By Samuel Horelick
This textbook is written for everybody who has skilled demanding situations studying Calculus. This e-book rather teaches you, is helping you recognize and grasp Calculus via transparent and significant causes of all of the rules, innovations, difficulties and approaches of Calculus, potent challenge fixing talents and techniques, totally labored issues of whole, step by step motives.
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Extra info for Calculus Fundamentals Explained
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.