Chapter 5 Logarithmic, Exponential, and Transcendental Functions Section 5.1 The Natural Logarithmic Function: Differentiation Objective: In this lesson you learned the properties of the natural logarithmic function and how to find the derivative of the natural logarithmic function. Course Number Instructor Date Important Vocabulary Define each term or concept. Natural logarithmic function The natural logarithmic function is defined by ln x = x 1/t dt, x > 0. 1 e The letter e denotes the positive real number such that e ln e = 1/t dt = 1. 1 The Natural Logarithmic Function (Pages The domain of the natural logarithmic function is the set of all positive real numbers . The value of ln x is positive for x > 1 and negative for 0 < x < 1 . Moreover, ln (1) = 0 , because the upper and lower limits of integration are equal when x = 1 . The natural logarithmic function has the following properties: The domain is (0, and the range is The function is continuous, increasing, and one-to-one. The graph is concave downward. If a and b are positive numbers and n is rational, then the following properties are true: 1. ln (1) = 0 . ln (ab) = ln a + ln b . ln(an) = n ln a . What you should learn How to develop and use properties of the natural logarithmic function Larson/Edwards Calculus 9e Notetaking Guide IAE Copyright © Cengage Learning. All rights reserved. 95
ln a = ln a ln b . Example 1: Expand the logarithmic expression ln ln x + 4 ln y ln 2 xy 4 . 2 The Number e (Page 327) The base for the natural logarithm is defined using the fact that the natural logarithmic function is continuous, is one-to-one, and has a range of So, there must a unique real number x such that ln x = 1 . This number is denoted by the letter e , which has the decimal approximation 2.71828182846 . The Derivative of the Natural Logarithmic Function (Pages Let u be a differentiable function of x. Complete the following rules of differentiation for the natural logarithmic function: d [ln x] = 1/x , x > 0 dx d [ln u] = 1/u [du/dx] = , u > 0 dx What you should How to understand the definition of the number e What you should learn How to find derivatives of functions involving the natural logarithmic function Example 2: Find the derivative of x + 2x lnx f (x) = x2 ln x . If u is a differentiable function of x such that u 0 , then d [ln u ] = . In words, functions of dx the form y = ln u can be differentiated as if the absolute value signs were not present . Homework Assignment Page(s) Exercises trial
Section 5.2 The Natural Logarithmic Function: Integration 97 Section 5.2 The Natural Logarithmic Function: Integration Objective: In this lesson you learned how to find the antiderivative of the natural logarithmic function. Course Number Instructor Date Log Rule for Integration Let u be a differentiable function of x. 1 dx = ln| x | + C x 1 u dx = u du = ln| u | + C What you should learn How to use the Log Rule for Integration to integrate a rational function Example 1: Find 1 dx . x x ln| x | + C Example 2: Find 2 3 x3 dx . (1/3) ln|3 x3| + C Example 3: Find x2 4x + 1 x dx . (1/2)x2 4x + ln| x | + C If a rational function has a numerator of degree greater than or equal to that of the denominator , division may reveal a form to which you can apply the Log Rule.
Solve for x as a function of y: x = g(y) = f Interchange x and y. The resulting equation is y = f Define the domain of f to be the of f. Verify that f(f = x and f = x. Example 3: Find the inverse (if it exists) of f = 0.25x + 1.25 f (x) = 4x 5 . Derivative of an Inverse Function (Pages Let f be a function whose domain is an interval I. If f has an inverse function, then the following statements are true. If f is continuous on its domain, then f is continuous on its domain. If f is increasing on its domain, then f is increasing on its domain. If f is decreasing on its domain, then f is decreasing on its domain. If f is differentiable on an interval containing c and f 0, then f is differentiable at f(c). Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f 0 . Moreover, What you should learn How to find the derivative of an inverse function = 1 , f f 0 . This last theorem can be interpreted to mean that graphs of inverse functions have reciprocal slopes .
Section 5.5 Bases Other Than e and Applications Objective: In this lesson you learned the properties, derivatives, and antiderivatives of logarithmic and exponential functions that have bases other than e. Other than e (Pages Course Number Instructor Date What you should learn How to define If a is a positive real number (a 1) and x is any real number, exponential functions that then the exponential function to the base a is denoted by ax and is defined by ax = e(ln a)x . If a = 1, then y = 1x = 1 is a constant function . In a situation of radioactive decay, half-life is the number of years required for half of the atoms in a sample of radioactive material to decay . have bases other than e If a is a positive real number (a 1) and x is any positive real number, then the logarithmic function to the base a is denoted by loga x and is defined by loga x = 1/(ln a) ln x . Complete the following properties of logarithmic functions to the base a. loga 1 = 0 loga (xy) = loga x + loga y xn = n loga x x loga y = loga x loga y State the Properties of Inverse Functions y = ax if and only if x = loga y aloga x = x for x > 0 loga ax = x, for all x
The logarithmic function to the base 10 is called the common logarithmic function . Example 1: (a) Solve log8 x = 1 3 for x. (b) Solve 5 x = 0.04 for x. (a) x = 2 (b) x = 2 Differentiation and Integration To differentiate exponential and logarithmic functions to bases, you have three options: Use the definitions of ax and loga x and differentiate the rules for the natural exponential and logarithmic functions. Use logarithmic differentiation, or Use the differentiation rules for bases other than e. What you should How to differentiate and integrate exponential functions that have bases other than e Let a be a positive real number (a 1) and let u be a differentiable function of x. Complete the following formulas for the derivatives for bases other than e. d = (ln a) ax . dx d = (ln a) au [du/dx] . dx d [log x] = 1 / [(ln a) x ] . dx a d [log u] = 1 / [(ln a) u ] [du/dx] . dx a Occasionally, an integrand involves an exponential function to a base other than e. When this occurs, there are two options: (1) convert to base e the formula ax = e(ln a)x and then integrate or (2) integrate directly using the integration formula dx = (1/ln a) ax + C .
Let n be any real number and let u be a differentiable function of x. The Power Rule for Real Exponents gives. d [xn ] = . dx d [un ] = [du/dx] . dx Applications of Exponential Functions (Pages Complete the following limit statement: x x What you should How to use exponential functions to model compound interest and lim + 1 = lim x + 1 = e . exponential growth x x Let P be the amount deposited, t the number of years, A the balance after t years, and r the annual interest rate (in decimal form), and n the number of compounding per year. Complete the following compound interest formulas: Compounded n per year: A = P(1 + r/n)nt Compounded continuously: A = Pert Example 2: Find the amount in an account after 10 years if $6000 is invested at an interest rate of 7%, compounded monthly. compounded continuously. (a) $12,057.97 (b) $12,082.52
Section 5.6 Inverse Trigonometric Functions: Differentiation 109 Section 5.6 Inverse Trigonometric Functions: Differentiation Objective: In this lesson you learned about the properties of inverse trigonometric functions and how to find derivatives of inverse trigonometric functions. Course Number Instructor Date Inverse Trigonometric Functions None of the six basic trigonometric functions has an inverse function . This is true because all six trigonometric functions are periodic and therefore are not one-to-one . However, their domains can be redefined in such a way that they will have inverse functions on the restricted domains . For each of the following definitions of inverse trigonometric functions, supply the required restricted domains and ranges. What you should learn How to develop properties of the six inverse trigonometric functions An alternative notation for the inverse sine function is x . Example 1: Evaluate the function: Example 2: Evaluate the function: arcsin . arccos 1 . 2 Example 3: Evaluate the function: arcos (0.85). 0.5548
the Inverse Property for the Sine function. If 1 x 1 and y then sin(arcsin x) = x and arcsin(sin y) = y. the Inverse Property for the Cosine function. If 1 x 1 and 0 y then cos(arccos x) = x and arccos(cos y) = y. State the Inverse Property for the Tangent function. If x is a real number and < y < then tan(arctan x) = x and arctan(tan y) = y. Derivatives of Inverse Trigonometric Functions (Pages Let u be a differentiable function of x. What you should learn How to differentiate an inverse trigonometric function d [arcsin u] = dx d [arccos u] = dx d [arctan u] = dx d [arc cot u] = dx d [arc secu] = 1 u2 u2 1 + u2 1 + u2 dx d [arc cscu] = dx | u | | u | u2 1 u2 1 Homework Assignment
Section 5.7 Inverse Trigonometric Functions: Integration 111 Section 5.7 Inverse Trigonometric Functions: Integration Objective: In this lesson you learned how to find antiderivatives of inverse trigonometric functions. Course Number Instructor Date Integrals Involving Inverse Trigonometric Functions (Pages Let u be a differentiable function of x, and let a > 0. du What you should learn How to integrate functions whose antiderivatives involve inverse trigonometric functions a2 u2 = arcsin (u/a) + C . du a2 + u2 = (1/a) arctan (u/a) + C . du u u2 a2 Example 1: = (1/a) arcsec (| u | /a) + C . 6x dx 4 + 9x4 ½ arctan(3x2/2) + C Completing the Square (Pages Completing the square helps when quadratic functions are involved in the integrand . Example 2: Complete the square for the polynomial: x2 + 6x + 3 . (x + 3)2 6 Example 3: Complete the square for the 2x2 + 16x . 2[(x + 4)2 16] What you should How to use the method of completing the square to integrate a function
Review of Basic Integration Rules (Pages Complete the following selected basic integration rules. 1 u dx = u du = ln| u | + C What you should learn How to review the basic integration rules involving elementary functions = u + C u du = ln|sin u| + C du a2 + u2 = (1/a) arctan (u/a) + C u du = sin u + C u du = tan x + C Homework Assignment Page(s) Exercises
Section 5.8 Hyperbolic Functions Objective: In this lesson you learned about the properties of hyperbolic functions and how to find derivatives and antiderivatives of hyperbolic functions. Hyperbolic Functions (Pages Complete the following definitions of the hyperbolic functions. Course Number Instructor Date What you should learn How to develop properties of hyperbolic functions tanh x = (sinh x) / (cosh x) . csch x = 1 / (sinh x), x 0 . sech x = 1 / (cosh x) . coth x = 1 / (tanh x), x 0 . Complete the following hyperbolic identities. cosh2 x sinh2 x = 1 . tanh2 x + sech2 x = 1 . coth2 x csch2 x = 1 . + cosh 2 x 2 = sinh2 x . 1 + cosh 2 x 2 = cosh2 x . sinh x cosh x = sinh 2x . cosh2 x + sinh2 x = cosh 2x . sinh (x + y)= sinh x cosh y + cosh x sinh y . sinh (x y)= sinh x cosh y cosh x sinh y . cosh (x + y)= cosh x cosh y + sinh x sinh y . cosh (x y)= cosh x cosh y sinh x sinh y .
Inverse Hyperbolic Functions (Pages the inverse hyperbolic function given by each of the following definitions and give the domain for each. Domain What you should How to develop properties of inverse hyperbolic functions ln (x + ln (x + x2 + 1) = x , . x2 = x , [1, . ln 1 + x 1 x = x , 1) . 1 ln x + 1 = x , (1, . 2 x ln 1 + 1 x2 x = x , (0, 1] . 1 1 + x2 ln + = x , 0) (0, . x x Differentiation and Integration of Inverse Hyperbolic Functions (Pages Let u be a differentiable function of x. Complete each of the following rules of differentiation and integration. What you should learn How to differentiate and integrate functions involving inverse hyperbolic functions d [ u ] = dx u2 + 1 d [ u ] = dx u2 d [ u ] = dx 1 u2
d [ u ] = dx u 1 + u2 du u2 ± a2 = ln(u + ± a2) + C . du a2 u2 du = 1 / (2a) ln| (a + u) / (a u) | + C . ln( (a + ± u2) / |u|) + C . u a2 ± u2 = Homework Assignment Page(s) Exercises