Chapter 7:The Transcendental Functions WILEY Section 7.1 One-to-One Functions,Inverses Section 7.Exponential Growth and Decay Definition One-to-One a.Theorem Theorem:Inverse Functions b.Population Growth One-to-One Functions,Properties and Graphs e.Radioactive Decay d.Graphs of fand f d.Compound Interest e.Continuity and Diflerentiability of Inverses e.Rule of 72 f.Theorem,Inverses Section 7.2 The Logarithm,Part I Arc Sine a.The Logarithm Function b.Properties e.The Number e Graph of the Logarithm Function f.Arc Tangent Derivatives Section 7.3 The Logarithm Function,Part II Section 7.8 The Hyperbolie Sine and Cosine c Integration of Trigonometric Functions a.Hyperbolic Sine and Hyperbolic Cosine b.The Graphs Section 7.4 The Exponential Function e.Identities & Properties 1-3 Section 7.9 The Other Hyperbolic Functions C. Properties 4-6 a.Hyperbolic Tangent.Hyperbolic Cotangent Hyperbolic Secant. d. Theorem Hyperbolic Cosecant e.The Derivative and Integral b.Hyperbolic Inverses Section 7.5 Arbitrary Powers:Other Bases a Properties b.Base p:The Function Ax)=p* e.Base p:The Function Ax)=logx 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Section 7.1 One-to-One Functions; Inverses a. Definition; One-to-One b. Theorem; Inverse Functions c. One-to-One Functions; Properties and Graphs d. Graphs of f and f -1 e. Continuity and Differentiability of Inverses f. Theorem; Inverses Section 7.2 The Logarithm, Part I a. The Logarithm Function b. Properties c. The Number e d. Natural Log Function e. Graph of the Logarithm Function Section 7.3 The Logarithm Function, Part II a. Differentiation and Graphing b. Integration c. Integration of Trigonometric Functions Section 7.4 The Exponential Function a. Definition b. Properties 1-3 c. Properties 4-6 d. Theorem e. The Derivative and Integral Section 7.5 Arbitrary Powers; Other Bases a. Properties b. Base p: The Function f(x) = p x c. Base p: The Function f(x) = logp x Chapter 7: The Transcendental Functions Section 7.6 Exponential Growth and Decay a. Theorem b. Population Growth c. Radioactive Decay d. Compound Interest e. Rule of 72 Section 7.7 Inverse Trigonometric functions a. Arc Sine b. Arc Sine; Inverses c. Arc Sine; Integrals d. Arc Tangent e. Arc Tangent: Inverses f. Arc Tangent; Derivatives g. Arc Cosine, Arc Cotangent, Arc Secant, Arc Cosecant h. Relations to 1/2π and Derivatives Section 7.8 The Hyperbolic Sine and Cosine a. Hyperbolic Sine and Hyperbolic Cosine b. The Graphs c. Identities Section 7.9 The Other Hyperbolic Functions a. Hyperbolic Tangent, Hyperbolic Cotangent, Hyperbolic Secant, Hyperbolic Cosecant b. Hyperbolic Inverses
One-to-One Functions:Inverses DEFINITION 7.1.1 A function f is said to be one-to-one if there are no two distinct numbers in the domain of f at which f takes on the same value. f()=f(x2) implies xI=x2. Main Menu C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. One-to-One Functions: Inverses
f is one-to-one f is not one-to-one:f(x)=f(x2) Figure 7.1.2 Figure 7.1.1 Main Meny r
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
One-to-One Functions:Inverses Inverses THEOREM 7.1.2 If f is a one-to-one function,then there is one and only one function g defined on the range of f that satisfies the equation f(g(x))=x for all x in the range of f. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. One-to-One Functions: Inverses Inverses
DEFINITION 7.1.3 INVERSE FUNCTION Let fbeaon-to-oe fuction.The ierse ofdenoted byis the unique function defined on the range of fthat satisfies the equion f(f(x)=x for all x in the range of f. (Figure 7.1.3) Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
f1() x =f(f-1(x)) i Figure7.1.3 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
One-to-One Functions:Inverses Figure 7.1.4 Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. One-to-One Functions: Inverses
One-to-One Functions:Inverses Example 1 You have seen that the cubing function f(x)=x3 is one-to-one.Find the inverse Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. One-to-One Functions: Inverses Example 1 You have seen that the cubing function f (x) = x 3 is one-to-one. Find the inverse
One-to-One Functions:Inverses Example 2 Show that the linear function y=3x-5 is one-to-one. Find the inverse. Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. One-to-One Functions: Inverses Example 2 Show that the linear function y = 3x – 5 is one-to-one. Find the inverse
One-to-One Functions:Inverses 5/3 Figure 7.1.5 Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. One-to-One Functions: Inverses