By David A. Sanchez
The writer starts off with a common dialogue of the linear equation and develops the notions of a primary process of options, the Wronskian, and the corresponding basic matrix. He then introduces the nonhomogeneous linear equation and the real edition of parameters formulation, during which, following a attention of the nth-order linear equation, the answer of the nonhomogeneous nth-order linear equation is received. A bankruptcy is then dedicated to the linear equation with consistent coefficients.
The following chapters are introductory discussions of balance idea for self reliant and nonautonomous structures. incorporated listed here are effects for nonlinear platforms, Liapunov’s direct process, and a few effects for the second-order linear equation. the ultimate bankruptcy takes up the issues of the lifestyles and area of expertise of ideas and comparable subject matters. appendixes — “Series options of Second-Order Linear Equations” and “Linear platforms with Periodic Coefficients” — also are supplied, and there are difficulties on the finish of every chapter.
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Extra info for Ordinary Differential Equations and Stability Theory: An Introduction
Example text
Arithmetic Combinations and Composition of Functions Inverse Functions Families of Functions Symmetry, Even and Odd Functions CHAPTER 2 – LIMITS Limits and Continuity Infinite Limits and Vertical Asymptotes Limits at Infinity and Horizontal Asymptotes Continuity Tangent Lines, Areas and Limits Derivative as a Limit CHAPTER 3 – DERIVATIVES Writing Equations of Lines Derivatives and Rules of Differentiation Writing Equations of Lines Tangent to a Function Average and Instantaneous Velocity Chain Rule Increasing and Decreasing Functions, Mean–Value Theorem First Derivative Test Second Derivative Test Concavity and Inflection Points Absolute Maximum and Minimum, Extreme–Value Theorem Business Applications CHAPTER 4 – INTEGRALS Indefinite Integral: Integration as Anti–differentiation Definite Integral: Area Under the Graph Area Between Two Graphs Volume of a Solid of Revolution Business Applications Appendix A: Logarithms, Derivatives of Logarithmic Functions Appendix B: Trigonometry, Derivatives and Integrals of Trigonometric Functions Appendix C: Parametric Equations, Polar Coordinates, Area in Polar Coordinates Introduction This textbook is a product of 15 years experience teaching Calculus to students in Colleges and high schools.
The line through P and Q is called the secant line. If we move the point Q along the graph towards the point P, then the secant line will move upwards toward a limiting position when the points P and Q coincide. The line in this limiting position is the tangent line to the graph at the point P. As we will see, the slope of this line evaluated at P is the derivative of the function f(x). Areas and Limits: The areas of figures with straight–line boundaries can be found by subdividing the figures into various rectangles and triangles and adding the areas of the constituent parts.
The slope m of the line between two points is simply the distance “up” divided by the distance “across”, or, equivalently, the slope m of the line between two points (x1, x2) and (y1, y2) is rise/run which is (y2 – y1)/(x2 – x1). 3. Power Function: Power (or exponential) functions are of the form f(x) = xn, where n is a real number. The graph of such function is a curve. For example: Degree n = 2 Degree n = 3 Degree n = 4 Degree n = 5 4. Polynomial Function: A polynomial function is expressible as a sum of several terms cxn, where c and n are numbers, n is 0 or positive.