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Unformatted text preview: Portillo, Tom Final 1 Due: Dec 19 2006, 11:00 pm Inst: David Benzvi 1 This printout should have 25 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Estimate the area under the graph of f ( x ) = 21 x 2 on [0 , 4] by dividing [0 , 4] into four equal subintervals and using right endpoints as sam ple points. 1. area 52 2. area 54 correct 3. area 55 4. area 56 5. area 53 Explanation: With four equal subintervals and right end points as sample points, A n f (1) + f (2) + f (3) + f (4) o 1 , since x i = x * i = i . Consequently, A 54 . keywords: estimate area, quadratic function Riemman sum 002 (part 1 of 1) 10 points Use properties of integrals to determine the value of I = Z 4 f ( x ) dx when Z 6 f ( x ) dx = 9 , Z 6 4 f ( x ) dx = 4 . 1. I = 8 2. I = 13 3. I = 13 4. I = 5 correct 5. I = 8 6. I = 5 Explanation: Since Z a b f ( x ) dx = Z b a f ( x ) dx and Z b a f ( x ) dx + Z c b f ( x ) dx = Z c a f ( x ) dx, we see that Z a b f ( x ) dx = Z c a f ( x ) dx Z c b f ( x ) dx . Consequently, I = (9 4) = 5 . keywords: integral, properties of integrals 003 (part 1 of 1) 10 points Determine g ( x ) when g ( x ) = Z 5 x 2 t 2 sec t dt. 1. g ( x ) = 4 x sec 2 x 2. g ( x ) = 2 x 2 tan x Portillo, Tom Final 1 Due: Dec 19 2006, 11:00 pm Inst: David Benzvi 2 3. g ( x ) = 2 x 2 tan x 4. g ( x ) = 4 x sec 2 x 5. g ( x ) = 4 x sec x tan x 6. g ( x ) = 4 x sec x tan x 7. g ( x ) = 2 x 2 sec x correct 8. g ( x ) = 2 x 2 sec x Explanation: By Properties of integrals and the Funda mental Theorem of Calculus, d dx Z a x f ( t ) dt = d dx  Z x a f ( t ) dt = f ( x ) . When g ( x ) = Z 5 x f ( t ) dt, f ( t ) = 2 t 2 sec t, therefore, g ( x ) = 2 x 2 sec x . keywords: Stewart5e, FTC, derivative form, properties of integrals, 004 (part 1 of 1) 10 points Find the volume, V , of the solid obtained by rotating the region bounded by y = 2 x , x = 2 , x = 3 , y = 0 about the xaxis. 1. V = 1 6 2. V = 1 3 3. V = 1 6 4. V = 1 3 5. V = 2 3 6. V = 2 3 correct Explanation: The volume of the solid of revolution ob tained by rotating the graph of y = f ( x ) on [ a, b ] about the xaxis is given by volume = Z b a f ( x ) 2 dx. When f ( x ) = 2 x , a = 2 , b = 3 , therefore, V = Z 3 2 4 x 2 dx. Consequently, V =  4 x 3 2 = 2 3 . keywords: volume, integral, solid of revolu tion 005 (part 1 of 1) 10 points Evaluate the integral I = Z ln 3 xe x 2 e x 2 + 2 dx. 1. I = 1 2 5 3 2. I = 2 5 3 3. I = 1 2 7 2 Portillo, Tom Final 1 Due: Dec 19 2006, 11:00 pm Inst: David Benzvi 3 4. I = 7 2 5. I = 5 3 correct 6. I = 2 7 2 Explanation: The presence of the term xe x 2 in the nu merator suggests the substitution u = e x 2 + 2 , for then du = 2 xe x 2 dx....
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 Spring '06
 Benzvi

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