Calculus and Analytic Geometry II Cloud County Community College Spring, 2011 Instructor: Timothy L. Warkentin

Chapter 07: Integrals and Transcendental Functions 7.1 The Logarithm Defined as an Integral 7.2 Exponential Change and Separable Differential Equations 7.3 Hyperbolic Functions 7.4 Relative Rates of Growth

Chapter 07 Overview This chapter formally develops the natural logarithmic function and its inverse, the exponential function, from basic definitions. There are two fundamental and mutually exclusive classes of mathematical functions: Algebraic and Transcendental. Algebraic functions are those that can be constructed using a finite number of Algebraic operations (addition, subtraction, multiplication, division and root extraction). Algebraic functions include polynomial, rational and power functions with rational exponents. Transcendental functions are constructed using an infinite number of Algebraic operations. Transcendental functions include the logarithmic, trigonometric, and hyperbolic functions along with their inverses.

07.01: The Logarithm Defined as an Integral 1 The basis for the study of logarithms and exponentials is the Definition of natural logarithms. Natural Logarithms are defined in terms of the following integral function. The number e is the solution of the equation ln[x] = 1. The derivative of ln[x] is computed using the FTC. The antiderivative of ln[x] is not found in this chapter.

07.01: The Logarithm Defined as an Integral 2 The domain, range and properties of the natural logarithm function. plugs the hole in the power law when n = -1. Example 1 Deriving the antiderivatives of tangent, cotangent, secant and cosecant.

07.01: The Logarithm Defined as an Integral 3 The inverse of ln[x] is ex since ln[ex]= x ln[e] = x. The natural exponential function is its own derivative/antiderivative. The domain, range and properties of the natural exponential function Defining all exponential functions in terms of the natural exponential function. (ax = e(ln a) x). Review: exponential derivatives/antiderivatives. Defining all logarithmic functions in terms of the natural logarithmic function. (logbx = ln[x]/ln[b] by the change of base formula). Review: logarithmic derivatives. Example 2

07.02: Exponential Change and Separable Differential Equations 1 When a quantity increases or deceases at a rate proportional to its current size it is said to be changing exponentially. Derivation of exponential change model. Separable Differential Equations Derivation of exponential change model. Examples 1 & 2 Contrasting y = y0 bt with y = y0 ekt. Exponential equations are solved using logarithms. Example 3 Extra Topic: Compound interest and doubling times. Half-life. Example 4 Newton’s Law of Cooling. Example 5

07.03: Hyperbolic Functions 1 Hyperbolic trigonometric functions are combinations of the ex and e-x functions that are particularly useful in solving differential equations. Hyperbolic functions behave similarly to ordinary trigonometric functions. The parameter u in each case is twice the area of the shaded segment in the following slide. Homework Problem 86

07.03: Hyperbolic Functions 2

07.03: Hyperbolic Functions 3 Hyperbolic function definitions. Hyperbolic function identities. Derivatives of hyperbolic functions. Example 1a Integrals of hyperbolic functions. Examples 1b, 1c, & 1d Inverse hyperbolic function definitions. Inverse hyperbolic function identities. Derivatives of inverse hyperbolic functions. Example 2 Integrals of inverse hyperbolic functions. Example 3

07.04: Relative Rates of Growth 1 This section is not covered.