Chapter 5 Logarithmic Exponential and Other Transcendental Functions

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

Chapter 5 n Section 5: Bases other than e and Applications n MA. C. 2. 3 2000 Find the derivatives of functions, including algebraic, trigonometric, logarithmic, and exponential functions.

Base a n

Log base a n

Inverse Functions n n

Other Bases n n n Bases are used to solve: Earthquake measurement Half-life Depreciation Interest Rates And many more

Half Life n

Depreciation n Depreciation works in the same way. q q n It is estimated that a computer depreciates to one half its purchase price in one year. Thus the model is (½)t/1. In 3 years, the computer will be work (½)3 or 1/8 of its original cost. Problem: The time in which a machine depreciates to one-half its purchase price is 2 years. Find a model which yields the fraction of the purchase price as a function of time and determine that fraction after 5 years.

Annual Interest Rates n

Limit Involving e n

Comparing Rates n The government decides to give $1000 to students for graduating from high school. You decide to save the money for four years in case you cannot find a job immediately. The Bank of Newton is offering the following savings account options. In which account should you invest your money? q q Savings account at 5. 21% compounded annually CD at 5. 12% compounded continuously Money Market at 5. 13% compounded daily Checking account at 4. 98% compounded monthly

Differentiation and Integration n (1) You can use the definitions of ax and loga x as well as the rules already learned for natural log and exponential functions to differentiate, (2) use logarithmic differentiation (3) use the differentiation rules for bases other than e n

Differentiation and Integration n Find the derivative of: q y = 23 x q y = log 5 (sin x) q y = x(7 -3 x) n Tractor example: Derivative is the rate of change of the cost. Integral tells you how much money you have “lost” in a time frame.

Differentiation and Integration n

Comparing Constants and Variables n

Homework n n n Page 368 -370 3 -33 every three, 97, 101 42 -54, 64, 75 -87 every three, 95, 107