{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Practice_Exam - Practice Test For Extraordinary People And...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ( θ ) cos ( sin ( θ )) d θ (3) sec 4 ( θ ) (4) sin 4 ( θ ) (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 you’re 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) θ tan 2 ( θ ) d θ (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, 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
(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 cos ( θ ) d θ = sin ( θ ) , 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 ( θ ) d θ = ln | sec ( θ ) + tan ( θ ) | + C (Common 4): tan ( θ ) d 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): sec 2 ( θ ) d θ = tan ( θ ) + C (Common 6): sec ( θ ) tan ( θ ) d θ = sec ( θ ) + C 2.2 1. Substituting Out For a Better You Q: HOW DO I KNOW WHAT TO SUBSTITUTE FOR?
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}