Math 222 – 2nd Semester Calculus Lecture notes version 1 1 (Fall 2015) „is is a self contained set of lecture notes for Math 222 „e notes were wri−en by Sigurd Angenent, as part ot the MIU calculus project Some problems were contributed by A Miller „eLATEX•les, aswellastheInkscapeandOctave•leswhichwereusedtoproduce
Second Fundamental Theorem of Calculus Let f be a continuous function de ned on an interval I Fix a point a in I and de ne a function F on I by F(x) = Z x a f(t)dt: Then F is an antiderivative of f on the interval I, i e F0(x) = f(x) on I A proof of the Second Fundamental Theorem of Calculus is given on pages 318{319 of the textbook We
This tutorial manual is intended as a supplement to Rogawski's Calculus textbook and aimed at students looking to quickly learn Mathematica through examples It also includes a brief summary of each calculus topic to emphasize important concepts Students should refer to their textbook for a further explanation of each topic ü1 1 Getting Started
calculus lectures 3 continuity; I’ll talk about it later The shorthand is "mixed partials are equal" Mostly Worked Problems Let f(x,y) = sin(y2 x2) Compute all partials; show the mixed partials are equal ¶f ¶x = cos(y 22x ) ¶ ¶x (y 2 x ) = ( 2x)cos(y x2) ¶f ¶y = cos(y 22x 2) ¶ ¶y (y x 2) = (2y)cos(y x ) Now for the second
of by a constant fails to produce a second linearly independent solution, suppose we try multiplying by a functioninstead The simplest such function would be , so let’s see if is also a solution Substituting into the differential equation gives The first term is zero because ; the second term is zero because solves the auxiliary equation
This Second Edition incorporates improvements in exposition, several new and clearer proofs, expansion of some problem sets, a review of Analytic Trigonometry, a few remarkable airbrush figures, and bonuses on pages previously only partially used A quick evaluation of the alterations may be
Oct 09, 2012 · CALCULUS MADE EASY 6 took dxto mean numerically, say, 1 60 of x, then the second term would be 2 60 of x, whereas the third term would be 1 3600 of x2 This last term is clearly less important than the second But if we go further and take dxto mean only 1 1000 of x, then the second term will be 2 1000 of x2, while the third term will be only 1
BASIC CALCULUS REFRESHER Ismor Fischer, Ph D Dept of Statistics UW-Madison 1 Introduction This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in Mathematics, Statistics, Engineering, Pharmacy, etc It is not comprehensive, and
Seconde/Ensembledenombresetcalculs 1 Calcul dans N, Z et Q: Exercice 8270 Définitions: On classe les nombres suivants leurs natures: Tous les nombres admettant une écriture décimale for-
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Mathematica
for Rogawski's
Calculus
2nd Edition 2010
Based on Mathematica Version 7
Abdul Hassen, Gary Itzkowitz, Hieu D. Nguyen, Jay Schiffman
W. H. Freeman and Company
New York
© Copyright 2010
2 Mathematica for Rogawski's Calculus 2nd Editiion.nb
Table of Contents
Chapter 1 Introduction
1.1 Getting Started
1.1.1 First-Time Users of Mathematica 7
1.1.2 Entering and Evaluating Input Commands
1.1.3 Documentation Center (Help Menu)
1.2Mathematica's Conventions for Inputting Commands
1.2.1 Naming
1.2.2 Parentheses, Brackets, and Braces
1.2.3 Lists
1.2.4 Equal Signs
1.2.5 Referring to Previous Results
1.2.6 Commenting
1.3 Basic Calculator Operations
1.4 Functions
1.5 Palettes
1.6 Solving Equations
Chapter 2 Graphs, Limits, and Continuity of Functions
2.1 Plotting Graphs
2.1.1 Basic Plot
2.1.2 Plot Options
2.2 Limits
2.2.1 Evaluating Limits
2.2.2 Limits Involving Trigonometric Functions
2.2.3 Limits Involving Infinity
2.3 Continuity
Chapter 3. Differentiation
3.1 The Derivative
3.1.1 Slope of Tangent
Mathematica for Rogawski's Calculus 2nd Editiion.nb 3
3.1.2 Derivative as a Function
3.2 Higher-Order Derivatives
3.3 Chain Rule and Implicit Differentiation
3.4 Derivatives of Inverse, Exponential and Logarithmic Functions
3.4.1 Inverse of a Function
3.4.2 Exponential and Logarithmic Functions
Chapter 4 Applications of the Derivative
4.1 Related Rates
4.2 Extrema
4.3 Optimization
4.3.1 Traffic Flow
4.3.2 Minimum Cost
4.3.3 Packaging (Minimum Surface Area)
4.3.4 Maximum Revenue
4.4 Newton's Method
4.4.1 Programming Newton's Method
4.4.2 Divergence
4.4.3 Slow Convergence
Chapter 5 Integration
5.1 Antiderivatives (Indefinite Integral)
5.2 Riemann Sums and the Definite Integral
5.2.1 Riemann Sums Using Left Endpoints
5.2.2 Riemann Sums Using Right Endpoints
5.2.3 Riemann Sums Using Midpoints
5.3 The Fundamental Theorem of Calculus
5.4 Integrals Involving Trigonometric, Exponential, and Logarithmic Functions
Chapter 6 Applications of the Integral
6.1 Area Between Curves
6.2 Average Value
4 Mathematica for Rogawski's Calculus 2nd Editiion.nb
6.3 Volumes of Solids of Revolution
6.3.1 The Method of Discs
6.3.2 The Method of Washers
6.3.3 The Method of Cylindrical Shells
Chapter 7 Techniques of Integration
7.1 Numerical Integration
7.1.1 Trapezoidal Rule
7.1.2 Simpson's Rule
7.1.3 Midpoint Rule
7.2 Techniques of Integration
7.2.1 Substitution
7.2.2 Trigonometric Substitution
7.2.3 Method of Partial Fractions
7.3 Improper Integrals
7.4 Hyperbolic Functions
7.4.1 Hyperbolic Functions
7.4.2 Identities Involving Hyperbolic Functions
7.4.3 Derivatives of Hyperbolic Functions
7.4.4 Inverse Hyperbolic Functions
Chapter 8 Further Applications of Integration
8.1 Arc Length and Surface Area
8.1.1 Arc Length
8.1.2 Surface Area
8.2 Center of Mass
Chapter 9 Introduction to Differential Equations
9.1 Solving Differential Equations
9.2 Models of the Form y'kyb
9.2.1 Bacteria Growth
9.2.2 Radioactive Decay
Mathematica for Rogawski's Calculus 2nd Editiion.nb 5
9.2.3 Annuity
9.2.4 Newton's Law of Cooling
9.3 Numerical Methods Using Slope Fields
9.3.1 Slope Fields
9.3.2 Euler's Method
9.4 The Logistic Equation
Chapter 10 Infinite Series
10.1 Sequences
10.2 Infinite Series
10.2.1 Finite Sums
10.2.2 Partial Sums and Convergence
10.3 Tests for Convergence
10.3.1 Comparison and Limit Comparison Tests
10.3.2 The Integral Test
10.3.3 Absolute and Conditional Convergence
10.3.4 Ratio Test
10.3.5 Root Test
10.4 Power Series
10.4.1 Taylor Polynomials
10.4.2 Convergence of Power Series
10.4.3 Taylor Series
Chapter 11 Parametric Equations, Polar Coordinates, and Conic Sections
11.1 Parametric Equations
11.1.1 Plotting Parametric Equations
11.1.2 Parametric Derivatives
11.1.3 Arc Length and Speed
11.2 Polar Coordinates and Curves
11.2.1 Conversion Formulas
11.2.2 Polar Curves
6 Mathematica for Rogawski's Calculus 2nd Editiion.nb
11.2.3 Calculus of Polar Curves
11.3 Conic Sections
Chapter 12 Vector Geometry
12.1 Vectors
12.2 Matrices and the Cross Product
12.3 Planes in 3-Space
12.4 A Survey of Quadric Surfaces
12.4.1 Ellipsoids
12.4.2 Hyperboloids
12.4.3 Paraboloids
12.4.4 Quadratic Cylinders
12.5 Cylindrical and Spherical Coordinates
12.5.1 Cylindrical Coordinates
12.5.2 Spherical Coordinates
Chapter 13 Calculus of Vector-Valued Functions
13.1 Vector-Valued Functions
13.2 Calculus of Vector-Valued Functions
13.3 Arc Length
13.4 Curvature
13.5 Motion in Space
Chapter 14 Differentiation in Several Variables
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MATH 222 Second Semester Calculus