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Calculus

AP Calc Study Guide

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Copyright 1996,1997 Elaine Cheong All Rights Reserved Study Guide for the Advanced Placement Calculus AB Examination By Elaine Cheong 1 Table of Contents INTRODUCTION 2 TOPICS TO STUDY 3 ? Elementary Functions 3 ? Limits 5 ? Differential Calculus 7 ? Integral Calculus 12 SOME USEFUL FORMULAS 16 CALCULATOR TIPS AND PROGRAMS 17 BOOK REVIEW OF AVAILABLE STUDY GUIDES 19 ACKNOWLEDGEMENTS 19 2 Introduction Advanced Placement1 is a program of college-level courses and examinations that gives high school students the opportunity to receive advanced placement and/or credit in college. The Advanced Placement Calculus AB Exam tests students on introductory differential and integral calculus, covering a full-year college mathematics course.

AP Calculus Cheat Sheet

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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. ? 2005 Paul Dawkins Limits Definitions Precise Definition : We say ( )limx a f x L? = if for every 0e > there is a 0d > such that whenever 0 x a d< - < then ( )f x L e- < . ?Working? Definition : We say ( )limx a f x L? = if we can make ( )f x as close to L as we want by taking x sufficiently close to a (on either side of a) without letting x a= . Right hand limit : ( )limx a f x L+? = . This has the same definition as the limit except it requires x a> . Left hand limit : ( )limx a f x L-? = . This has the same definition as the limit except it requires x a< . Limit at Infinity : We say ( )limx f x L?? = if we

Ch9 SG

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372 CHAPTER 9 Mathematical Modeling with Differential Equations EXERCISE SET 9.1 1. y? = 2x2ex 3/3 = x2y and y(0) = 2 by inspection. 2. y? = x3 ? 2 sinx, y(0) = 3 by inspection. 3. (a) ?rst order; dy dx = c; (1 + x) dy dx = (1 + x)c = y (b) second order; y? = c1 cos t? c2 sin t, y?? + y = ?c1 sin t? c2 cos t+ (c1 sin t+ c2 cos t) = 0 4. (a) ?rst order; 2 dy dx + y = 2 ( ? c 2 e?x/2 + 1 ) + ce?x/2 + x? 3 = x? 1 (b) second order; y? = c1et ? c2e?t, y?? ? y = c1et + c2e?t ? ( c1et + c2e?t ) = 0 5. 1 y dy dx = x dy dx + y, dy dx (1? xy) = y2, dy dx = y2 1? xy 6. 2x+ y2 + 2xy dy dx = 0, by inspection. 7. (a) IF: ? = e3 ? dx = e3x, d dx [ ye3x ] = 0, ye3x = C, y = Ce?3x separation of variables: dy y = ?3dx, ln |y| = ?3x+ C1, y = ?e?3xeC1 = Ce?3x

Ch8 SG

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317 CHAPTER 8 Principles of Integral Evaluation EXERCISE SET 8.1 1. u = 3? 2x, du = ?2dx, ? 1 2 ? u3 du = ?1 8 u4 + C = ?1 8 (3? 2x)4 + C 2. u = 4 + 9x, du = 9dx, 1 9 ? u1/2 du = 2 3 ? 9u 3/2 + C = 2 27 (4 + 9x)3/2 + C 3. u = x2, du = 2xdx, 1 2 ? sec2 u du = 1 2 tanu+ C = 1 2 tan(x2) + C 4. u = x2, du = 2xdx, 2 ? tanu du = ?2 ln | cosu |+ C = ?2 ln | cos(x2)|+ C 5. u = 2 + cos 3x, du = ?3 sin 3xdx, ? 1 3 ? du u = ?1 3 ln |u|+ C = ?1 3 ln(2 + cos 3x) + C 6. u = 3x 2 , du = 3 2 dx, 2 3 ? du 4 + 4u2 = 1 6 ? du 1 + u2 = 1 6 tan?1 u+ C = 1 6 tan?1(3x/2) + C 7. u = ex, du = exdx, ? sinhu du = coshu+ C = cosh ex + C 8. u = lnx, du = 1 x dx, ? secu tanu du = secu+ C = sec(lnx) + C 9. u = cotx, du = ? csc2 xdx, ? ? eu du = ?eu + C = ?ecot x + C

Multi-Variable Calculus summary

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AP CALC integrations

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Review 6 Name: __________________________________________________ AP Calculus BC Date: ______________________________ Period: _____ Find the area between and . Find the area between , and . (Use a single integral!) Find the volume of the region bounded by , and revolved about the -axis. (Use the method of disks!) Find the volume of the region bounded by , and revolved about the -axis. (Use the method of washers!) Find the volume of the region bounded by from , and revolved about . (Use the method of shells!) Find the arc length for the specified portion of each curve. An object is dropped from a height of 100 feet. Find its velocity when it hits the ground.

Derivative of x^2

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A derivative of X^2 or any other simple exponential expression is very easy. Bring the exponent value to the front and subtract one from the exponent. For example, the derivative of X^2 is 2X(^1)
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calc formulas

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CSSS 505 Calculus Summary Formulas Differentiation Formulas 1. 1 ( ) ? = n n x nx dx d 17. dx du dx dy dx dy = ? Chain Rule 2. fg fg gf dx d ( ) = ? + ? 3. 2 ( ) g gf fg g f dx d ? ? ? = 4. f (g(x)) f (g(x))g (x) dx d = ? ? 5. x x dx d (sin ) = cos 6. x x dx d (cos ) = ?sin 7. x x dx d 2 (tan ) = sec 8. x x dx d 2 (cot ) = ?csc 9. x x x dx d (sec ) = sec tan 10. x x x dx d (csc ) = ?csc cot 11. x x e e dx d ( ) = 12. a a a dx d x x ( ) = ln 13. x x dx d 1 (ln ) = 14. 2 1 1 ( sin ) x Arc x dx d ? = 15. 2 1 1 ( tan ) x Arc x dx d + = 16. | | 1 1 ( sec ) 2 ? = x x Arc x dx dTrigonometric Formulas 1. sin cos 1 2 2 ? + ? = 2. ? ? 2 2 1+ tan = sec 3. ? ? 2 2 1+ cot = csc 4. sin(?? ) = ?sin? 5. cos(?? ) = cos? 6. tan(?? ) = ? tan? 7. sin(A + B) = sin Acos B + sin Bcos A

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