p351 Section 5.5: Bases Other than e and Applications

Transcription

1 p351 Section 5.5: Bases Other than e and Applications Definition of Exponential Function to Base a If a is a positive real number (a 1) and x is any real number, then the exponential function to the base a is denoted by a x and is defined by If a = 1, then y = 1 x = 1 is a constant function These functions obey the usual laws of exponents. Here are some familiar properties: ** When modeling half life of a radioactive sample, it is convenient to use 1/2 as the base of the exponential model. Example 1: Radioactive Half Life Model The half life of carbon 14 is about 5730 years. If 1 gram of carbon 14 is present in a sample, how much will be present in 10,000 years? Let t = 0 represent the present time and let y represent the amount (in grams) of carbon 14 in the sample. Using a base of 1/2, you can model y by the equation **Do you remember the equation for exponential growth/decay? a = initial amount (1 gram) b = growth/decay factor (b = 1 + r) (1/2 for half life) Notice that when t = 5730, the amount is reduced to half the original amount When t = 11,460, the amount is reduced to a quarter of the original amount, and so on. To find the amount of carbon 14 after 10,000 years, substitute 10,000 for t. 1

2 Evaluate the expression without using a calculator Use the circle rule to convert each logarithm to an exponential expression #5. #7. Write the exponential equation as a logarithmic equation or vice versa #9. (a). (b). #11. (a). (b). 2

3 #1. The time in which a machine depreciates to one half its purchase price is given. Find a model that yields the fraction of the purchases price as a function of time and determine that fraction at time t 0. Depreciation time = 3 years t 0 = 6 years Definition of Logarithmic Function to Base a 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 log a x and is defined as (1). (2). (3). (4). Log of 1 Log of a Product Log of a Power Log of a Quotient 3

4 and are inverse functions of each other. Properties of Inverse Functions (1). if and only if (2). (3). for x > 0 for all x The logarithmic function to the base 10 is called the common logarithmic function. For common logs, if and only if Example 2: Bases Other than e Solve for x in each equation (a). log both sides power property (b). Change to an exponential by the circle rule 4

5 Solve for x or b #23. (a). (b). #24. (a). (b). 5

6 Solve the equation accurate to three decimal places #29. You could simplify inside the parenthesis, but I'm not going to this time 6

7 Theorem 5.13: Derivatives for Bases Other than e Let a be a positive real number (a 1) and let u be a differentiable function of x (1). (2). (3). (4). Example 3: Differentiating Functions to Other Bases Find the derivative of each of the following: (a). (b). (c). 7

8 Sometimes an integrand involves an exponential function to a base other than e. When this occurs, there are two options: (1) convert to base e using the formula and then integrate, or (2) integrate directly, using the integration formula: Example 4: Integrating an Exponential Function to Another Base Theorem 5.14: The Power Rule for Real Exponents Let n be any real number and let u be a differentiable function of x (1). (2). 8

9 Example 5: Comparing Variables and Constants Each functions uses a different differentiation formula, depending on whether the base and exponent are constants or variables (a). (b). (c). Constant Rule Exponential Rule Power Rule (d). Logarithmic Differentiation log (LN) both sides Power Property Use Product Rule to find the derivative Solve for y' Substitute for y = 9

10 Some more examples of finding the derivative of the function #45. Use the Product Rule to find the derivative Factor out a GCF #51. Rewrite the logarithm using the quotient property Use the Power property #55. Rewrite the logarithm, changing it to a natural log Find the derivative using the quotient rule Pull t 2 to the front #59. Use logarithmic differentiation to find dy/dx LN both sides Power rule bring the exponent to the front Use the Product rule to find the derivative Solve the equation for dy/dx Substitute for y = 10

11 Find or evaluate the integral #61. #65. #67. 11

12 Applications of Exponential Functions Theorem 5.15: A Limit Involving e Summary of Compound Interest Formulas Let P = amount of deposit, t = number of years, A = balance after t years, r = annual interest rate (in decimal form), and n = number of compoundings per years. 1. Compounded n times per year: 2. Compounded continuosly: Example 6: Comparing Continuous and Quarterly Compounding A deposit of $2500 is made in an account that pays an annual interest rate of 5%. Find the balance in the account at the end of 5 years if the interest is compounded (a) quarterly, (b) monthly (c) continuously P = 2500 r = 0.05 t = 5 (a) Quarterly (n = 4): (b) Monthly (n = 12): (c) Continuously: 12

13 Example 7: Bacterial Culture Growth A bacterial culture is growing according to the logistics growth function where y is the weight of the culture in grams and t is the time in hours. Find the weight of the culture after (a) 0 hours, (b) 1 hour, and (c) 10 hours (d) What is the limit as t approaches infinity? (a). When t = 0: (b). When t = 1: (c). When t = 10: (d). Limit as t approaches infinity: y =

14 HW p357 #6, 10, 12, 22, 34, 44, 46, 52, 62, 66, 80 14