Linear Function and its Application - Lecture Notes | MATH 250A, Exams of Differential Equations

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MATH 250a Fall Semester 2007 Section 2 (J. M. Cushing)

Tuesday, August 21

http://math.arizona.edu/~cushing/250a.html

TENTATIVE SCHEDULE FOR MATH 250A, Section 2, FALL 2007

August

21 Introduction; Linear Functions and Linear Approximations23 Exponential Functions28 Power, Polynomial, and Rational Functions30 Trigonometric Functions

September

4 Review: Fundamental Theorem of Calculus6 Integration Techniques I: Substitution, Integration by Parts11 Integration Techniques II: Partial Fractions, Trigonometric Substitution13 Integration Techniques III: Tables and Numerical Approximation18 Error Estimates for Numerical Approximation20 EXAM 125 Improper Integrals27 Comparison of Improper Integrals; Numerical Approximation

October

2 Applications of Integration: Area and Volume4 Applications of Integration: Arc Length and Surface Area9 Integration in Polar Coordinates11 Review of Geometric Applications of Integration16 Applications of Integration: Mass, Density, and Center of Mass18 EXAM 223 Discrete Dynamical Systems and Introduction to Sequences; Convergence25 Introduction to Series; Geometric Series and the Ratio Test30 Convergence of Series; Improper Integrals and the Integral Test

November

1 Power Series and Radius of Convergence6 Finding and Using Taylor Series; Taylor Series for Well-Known Functions8 Error Estimates for Taylor Series13 Introduction to Fourier Series15 EXAM 320 Linear Algebra: Vectors and Matrices, Matrix Algebra, Inverses22 Thanksgiving Day – No class27 Linear Algebra: Linear Systems of Equations, Determinants,29 Linear Algebra: Eigenvalues and Eigenvectors

December

4 Review

Chapters1 - 6^ Chapter7.1 – 7.6^ Chapter7.7 – 7.8 Chapter 8 Chapter 9Chapter 10SupplementalMaterial

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Chapter 1.1^ Allowed

Mathematical Functions

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Chapter 1.1 Not Allowed

Mathematical Functions

Define by means of algebraic formula :

Chapter 1.

Mathematical Functions

Define by means of algebraic formula :

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Chapter 1.

Mathematical Functions

Define by means of graph :

x

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Chapter 1.

Mathematical Functions

Define by means of graph :

Cartesian coordinate system

x

y

Chapter 1.

Mathematical Functions

domain

range

Define by means of graph :

x

y

Chapter 1.

Mathematical Functions

Define by means of graph :

x

y

Chapter 1.

Mathematical Functions

Define by means of graph :

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Curve defines function:

Chapter 1.

Mathematical Functions

Define by means of graph :

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Not all graphsdefine a function

Chapter 1.

Mathematical Functions

Two general problems in analytic geometry :

(1) Given an algebraic formula for a function, draw its graph(2) Given a graph, find an algebraic formula for the function

defined by the graph.

Two general problems in analytic geometry : Each problem has its difficulties (especially (2) ). One goal is to develop a repertoire of special functions

for which one can do both tasks.^ Linear functionsPower functions (& polynomials)Rational functionsExponential/Logarithm functionsTrigonometric functions

(1) Given an algebraic formula for a function, draw its graph(2) Given a graph, find an algebraic formula for the function

defined by the graph.