Practice_Exam - Practice Test For Extraordinary People And...

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Practice Test For Extraordinary People And a Few Species of Swallow And Dolphin; Also “Calculus And You“: A Self-Help Guide David Imberti March 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) R x x 2 +1 dx (2) R cos ( θ ) cos ( sin ( θ )) (3) R sec 4 ( θ ) (4) R sin 4 ( θ ) (5) R 1 x 2 +2 x dx (6) R x 3 x 2 - 1 dx Compute the following integrals (take your time, relax) (I have the solutions on my person, ask me if you’re unsure): (1) R ln ( x ) dx (2) R arcsin ( x ) dx (3) R x 4 +1 x 3 + x 2 - x - 1 dx (4) R ( p y 2 - y ) - 1 dy (5) R θtan 2 ( θ ) (6) R 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 R 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, THERE’S THIS INTEGRAL AND I DON’T KNOW HOW TO . . . WHAT’S THIS . . . I DON’T EVEN- A: O.k., don’t panic, grab your towel. Remember, you’ve got only a handful of methods to integrate with that you need to remember: (0) It’s an integral you already know. 1
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(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 I’ve listed. Q: COULD YOU EXPLAIN EVERYTHING IN THAT LIST? A: Sure, what specifically? 2.1 0. Integrals We Know: You’re Half-way There! Q: WHAT INTEGRALS SHOULD I JUST MEMORIZE? A: O.k., I’m going to talk a little bit about the more complicated integrals we’ve already done. (you should know things like R cos ( θ ) = sin ( θ ) , R e x dx = e x , R x a dx = 1 a +1 x a +1 (for a 6 = - 1, else R x - 1 dx = - x - 2 )) Here are the common integrals that have funky forms that you might forget/should memorize: (Common 1): R 1 1 - x 2 dx = arcsin ( x ) + C (Common 2): R 1 1+ x 2 dx = arctan ( x ) + C (Common 3): R sec ( θ ) = ln | sec ( θ ) + tan ( θ ) | + C (Common 4): R tan ( θ ) = - ln | cos ( θ ) | + C Also, these are common trigonometric integrals, but they’re easy to forget when you’re chugging down in the mood of trigonometric substitution: (Common 5): R sec 2 ( θ ) = tan ( θ ) + C (Common 6): R sec ( θ ) tan ( θ ) = sec ( θ ) + C 2.2 1. Substituting Out For a Better You
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