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Unformatted text preview: Funding provided by MATH 138
Spring Term 2001
Final Examination L n (
( 61 a b (c )
)
) (d) 2. [12]
(a)
(b)
(C) State what is meant by a general ﬁrst order linear differential equation.
State what is meant by a general ﬁrst order separable differential equation. In each of the following cases, determine whether the differential equations are separable, linear, both or
neither (do NOT attempt to solve). i (x 1)2%=(—1)/(y+ 1)
ii. (1—c”)—1: (cos mpg—(eh
iii. (1—sinm)%:z3 y+x3
iv. 631% = myey + (sin 11:)e‘y Solve the initial value problem
12% —— y: m3 sin :c
y(27r) :0 Let {an} be a sequence. Deﬁne what is meant by the statement ’{an} is increasing’.
State (but do not prove) a theorem giving conditions under which an increasing sequence will converge. The sequence {an} is deﬁned by (11 = ﬂ, an+1 : 1/2 +117, (71 2 1). Using mathematical induction (or
otherwise), prove that {an} is increasing, and that (1,, < 2 for all n 2 1. Hence deduce its limit. 3.[12] Determine with justiﬁcation, which of the following series are convergent or diverge. (a ( ( b (c d (e 4[.12] Consider the series Zn_1(— 1)”
(a
( b) Is the series absolutely convergent or conditionally convergent? Brieﬂy justify. )
)
)
)
) ) 1
2:0: 1n2+1 1
$312,111 2n+1 Prove that the series is convergent. (c) How many terms of the series are required to approximate the sum of the series, with an error which is 5. [14]
(a) less than 0.01? i. Find the second degree Taylor polynomial, T2(:(:), at the origin for f(x) = (1 + m)‘1/3.
ii. Show that f(a:)— Tzzr( :1:_) > 1—14123 ,1‘2 0. iii. Use part (ii) to estimate 01/821 W. Show that the error in the estimate is less than 1/5000. (b) If ﬁx): T12. ﬁnd f(5 (0) 6. [10] Find the radius of convergence and the interval of convergence of the series 22:0 —3)"$"
x/n+1 '
1 7. [12] The position of a particle at time t is given by
1 .
a: :2cost,y: §s1nt,()g t S 211' (a) Sketch the path followed by the particle and indicate with an arrow the direction of motion of the particle
as t increases. (b) Calculate the velocity vector of the particle at time t, and ﬁnd the positions of the particle when its
velocity vector is horizontal. (c) Obtain an expression (but do no evaluate) for the distance travelled by the particle from t = 0 to
t = 7r/6. (d) Find the equation of the tangent line to this curve when t = 7r/ 6. 8. [12] Consider the function
f(x, y) = «9—7—7.
(a) Sketch the domain of f.
(b) Determine the range of f.
(c) Sketch a contour map (i.e., level curves) and a graph of f.
(d) Calculate fm(1,2) and fy(l,2).
(e) Find the linearization of f at (1, 2). ...
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 Calculus

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