sol for exam 2 take home

# sol for exam 2 take home - Math 116 CALCULUS II SOLUTION...

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Math 116 CALCULUS II SOLUTION FOR MIDTERM EXAM – II (Take-home; 11/14) November 19 (Tue), 2013 Instructor: Yasuyuki Kachi Line #: 23590. [I] (Take-home; 10pts) (a) f ( x ) = m s n =0 f ( n ) ( 0 ) n ! x n + i x t =0 f ( m +1) p x - t P m ! t m dt. (b) If lim m -→ i x t =0 f ( m +1) p x - t P m ! t m dt = 0 , then f ( x ) = s n =0 f ( n ) ( 0 ) n ! x n . This is identical to “Review of Lectures – XX”, page 10, Exercise. [II] (Take-home; 10pts) Madam Rich has recently received a letter from the President and CEO of The University X, her alma mater. It goes: Dear Madam Rich, We are so pround of you as being the most prominent alumna of The University X. Could you please help us by donating us some money. We will proudly name our new football stadium “Madam Rich Stadium”. Please agree to sign the enclosed sheet. Sincerely yours, *** , President & CEO of The University X 1

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a Enclosed sheet A I, Madam Rich, hereby agree to consecutively pay to The University X, exactly as dictated by the payment schedule below. I agree that, on each payment, the amount will be automatically deducted from my bank account, with account number xxx, and it will be automatically transferred to The University X account. Payment Schedule A. The amount of payment on the payment # n is exactly one n -th of a dollar p = \$ 1 n P . ( See Table 1. ) B. As for the timing of payments, the interval between two consecutive payments is exactly one half of the previous interval. ( See Table 2. ) C. The payment will conclude 48 hours after the payment # 1, but not before. payment # v v v v amount #1 v v v v v v v v \$ 1 #2 v v v v v v v v \$ 1 / 2 #3 v v v v v v v v \$ 1 / 3 #4 v v v v v v v v \$ 1 / 4 #5 v v v v v v v v \$ 1 / 5 #6 v v v v v v v v \$ 1 / 6 #7 v v v v v v v v \$ 1 / 7 . . . . . . Table 1. 2
payment # v v v v timing of payment #1 v v v v v v v v −− #2 v v v v v v v v 24 hours after payment #1 #3 v v v v v v v v 24/2 hours after payment #2 #4 v v v v v v v v 24/4 hours after payment #3 #5 v v v v v v v v 24/8 hours after payment #4 #6 v v v v v v v v 24/16 hours after payment #5 #7 v v v v v v v v 24/32 hours after payment #6 . . . . . . Table 2. Suppose you are Madam Rich’s fnancial advisor. Your job is not to let her lose all her money. Should you tell her that it is okay to sign? Explain. b Answer B : Observe the Football series 24 2 0 + 24 2 1 + 24 2 2 + 24 2 3 + 24 2 4 + 24 2 5 + 24 2 6 + ··· = 48.

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