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finalsol - Final Exam Solutions 1. Find the derivative of f...

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Final Exam Solutions 1 . Find the derivative of f ( x )= x tan x . Using a b = e b ln a , this function is f ( x e (tan x )ln x .So f ± ( x e (tan x x ± (sec 2 x x + (tan x ) 1 x ² = x tan x ± (sec 2 x x + tan x x ² . (Here, we used the chain rule: d dx e u = e u du dx , where u = (tan x x has derivative du dx = (sec 2 x x + (tan x ) 1 x by the product rule.) 2 . Suppose we initially had 10 milligrams of Zirconium-95. How much will we have left in 100 days, assuming that it decays at a rate proportional to the amount present, with a half-life of 64 days? If y is the amount of the isotope, we have dy dx = ky where k is the unknown decay constant. This solves to give y = y 0 e kt . The initial amount is 10 milligrams, so y (0) = 10 = y 0 e k · 0 = y 0 .S o y =10 e . The half-life is 64 days, so y (64) = 5 or 10 e 64 k = 5. We solve e 64 k =0 . 5 to obtain 64 k =ln0 . 5or k = (ln0 . 5) / 64 = - 0 . 0108. Therefore y e - 0 . 0108 t and y (100) = 10 e - 1 . 08 =3 . 38 milligrams. 3 . Find the limit: lim t 0 1 - cos t t 2 . Use L’Hospital’s rule twice: lim t 0 1 - cos t t 2 = lim t 0 sin t 2 t = lim t 0 cos t 2 = cos0 2 = 1 2 . 4 . Find ³ (2 x + 3)sin xdx . Use integration by parts: ³ udv = uv - ³ vdu , with u =2 x +3, du dx , dv = sin and v = - cos x . We have ³ (2 x +3)sin =(2 x +3)( - cos x ) - ³ 2( - cos x ) dx = - (2 x +3)cos x +2sin x + C. 5 . Find ³ sec x tan 3 . This is ³ tan 2 x sec x tan = ³ (sec 2 x - 1)sec x tan (using the identity 1+tan 2 x = sec 2 x ). Let u = sec x so du = sec x tan . The integral becomes ³ (sec 2 x - x tan = ³ ( u 2 - 1) du = 1 3 u 3 - u + C = 1 3 sec 3 x - sec x + C .
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6 . Find ± dx x 4 - x 2 . Use the trig substitution x = 2sin θ ,s o dx = 2cos θdθ . The integral becomes ± dx x 4 - x 2 = ± 2cos 2sin θ ² 4 - 4sin 2 θ = ± θ (2cos θ ) = 1 2 ± csc = 1 2 ln | csc θ - cot θ | + C. We can use a right triangle to fnd csc θ and cot θ .N o w sin θ = x 2 . So the triangle will have angle θ , opposite side x and hypotenuse 2, since sin θ = opp hyp . The missing side o± the triangle is (by Pythagoras)
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This note was uploaded on 04/02/2008 for the course MATH 31B taught by Professor Valdimarsson during the Fall '08 term at UCLA.

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finalsol - Final Exam Solutions 1. Find the derivative of f...

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