PracticeExam3-soln

PracticeExam3-soln - Math 105 Practice Exam 3 Solutions...

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Math 105 Practice Exam 3 Solutions 1. (a) Evaluate lim ( x,y ) (1 , - 1) sin( x 2 + y ) x 2 + y or show that it doesn’t exist. We can use the substitution u = x 2 + y , so u 1 2 - 1 = 0: lim ( x,y ) (1 , - 1) sin( x 2 + y ) x 2 + y = lim u 0 sin( u ) u = lim u 0 sin( u ) - sin(0) u - 0 = d dt sin( t ) ± ± ± t =0 = cos(0) = 1 . Here we used the definition of derivative, f 0 ( a ) = lim x a f ( x ) - f ( a ) x - a , in reverse. (b) Consider the area function A ( x ) = R x 1 f ( t ) dt , with A (2) = 6 and A (3) = 5 . Compute Z 2 3 f ( t ) dt . By the Fundamental Theorem of Calculus, Z 2 3 f ( t ) dt = A (2) - A (3) = 6 - 5 = 1 . (c) A self-employed software engineer estimates that her annual income over the next 10 years will steadily increase according to the formula 70 , 000 e 0 . 1 t , where t is the time in years. She decides to save 10% of her income in an account paying 6% annual interest, compounded continuously. Treating the savings as a continuous income stream over a 10-year period, find the present value. PV = Z 10 0 7000 e 0 . 1 t e - 0 . 06 t dt = 7000 Z 10 0 e 0 . 04 t dt = 7000 · 1 0 . 04 e 0 . 04 t ± ± ± 10 0 = 175000( e 0 . 4 - 1) ( 86 , 000) (d) Draw the level curves of the graph of f ( x,y ) = 2 x 2 + y 2 at the heights 0 , 1 , 2 . For 1 and 2 it’s an ellipse, for 0 it’s just the point (0
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PracticeExam3-soln - Math 105 Practice Exam 3 Solutions...

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