1337Exam2solutions

1337Exam2solutions - . . sin (.irx) 9 1- llms+l ln 3 z -8...

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Math 1337 Section 5 T. Hagstrom sbIdi;dt% Fall 2008 Exam 2 There are 4 problems each worth 25 points plus an extra credit problem. Show all your work. Good luck! 1. Compute the derivative of the following functions: 9 i. w (x) = cos-I (+) sin t 8 ii. g(t) = 8 iii. f(z) = z2 in. (1 + z) - 1 )+- l - cl+ ra) - a? i. + - (1tx")Y-i5F
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2. Let f (x) = In (T + 2tan-' x). . - a. Find the domain and range of f. b. Compute ff(x). 8 c. Sketch the graph of f.
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3. Bismuth-210 has a half-life of 5 days. 7 i. Assuming an initial sample of 2 grams, write down a formula for the mass, A(t), re- maining after t days. ii. After how many days will there be less than 1 milligram? (A milligram is one one thousandth of a gram.) iii. Compute the relative growth rate, #.
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4. Compute the following limits:
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Unformatted text preview: . . sin (.irx) 9 1- llms+l ln 3 z -8 9s i. 1imZ+2- cos (2) In t 8 ii. limt+m 5. Extra Credit To solve calculus problems using computers it is common to replace derivatives by approx- . - imate differentiation formulas called difference formulas. One such formula which is often used is the following "second order backwards differentiation" (BDF2) formula: i. Use llHospital's rule to prove that if f is differentiable at x then the BDF2 formula \ O approximates f l ( x ) in the sense that: fr(x) = lim 3 f (x) - 4f (x - h) + f (x - 2h) 2h h+O \o ii. Now use 1'Hospital's rule to prove that if f is thrice differentiable at x then the BDF2 formula approximates f l ( x ) to second order in h in the sense that: 3 f ( x ) - 4 f ( x - h ) + f (x-2h) lim 2h h+O h2...
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1337Exam2solutions - . . sin (.irx) 9 1- llms+l ln 3 z -8...

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