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Practice Exam January 25, 2009 π 1 Estimate the area under the graph f (x) = cos x from 0 to 2 using four approximating rectangles and midpoints. Sketch the graph and the rectangles. Solution : π π π 3π π 5π π 7π π π π 3π π 5π π 7π f ( ) + f ( ) + f ( ) + f ( ) = cos ( ) + cos ( ) + cos ( ) + cos ( ) 8 16 8 16 8 16 8 16 8 16 8 16 8 16 8 16 2 Express the following limit as a deﬁnite integral in the interval [2, 6]: n lim xi ln(1 + x2 ) ∆x i n→∞ i=1 Solution : n 6 lim xi ln(1 + x2 ) ∆x = i x ln (1 + x2 ) dx n→∞ 2 i=1 3 Evaluate the following integral: 10 x − 5 dx 0 Solution : 10 5 10 x2 5 x2 10 25 25 x−5 dx = (5−x) dx+ (x−5) dx = 5x− + −5x = 25− +50−50− +25 = 25 0 0 5 2 0 2 5 2 2 Note you can also solve it by drawing the function and interpreting the area as the sum of the areas of two triangles each with base 5 and height 5. 4 Find the derivative of the following function: x2 1 g(x) = √ dt tan x 2 + t4 Solution : x2 tan x 1 1 2x sec2 x g(x) = √ dt − √ dt , therefore g (x) = √ −√ 0 2 + t4 0 2 + t4 2 + x8 2 + tan4 x 1 √ x t2 1+u4 5 If F (x) = 1 f (t) dt, where f (t) = 1 u du, ﬁnd F (2). Solution : √ √ 1 + (x2 )4 1 + x8 1 + 28 √ F (x) = f (x) , therefore F (x) = f (x) = 2x 2 =2 , hence F (2) = 2 = 257 x x 2 4 6 If f (1) = 12, f is continuous and 1 f (x) dx = 17, what is the value of f (4). Solution : 4 17 = f (x) dx = f (4) − f (1) = f (4) − 12 , therefore f (4) = 17 + 12 = 29 1 7 Evaluate the following integral: ex dx ex + 1 Solution : ex du Let u = ex + 1 then du = ex dx hence dx = = ln u + C = ln(ex + 1) + C ex + 1 u 8 Evaluate the following integral: π sin5 x dx −π Solution : π Since sin x is an odd function, then sin5 x is an odd function and therefore −π sin5 x dx = 0 9 Evaluate the following integral: 10 (3x2 + x − 10) dx 0 Solution : 10 x3 x2 10 100 (3x2 + x − 10) dx = 3 + − 10x = 103 + − 100 = 1000 + 50 − 100 = 950 0 3 2 0 2 10 Who is the best superhero? (Circle the answer or write in another one) Spider − M an , Superman , Batman , Captain America , Daredevil Green Lantern , W olverine , Buf f y , Angel , T im Gunn Ozymandias , Rorschach , M r. F antastic , El Santo , Other Solution : Gauss 2

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posted: | 4/24/2010 |

language: | English |

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