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.
x
ln[ x ] 
1
 t dt ,
x0
1
• The number e is the solution of the equation ln[x] = 1.
• The derivative of ln[x] is computed using the FTC.
d
1
ln x 
dx
x
• 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.
•
1
 x dx
 ln x  C 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
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•
•
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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.