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Winter2003 Midterm

# Winter2003 Midterm - (Math 138 —(Winter 2003...

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Unformatted text preview: (Math 138) — (Winter 2003) —— (Midterm)Page 1 0f (2) REPRINTED BY MATHSOC Dd Instructors: Winkler, Kempf, Sphigel, Cutler, Struthers, Lam, Harmsworth [5] 1. (a) Show that every member of the family of functions 3/ = Cerf/2, C e R is a solution of the DE. (differential equation) d_y_ d2: _ —xy (b) On the axes provided sketch the graph of the particular solutions which satisfy 1. 31(0) : 1 ii. y(\/ln 4): —1 [7] 2. Consider the following D.E. (a) Write done the general form of a ﬁrst order linear D.E. Is the DB. in (1) linear? (b) Write down the general form a ﬁrst order separable D.E. Is the DE. in (1) separable? (c) Show that the subsitution u : % transforms the DE. (1) into a ﬁrst order linear D.E. (Do not solve the DE.) [8] 3. Find all solutions of the following differential equations. Express the solutions in the form y : f (as) (a) d str:—y—+2y:cos:137 1:750 da: (b) [10] 4. (a) Does the improper integral converge or diverge? Justify your answer. (b) Show that the improper integral converges by its values. (c) Let 0 a: < 0 f(-T) : { 112:)2 (13 Z 0 Explain why f (1') is a probability density function. (d) Show that the mean of this probability density function is [r = 2 ln 2. :4 (Math 138) — (Winter 2003) — (Midterm)Page 2 of (2) REPRINTED BY MATHSOC |><1 [5] 5. (a) Give the precise statement of L’Hopitalis Rule. (b) Determine whether the following converge or diverge. If the sequence converges ﬁnd the limt. Justify your answer. _ ‘ _ T] 1. an _ (_3),, ii. an : (1+ g)” iii. an : sin ((n +%)1r) (a) Let an be a sequence deﬁned recursively by (11 : 1 and an“ :Mle Use a proof by induction to show that {an} is increasing and on < 5 for all n. Explain why {an} has a limit L and then compute L. (b) If lim an 2 0 and lim bn : ~00 then lim (1an : 0. n—ioo n—>OO 71—)00 Is this statement true or false? If true then prove it, if false than give a counterexample. [7] OO 6. (a) Deﬁne what is meant by the statement: The series 2 an is convergent. n21 (b) State and prove the Comparison Theorem for Series. [9] 7. (a) Determine, withjustiﬁcation, wether each of the following series is convergent or divergent. If convergent, state with justiﬁcation whether the convergence is conditional or absolute. 11:1 .. oo _5 n “.2 2+6) 71:1 00 (*2)” 111.; n! 10 (b) i. Show that the series converges. Determine the number of terms required in order that s — 5n 3 0.001 where s is the sum of the series and 5n is the sum of the ﬁrst n terms. ii. Show that the series i(“1)n x/ln n n 7122 is convergent. If the ﬁrst 25 terms are used to approximate the sum of the series, give an upper bound on the absolute value of the error of the estimate. ...
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