DOCSLIB.ORG

4.1 AP Calc Antiderivatives and Indefinite Integration.Notebook November 28, 2016

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
4.1 AP Calc Antiderivatives and Indefinite Integration.Notebook November 28, 2016 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 4.1 Antiderivatives and Indefinite Integration Learning Targets 1. Write the general solution of a differential equation 2. Use indefinite integral notation for antiderivatives 3. Use basic integration rules to find antiderivatives 4. Find a particular solution of a differential equation 5. Plot a slope field 6. Working backwards in physics Intro/Warm­up 1 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Note: I am changing up the procedure of the class! When you walk in the class, you will put a tally mark on those problems that you would like me to go over, and you will turn in your homework with your name, the section and the page of the assignment. Order of the day 2 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Vocabulary First, find a function F whose derivative is . because any other function F work? So represents the family of all antiderivatives of . The constant C is called the constant of integration. The family of functions represented by F is the general antiderivative of f, and is the general solution to the differential equation The operation of finding all solutions to the differential equation is called antidifferentiation (or indefinite integration) and is denoted by an integral sign: is read as the antiderivative of f with respect to x. Vocabulary 3 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Nov 28­9:38 AM 4 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 ∫ Use basic integration rules to find antiderivatives The Power Rule: where 1. Solve the differential equation So the solution to is 2. Evaluate (see tables on next slide) 3. Evaluate (see tables on next slide) I.Use basic integration rules to find antiderivatives 5 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Basic Integration Rules 6 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 ∫∫Initial Conditions and Particular Solutions 4. For the differential equation find the general equation. If the initial condition is find the particular solution. II. Initial Conditions and Particular Solutions 7 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 ∫∫∫ Slope Fields 5. Draw a slope field for III. Slope Fields 8 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 To find a program on Slope Fields: Slope field programs 9 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Your Homework AB Calculus 4.1 p.255 #2,3,6,15‐39(x3),45‐66(x3), 67,68,71,81,87* *=use calcchat.com BC Calculus 10.5 p.747 #12,24,30,39,44,47,57,67,73‐77 Homework 10.
Load more

Recommended publications
  • Notes on Calculus II Integral Calculus Miguel A. Lerma
    Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus.
    [Show full text]
  • Antiderivatives and the Area Problem
    Chapter 7 antiderivatives and the area problem Let me begin by defining the terms in the title: 1. an antiderivative of f is another function F such that F 0 = f. 2. the area problem is: ”find the area of a shape in the plane" This chapter is concerned with understanding the area problem and then solving it through the fundamental theorem of calculus(FTC). We begin by discussing antiderivatives. At first glance it is not at all obvious this has to do with the area problem. However, antiderivatives do solve a number of interesting physical problems so we ought to consider them if only for that reason. The beginning of the chapter is devoted to understanding the type of question which an antiderivative solves as well as how to perform a number of basic indefinite integrals. Once all of this is accomplished we then turn to the area problem. To understand the area problem carefully we'll need to think some about the concepts of finite sums, se- quences and limits of sequences. These concepts are quite natural and we will see that the theory for these is easily transferred from some of our earlier work. Once the limit of a sequence and a number of its basic prop- erties are established we then define area and the definite integral. Finally, the remainder of the chapter is devoted to understanding the fundamental theorem of calculus and how it is applied to solve definite integrals. I have attempted to be rigorous in this chapter, however, you should understand that there are superior treatments of integration(Riemann-Stieltjes, Lesbeque etc..) which cover a greater variety of functions in a more logically complete fashion.
    [Show full text]
  • A Brief Tour of Vector Calculus
    A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0.
    [Show full text]
  • Differentiation Rules (Differential Calculus)
    Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below.
    [Show full text]
  • Leonhard Euler: His Life, the Man, and His Works∗
    SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Notes Chapter 4(Integration) Definition of an Antiderivative
    1 Notes Chapter 4(Integration) Definition of an Antiderivative: A function F is an antiderivative of f on an interval I if for all x in I. Representation of Antiderivatives: If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x) = F(x) + C, for all x in I where C is a constant. Sigma Notation: The sum of n terms a1,a2,a3,…,an is written as where I is the index of summation, ai is the ith term of the sum, and the upper and lower bounds of summation are n and 1. Summation Formulas: 1. 2. 3. 4. Limits of the Lower and Upper Sums: Let f be continuous and nonnegative on the interval [a,b]. The limits as n of both the lower and upper sums exist and are equal to each other. That is, where are the minimum and maximum values of f on the subinterval. Definition of the Area of a Region in the Plane: Let f be continuous and nonnegative on the interval [a,b]. The area if a region bounded by the graph of f, the x-axis and the vertical lines x=a and x=b is Area = where . Definition of a Riemann Sum: Let f be defined on the closed interval [a,b], and let be a partition of [a,b] given by a =x0<x1<x2<…<xn-1<xn=b where xi is the width of the ith subinterval.
    [Show full text]
  • Calculus Formulas and Theorems
    Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9.
    [Show full text]
  • Reconciling Situation Calculus and Fluent Calculus
    Reconciling Situation Calculus and Fluent Calculus Stephan Schiffel and Michael Thielscher Department of Computer Science Dresden University of Technology stephan.schiffel,mit @inf.tu-dresden.de { } Abstract between domain descriptions in both calculi. Another ben- efit of a formal relation between the SC and the FC is the The Situation Calculus and the Fluent Calculus are successful possibility to compare extensions made separately for the action formalisms that share many concepts. But until now two calculi, such as concurrency. Furthermore it might even there is no formal relation between the two calculi that would allow to formally analyze the relationship between the two allow to translate current or future extensions made only for approaches as well as between the programming languages one of the calculi to the other. based on them, Golog and FLUX. Furthermore, such a formal Golog is a programming language for intelligent agents relation would allow to combine Golog and FLUX and to ana- that combines elements from classical programming (condi- lyze which of the underlying computation principles is better tionals, loops, etc.) with reasoning about actions. Primitive suited for different classes of programs. We develop a for- statements in Golog programs are actions to be performed mal translation between domain axiomatizations of the Situ- by the agent. Conditional statements in Golog are composed ation Calculus and the Fluent Calculus and present a Fluent of fluents. The execution of a Golog program requires to Calculus semantics for Golog programs. For domains with reason about the effects of the actions the agent performs, deterministic actions our approach allows an automatic trans- lation of Golog domain descriptions and execution of Golog in order to determine the values of fluents when evaluating programs with FLUX.
    [Show full text]
  • Basic Calculus I Fall 2007 Class 11 Monday September 24 Introduction to Continuity
    Math 110 Basic Calculus I Fall 2007 Class 11 Monday September 24 Introduction to Continuity An informal definition of continuity of a function is that if it’s graph can be drawn without picking up your drawing implement, then it is continuous at all the iput values that are graphed. DEFINITION: Continuity at a point A function f(x) is said to be continuous at a point x = a if ALL of the following statements are true: 1. f(a) is defined 2. lim f(x) exists x→a 3. lim f(x)=f(a) x→a If a function is not continuous at a point we say that it has a discontinuity at x = a. EXAMPLE Let’s think about the different ways that this defnition can fail for a function: 1. f(x) may be undefined at the point x = a 2. The value of f(a) and the value of the limit of f(x)asx approaches a may be different 3. The limit of f(x)asx approaches a may not exist because the one sided limits are different 4. The limit of f(x)asx approaches a may not exist for other reasons than those given in (3) GroupWork Let’s provide algebraic examples of functions which correspond to each of the “Discontinuity Scenarios” described above and draw a picture of each of the scenarios which depict the way a function can be discontinous at a apoint. DEFINITION: Continuity on an open interval If a function f(x) is continuous at every point in an open interval (a, b) we say that f(x) is continuous on the interval (a, b).
    [Show full text]
  • 12.4 Calculus Review
    Appendix 445 12.4 Calculus review This section is a very brief summary of Calculus skills required for reading this book. 12.4.1 Inverse function Function g is the inverse function of function f if g(f(x)) = x and f(g(y)) = y for all x and y where f(x) and g(y) exist. 1 Notation Inverse function g = f − 1 (Don’t confuse the inverse f − (x) with 1/f(x). These are different functions!) To find the inverse function, solve the equation f(x) = y. The solution g(y) is the inverse of f. For example, to find the inverse of f(x)=3+1/x, we solve the equation 1 3 + 1/x = y 1/x = y 3 x = . ⇒ − ⇒ y 3 − The inverse function of f is g(y) = 1/(y 3). − 12.4.2 Limits and continuity A function f(x) has a limit L at a point x0 if f(x) approaches L when x approaches x0. To say it more rigorously, for any ε there exists such δ that f(x) is ε-close to L when x is δ-close to x0. That is, if x x < δ then f(x) L < ε. | − 0| | − | A function f(x) has a limit L at + if f(x) approaches L when x goes to + . Rigorously, for any ε there exists such N that f∞(x) is ε-close to L when x gets beyond N∞, i.e., if x > N then f(x) L < ε. | − | Similarly, f(x) has a limit L at if for any ε there exists such N that f(x) is ε-close to L when x gets below ( N), i.e.,−∞ − if x < N then f(x) L < ε.
    [Show full text]
  • The Legacy of Leonhard Euler: a Tricentennial Tribute (419 Pages)
    P698.TP.indd 1 9/8/09 5:23:37 PM This page intentionally left blank Lokenath Debnath The University of Texas-Pan American, USA Imperial College Press ICP P698.TP.indd 2 9/8/09 5:23:39 PM Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. THE LEGACY OF LEONHARD EULER A Tricentennial Tribute Copyright © 2010 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-84816-525-0 ISBN-10 1-84816-525-0 Printed in Singapore. LaiFun - The Legacy of Leonhard.pmd 1 9/4/2009, 3:04 PM September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard Leonhard Euler (1707–1783) ii September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard To my wife Sadhana, grandson Kirin,and granddaughter Princess Maya, with love and affection.
    [Show full text]