Download Mathematics B (Elec Eng): revision exercises 1

Mathematics B (Elec Eng): revision exercises 1
1
1
3 1/3 , (iii)
(1 − x )
(8 − x3 )1/3
2. Find the Maclaurin series of (i) cos x sin 2x, (ii) x ln(1 + x3 ), (iii) sec x, as far as the x4
term
1. Expand (i) (2x − y)4 , (ii)
3. Use L’Hopital’s rule to find the limits
ex−1 − 1
x→1
ln x
8x2
x→0 cos x − 1
lim
1 − sin θ
θ→π/2 1 + cos 2θ
lim
lim
lim
x→π
sin2 x
1 + cos x
4. Find the exact values of (i) sinh(ln 3), (ii) cosh(− ln 2), (iii) tanh(2 ln 5).
5. If sinh x = −2 find the values of cosh x, tanh x, sech x, cosech x and coth x.
6. Show that cosh 21 x =
q
1
2 (1
+ cosh x).
7. Evaluate
Z
(i)
√
dx
,
1 + 2x2
Z
(ii)
p
dx
,
(x − 1)(x + 9)
Z e
(iii)
1
dx
x 1 + (ln x)2
p
Answers:
1. (i) (2x − y)4 = 16x4 − 32x3 y + 24x2 y 2 − 8xy 3 + y 4
1
9
(ii)
= 1 + 13 x3 + 29 x6 + 14
81 x + · · ·
(1 − x3 )1/3
1
1 3
1 6
7
(iii)
= 12 + 48
x + 576
x + 41472
x9 + · · ·.
(8 − x3 )1/3
5
2. (i) cos x sin 2x = 2x − 73 x3 + 61
60 x + · · ·
1 7
1 10
3
4
(ii) x ln(1 + x ) = x − 2 x + 3 x + · · ·
5 4
(iii) sec x = 1 + 21 x2 + 24
x + ···
1
3. 1, −16, 4 , 2
4. (i) 43 , (ii) 54 , (iii) 312
313 .
√
√
√
√
5. cosh x = 5, tanh
x = −2/ 5, sech x = 1/ 5, cosech x = −1/2, coth x = − 5/2.
√
7. (i) √12 sinh−1 ( 2x), (ii) cosh−1 ((x + 4)/5), (iii) sinh−1 (1).
Potentially useful information from formulae booklet
n(n − 1) 2 n(n − 1)(n − 2) 3
x +
x + ···
2!
3!
(finite series if n is a positive integer or zero. If not, infinite series convergent when |x| < 1).
(1 + x)n = 1 + nx +
3
2
ex = 1 + x + x2! + x3! + · · ·
2
3
ln(1 + x) = x − x2 + x3 − · · ·
5
3
sin x = x − x3! + x5! − · · ·
2
4
cos x = 1 − x2! + x4! − · · ·
arctan x = tan−1 x = x −
d
x
sinh−1
dx
a
−1<x≤1
x3
3
+
x5
5
− ···
−1≤x≤1
1
d
x
1
d
x
a
,
cosh−1
=√
,
tanh−1
= 2
2
2
2
2
dx
a
dx
a
a
−
x2
a +x
x −a
Note: the formulae booklet may use the notation arcsinh instead of sinh−1 ; similarly for
the other trigonometric and hyperbolic functions.
=√