The AP Calculus Problem Book - crunchy math

1 TheAPCalculusProblemBook ChuckGarner, AP Calculus Problem BookPublication history:First edition, 2002 Second edition, 2003 Third edition, 2004 Third edition Revised and Corrected, 2005 Fourth edition, 2006, Edited by Amy LanchesterFourth edition Revised and Corrected, 2007 Fourth edition, Corrected, 2008 This book was produced directly from the author s LATEX were drawn by the author using the TEXdraw screen-shots produced by a TI-83 Plus calculator using a TI-Graph (pronounced Lay-Tek ) is a document typesetting program (not a word processor) that is available free ,which also includes TEXnicCenter, a free and easy-to-use Graphs of Functions .. The Slippery Slope of Lines .. The Power of Algebra .. Functions Behaving Badly .. Take It to the Limit .. One-Sided Limits .. One-Sided Limits (Again) .. Limits Determined by Graphs .. Limits Determined by Tables .. The Possibilities Are .. Average Rates of Change: Episode I .. Exponential and Logarithmic Functions.

2 Average Rates of Change: Episode II .. Take It To the Limit One More Time .. Solving Equations .. Continuously Considering Continuity .. Have You Reached the Limit? .. Multiple Choice Questions on Limits .. Sample Problems on Limits .. 26 Last Year s Limits Test .. Negative and Fractional Exponents .. Logically Thinking About Logic .. The Derivative By Definition .. Going Offon a Tangent .. Six Derivative Problems .. Trigonometry: a Refresher .. 4112 The AP Calculus Problem Continuity and Differentiability .. The RULES: Power Product Quotient Chain .. Trigonometric Derivatives .. Tangents, Normals, and Continuity (Revisited) .. Implicit Differentiation .. The Return of Geometry .. Meet the Rates (They re Related) .. Rates Related to the Previous Page .. Excitement with Derivatives! .. Derivatives of Inverses .. D eriv e, Derivado, Ableitung, Derivative .. Sample Problems on Derivatives .. Multiple-Choice Problems on Derivatives.

3 56 Last Year s Derivatives Test .. The Extreme Value Theorem .. Rolle to the Extreme with the Mean Value Theorem .. The First and Second Derivative Tests .. Derivatives and Their Graphs .. Two Derivative Problems .. Sketching Functions .. Problems of Motion .. Maximize or Minimize? .. More Tangents and Derivatives .. More Excitement with Derivatives! .. Bodies, Particles, Rockets, Trucks, and Canals .. Even More Excitement with Derivatives! .. Sample Problems on Applications of Derivatives .. Multiple-Choice Problems on Applications of Derivatives .. 89 Last Year s Applications of Derivatives Test .. The ANTI derivative! .. Derivative Rules Backwards .. The Method of Substitution .. Using Geometry for Definite Integrals .. Some Riemann Sums .. The MVT and the FTC .. The FTC, Graphically .. Definite and Indefinite Integrals .. Integrals Involving Logarithms and Exponentials .. It Wouldn t Be Called the Fundamental Theorem If It Wasn t Fundamental.

4 Definite and Indefinite Integrals Part 2 .. Regarding Riemann Sums .. Definitely Exciting Definite Integrals! .. How Do I Find the Area Under Thy Curve? Let Me Count the .. Three Integral Problems .. Trapezoid and Simpson .. Properties of Integrals .. Sample Problems on Integrals .. Multiple Choice Problems on Integrals ..124 Last Year s Integrals Test .. Volumes of Solids with Defined Cross-Sections .. Turn Up the Volume! .. Volume and Arc Length .. Differential Equations, Part One .. The Logistic Curve .. Differential Equations, Part Two .. Slope Fields and Euler s Method .. Differential Equations, Part Three .. Sample Problems on Applications of Integrals .. Multiple Choice Problems on Application of Integrals ..147 Last Year s Applications of Integrals Test .. A Part, And Yet, .. Partial Fractions .. Trigonometric Substitution .. Four Integral Problems .. L H opital s Rule .. Improper Integrals! .. The Art of Integration.

5 Functions Defined By Integrals .. Sample Problems on Techniques of Integration .. Sample Multiple-Choice Problems on Techniques of Integration .. 173 Last Year s Techniques of Integration Test ..1757 SERIES,VECTORS, Sequences: Bounded and Unbounded .. It is a Question of .. Infinite Sums .. Tests for Convergence and Divergence .. More Questions of .. Power Series! .. Maclaurin Series .. Taylor Series .. Vector Basics .. Calculus with Vectors and Parametrics .. Vector-Valued Functions .. Motion Problems with Vectors .. Polar Basics .. Differentiation (Slope) and Integration (Area) in Sample Problems on Series, Vectors, Parametrics,and Polar .. 1984 The AP Calculus Problem Sample Multiple-Choice Problems on Series, Vectors, Parametrics, and Polar .. 201 Last Year s Series, Vectors, Parametrics, and Polar Test .. Hyperbolic Functions .. Surface Area of a Solid of Revolution .. Linear First Order Differential Equations .. Curvature.

6 Newton s Method .. Algebra .. Derivative Skills .. Can You Stand All These Exciting Derivatives? .. Different Differentiation Problems .. Again! .. Int egrale, Integrale, Integraal, Integral .. Calculus Is an Integral Part of Your Life .. Particles .. Areas .. The Deadly Dozen .. Two Volumes and Two Differential Equations .. Differential Equations, Part Four .. More Integrals .. Definite Integrals Requiring Definite Thought .. Just When You Thought Your Problems Were .. Interesting Integral Problems .. Infinitely Interesting Infinite Series .. Getting Serious About Series .. A Series of Series Problems ..24010 GROUP INVESTIGATIONS241 About the Group Investigations .. Finding the Most Economical Speed for Trucks .. Minimizing the Area Between a Graph and Its Tangent .. The Ice Cream Cone Problem .. Designer Polynomials .. Inventory Management .. Optimal Design of a Steel Drum ..24611 Calculus LABS247 About the Labs ..2481: The Intermediate Value Theorem.

7 2502: Local Linearity ..2523: Exponentials ..2544: A Function and Its Derivative ..2565: Riemann Sums and Integrals ..2596: Numerical Integration ..262 CONTENTS57: Indeterminate Limits and l H opital s Rule ..2678: Sequences ..2709: Approximating Functions by Polynomials ..27210: Newton s Method ..27412 TI-CALCULATOR LABS277 Before You Start ..2781: Useful Stuff..2792: Derivatives ..2813: Maxima, Minima, Inflections ..2834: Integrals ..2845: Approximating Integrals ..2866: Approximating Integrals II ..2877: Applications of Integrals ..2898: Differential Equations ..2929: Sequences and Series ..29313 CHALLENGE PROBLEMS295 Set A .. 296 Set B .. 297 Set C .. 299 Set D .. 301 Set E .. 303 Set F .. 305 AFORMULAS309 Formulas from Geometry ..310 Greek Alphabet ..311 Trigonometry ..312 BSUCCESSINMATHEMATICS315 Calculus BC Syllabus ..316 CANSWERS329 Answers to Last Year s Tests ..3436 The AP Calculus Problem BOOKCHAPTER1 LIMITS78 The AP Calculus Problem Graphs of FunctionsDescribe the graphs of each of the following functions usingonly one of thefollowing terms:line, parabola, cubic, hyperbola, +5x2 x + x3+ 9 + 3x 1 x2 Graph the following functions on your calculator on the window 3 x 3, 2 y to the graph at the origin: A) goes vertical; B) formsacusp;C)goeshorizontal; or D) stops at on the answers from the problems above, find a pattern forthebehavioroffunctionswith exponents of the following forms:xeven/odd,xodd/odd, the following functions on your calculator in the standard window andsketch what you see.

8 At what value(s) ofxare the functions equal to zero? |x 1| |x2 4| |x3 8| |4+x2| |x3| |x2 4x 5|In the company of friends, writers can discuss their books, economists the state of the economy, lawyers theirlatest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics atall. And the more profound their work, the less understandable it is. Alfred AdlerCHAPTER 1. The Slippery Slope of LinesThe point-slope form of a line ism(x x1)=y the first six problems, find the equation of the line with the given :23;passesthrough(2,1) : 14;passesthrough(0,6) through (3,6) and (2,7) through ( 6,1) and (1,1) through (5, 4) and (5,9) through (10,3) and ( 10,3) (1,2) and (2,5). Another line passes through (0,0) and ( 4,3). Findthe point where the two lines 25and passing through (2,4) is parallel to another line passing through( 3,6). Find the equations of both 3andpassingthrough(1,5) is perpendicular to another line passingthrough (1,1).

9 Find the equations of both (8,8) and ( 2,3). Another line passes through (3, 1) and ( 3,0).Find the point where the two lines functionf(x)isaline. Iff(3) = 5 andf(4) = 9, then find the equation of the linef(x). functionf(x)isaline. Iff(0) = 4 andf(12) = 5, then find the equation of the linef(x). functionf(x) (x)is3andf(2) = 5, then findf(7). functionf(x) (x)is23andf(1) = 1, then findf(32). (2) = 1 andf(b)=4,thenfindthevalueofbso that the linef(x) the equation of the line that hasx-intercept at 4 andy-intercept at the equation of the line with slope 3 which intersects thesemicircley= 25 x2atx= ,because I really want a credit card. John Nash10 The AP Calculus Problem The Power of AlgebraFa c t o r e a c h o f t h e f o l l o w i n g c o m p l e t e ly . 18y+ 37u+ +9c 848.(x 6)2 (x+9)2 36(x+9)+ r2+2 r+hr+ + + 155.(x+2)3+ 2x2+9x 5p3+8p2 40 Simplify each of the following (x 4) + 2(x+5)6(x 4) y 1y 5x 1y+y1 1xRationalize each of the following 3+9 7 2+ 52 +8 x+ 34+ 6 x 3+ 2x+3 x+5 5 6 34 5+ x+3 3 Incubation is the work of the subconscious during the waitingtime,whichmaybeseveralyears.

10 Illumination,which can happen in a fraction of a second, is the emergence ofthe creative idea into the conscious. This almostalways occurs when the mind is in a state of relaxation, and engaged lightly with ordinary matters. Illuminationimplies some mysterious rapport between the subconscious and the conscious, otherwise emergence would nothappen. What rings the bell at the right moment? John E. LittlewoodCHAPTER 1. Functions Behaving BadlySketch a graph of each function, then find its (x)={x2x 12x+3x< (t)={|t|t<1 3t+4t (x)=x+|x| (r)= 1 r2 1 r 11rr> (x)= 1/x x < 1x 1 x 11/x x > (x)=x|x|For the following, find a) the domain; b) they-intercept; and c) all verticaland horizontal 1(x 1) 2x +2x 2xx2 4x+3x 4 Choose the best of the following represents the graph off(x)movedtotheleft3units?A)f(x 3)B)f(x) 3C)f(x+3)D)f(x)+ of the following represents the graph ofg(x)movedtotheright2unitsanddown7units ?A)g(x 2) 7B)g(x+2)+7C)g(x+7) 2D)g(x 7) + 2Fa c t o r e a c h o f t h e f o l l o w i n g.}}