limx®0
2
sin 2 x cos x
Ix2 + 3 xM cos 3 x
23. limx®0
2
sin x cos 2 x
24. Use Taylor series to show that tan x can be approximated by x for small values of x
9
25. Use a second order Taylor series to solve cos Θ = 10
for Θ algebraically. Compare your answer with the answer a calculator
1
gives. How well would this method work for cos Θ = 4 ?
1 - cos x
sin 3 x
22. limx®0
2
sin 2 x cos x
Ix2 + 3 xM cos 3 x
MMES_HTRev.nb 5
23. limx®0
2
sin x cos 2 x
24. Use Taylor series to show that tan x can be approximated by x for small values of x
9
25. Use a second order Taylor series to solve cos Θ = 10
for Θ algebraically. Compare your answer with the answer a calculator
1
gives. How well would this method work for cos Θ = 4 ?
VII. Solve the following differential equations:
1
1. y’ + 2 y = 2 + t
2. y’ + 2 t y = 2 t ã-t
3. y’ = x2 y
2
4. y dx + H2 x - y ãy L dy = 0
5. y’’ - 2 y’ + y = ãx ln x
6. y’’ + 6 y’ + 13 y = 0
7. y’’ - 2 y’ - 3 y = -3 t ã-t
8. y’’ + y’ - 2 y = 2 t, yH0L = 0, y’ H0L = 1
9. y’’ + y = 2 sin x
x
10. dx + J y - sin yN dy = 0
11. y’’ + 9 y = t2 ã3 t + 6
12. 2 x + 3 + H2 y - 2L y’ = 0
13. y’’ + y = 3 sin 2 t + t cos 2 t
14. Hx + 2L sin y dx + x cos y dy = 0
15. I3 x2 - 2 x y + 2M dx + I6 y2 - x2 + 3M dy = 0
2 cos 2 x
16. y’ =
3+2y
1
17. y’ + t y = 3 cos 2 t, t > 0
18. y’’ + 2 y’ + 5 y = 4 ã-t cos 2 t, y(0) = 1, y’(0) = 0
ãt
19. y’’ - 2 y’ + y =

2
t +1
20. ãx dx + Hãx cot y + 2 y csc yL dy = 0
21. y’’ - 4 y’ + 3 y = ã3 x + x + 3
22. 2 y’’ + 18 y = 6 tan 3 x
23. 2 y’’ - 3 y’ + y = 0
24. y’ + y2 sin x = 0
x2
25. y’ = 2
26.
1+y
Hy ãx y cos 2 x - 2 ãx y
sin 2 x + 2 xL dx + Hx ãx y cos 2 x - 3L dy = 0
1
27. t y’ + 2 y = t2 - t + 1, yH1L = 2 , t > 0
y
28. I x + 6 xM dx + Hln x - 2L dy = 0
29. y’’ + 2 y’ = 3 + 4 sin 2 t
30. Hx + 2L sin y dx + x cos y dy = 0
31. y’’ + 2 y’ + 5 y = 3 sin 2 t
32. t3 y’ + 4 t2 y = ã-t , yH-1L = 0
33. I3 x2 y + 2 x y + y3 M dx + Ix2 + y2 M dy = 0
x2
34. y’ = 3
yI1 + x M
35. u’’ + Ωo 2 u = cos Ωo t
36. y’’ - 6 y’ + 9 y = 0
37. t y’ + 2 y = sin t, yHΠ 2L = 1
31. y’’ + 2 y’ + 5 y = 3 sin 2 t
32. t3 y’ + 4 t2 y = ã-t , yH-1L = 0
6 MMES_HTRev.nb
33. I3 x2 y + 2 x y + y3 M dx + Ix2
+ y2 M dy = 0
x2
34. y’ = 3
yI1 + x M
35. u’’ + Ωo 2 u = cos Ωo t
36. y’’ - 6 y’ + 9 y = 0
37. t y’ + 2 y = sin t, yHΠ 2L = 1
VIII. Substitutions
dy
x2 + 3 y2
1. Solve dx
=
using the substitution y = v x
2xy
dy
2. Transform y’’ HxL + y y’ HxL = 0 into a first order differential equation using the substitution v = dx
¹ 0

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Revision Questions - Department of Earth Sciences

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