Practice_Exam - Practice Test For Extraordinary People And...

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Unformatted text preview: Practice Test For Extraordinary People And a Few Species of Swallow And Dolphin; Also Calculus And You: A Self-Help Guide David Imberti April 6, 2010 1 A Small Practice Test The following have their answers worked out below (so, spoiler warning!), consider these as a warm-up: (1) x x 2 +1 dx (2) cos ( ) cos ( sin ( )) d (3) sec 4 ( ) (4) sin 4 ( ) (5) 1 x 2 +2 x dx (6) x 3 x 2- 1 dx Compute the following integrals (take your time, relax) (I have the solutions on my person, ask me if youre unsure): (1) ln ( x ) dx (2) arcsin ( x ) dx (3) x 4 +1 x 3 + x 2- x- 1 dx (4) ( y 2- y )- 1 dy (5) tan 2 ( ) d (6) 1 x +1+ x dx (7) n = 3, approximate f ( x ) = 1 x between 1 and 2 with the Trapezoidal rule. (8) Connect the following dots: (- 2 , 0) , (- 2 , 2) , (- 1 , 2) , (- 1 , 1) , (1 , 1) , (1 , 3) , (3 , 3) , (3 , 0) , (- 2 , 0). Find the coordinates of the centroid of the block region inside these points. (9) Find the surface area by rotating x = 1 3 ( y 2 + 2) 3 2 , 1 y 2 about the x-axis. (10) Is 1 ln | x + 1 |- ln | x | dx convergent or divergent? You might also want to take a look at: p. 448 25,73 p. 518 21,23,33,18,56,57,2,30,41,46,47 p. 549 34 p. 561 3ab, 12 p. 694 16, 19 p. 759 1-6, 8, 27 Or old exams. 2 Calculus and You Q: OMIGOSH, THERES THIS INTEGRAL AND I DONT KNOW HOW TO . . . WHATS THIS . . . I DONT EVEN- A: O.k., dont panic, grab your towel. Remember, youve got only a handful of methods to integrate with that you need to remember: (0) Its an integral you already know. 1 (1) Direct Substitution. (2) Trigonometric identities. (3) Trigonometic substitution. (4) Partial Fractions. (5) Integration by Parts. If you want to go about this methodically, I would go down the list in the order Ive listed. Q: COULD YOU EXPLAIN EVERYTHING IN THAT LIST? A: Sure, what specifically? 2.1 0. Integrals We Know: Youre Half-way There! Q: WHAT INTEGRALS SHOULD I JUST MEMORIZE? A: O.k., Im going to talk a little bit about the more complicated integrals weve already done. (you should know things like cos ( ) d = sin ( ) , e x dx = e x , x a dx = 1 a +1 x a +1 (for a =- 1, else x- 1 dx =- x- 2 )) Here are the common integrals that have funky forms that you might forget/should memorize: (Common 1): 1 1- x 2 dx = arcsin ( x ) + C (Common 2): 1 1+ x 2 dx = arctan ( x ) + C (Common 3): sec ( ) d = ln | sec ( ) + tan ( ) | + C (Common 4): tan ( ) d =- ln | cos ( ) | + C Also, these are common trigonometric integrals, but theyre easy to forget when youre chugging down in...
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Practice_Exam - Practice Test For Extraordinary People And...

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