M1014FE2009R

# D 1 point will r 4 provide an upper or lower bound

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d) [1 point] Will R 4 provide an upper or lower bound for ln 2? 6. Evaluate the following definite integrals. [5 points each] a) dx x x + + 2 1 2 2 3 1 b) θ θ θ π d 2 / 0 cos 7. [10 points] Consider the curve y = 1/x 2 , for x > 0. Consider the areas under the curve for 0 < x < 1 and x > 1. In each case determine whether the area is finite or infinite and if finite, determine the area. 8. Suppose that the probability density for the failure of a cell phone at time t after purchase is f(t) = 0.2e -t/5 per year (where t is in years). Note that all phones will fail eventually ( 1 2 . 0 0 5 / = - dt e t ) a) [5 points] What is the probability that it will fail in the first 3 years after purchase. b) [5 points] What is the probability that it will still be working after 10 years (without any repairs). 2

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EXPLAIN your analyses, i.e. we want a few words, not just the answers. 9. a) [7 points] Solve the differential equation: xy′ = y + x 2 sin x, x > 0 if y(π) = 0. b) [3 points] Determine the values y(π/2), y(3π/2) and y(2π) 10. [10 points] A fish population has been modelled by the differential equation: m P dt dP - = 1 . 0 where t is measured in years and m > 0 represents the constant rate of harvesting. The initial fish population in the lake is P 0 = 20,000. Find the rate of harvesting (number per year) that will decrease the fish population by 50% in 10 years. 11. For each of the following, determine whether the series is absolutely convergent, conditionally convergent, or divergent. Explain each case and indicate the test you have used. [4 points for parts (a); 3 points for each of parts (b) and (c)] (a) n n n 1 ) 1 ( 1 1 = + - (b) ! 2 ) 1 ( 2 1 n n n n n = - (c) = + 1 ) 3 1 ( n n n 12. (a) [4 points] Find the interval of convergence of the power series: ) ln( 4 ) 1 ( 2 n x n n n n = - (b) [6 points] Find a power series representation for f(x) = ln(1+x) and determine its radius of convergence. 3
• Fall '09
• ganong
• Calculus, Professor Taylor, Professor Guiasu, Professor Poliakov

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