Welcome to Week 3 College Trigonometry

Exponentials Functions that contain exponents: f (x) = x c x is the base c is the exponent

Exponentials IN-CLASS PROBLEMS

Exponentials Exponential functions: f (x) = c x c is the base x is the exponent

Exponentials Limitations: the base must be a constant the base must be >0 the exponent must be a variable

Functions containing an exponent or exponential functions? f (w) = w 9 f (y) = 72 y f (x) = x f (x) = x 50 f (z) = (8z+1) z f (x) = (x 2 +4)6 x Exponentials IN-CLASS PROBLEMS

Exponentials Graphs of Exponential Functions

Exponentials If the base is > 1, then the graph goes up to the right If 0<base<1, then the graph goes up to the left

Exponentials All pass through the point (0,1) because a 0 =1

Exponentials The domain (x s ) is (-∞,∞) The range (y s ) is (0,∞)

Questions?

Exponentials

Iteration Means repeating a process until you reach a goal

Iteration In math class, we are trying to solve an equation by closer and closer approximations to the “true” answer

Iteration So, we can manually tweak values of the coefficients until we get the best fit to our data This is actually how the computer calculates trig functions!

Iteration When we did the West Africa rainfall model, we tried different values for each of the constants in the equation until we got a “fit” we liked

Iteration That was iteration!

Iteration Iteration is used by your calculator and in computer software to do calculations

Iteration Some people think their calculators have a huge look-up table for all the answers

Iteration They don’t – it would be impossible for a calculator to have a look-up table for every possible value a user might enter for every possible function the user might want to use!

Iteration The calculator has a formula for every function that will zero in on the right answer after a short number of calculations

Iteration The calculator company wants a formula that zeros in on a stable number after the fewest number of calculations

Which is the better formula? Iteration IN-CLASS PROBLEMS

Iteration The one that gets there quickest!

Iteration IN-CLASS PROBLEMS

Iteration All the functions in your calculator are based on a long series that iterates back and forth and zeros in on the correct answer

Iteration

The formula selected by your calculator manufacturer determines how fast and accurate your calculator will be

Iteration It’s why different people get slightly different answers to calculator problems

Exponentials Irrational number e ≈

Exponentials The natural number e was first calculated by the Swiss Mathematician Jacob Bernoulli He called it “b”

Exponentials But it’s also called “Euler’s Number” after the Swiss Mathematician Leonhard Euler – he first called it “e”

Use a calculator to find: e 2.3 e 3.4 e e e 7 Exponentials IN-CLASS PROBLEMS

Exponentials e is used a lot in population models, compound interest, things that are growing "exponentially"

Grey Wolf Population f(x) = 1.85x x vs f(x) = 1.26e 0.264x Which is the better model? Exponentials IN-CLASS PROBLEMS

Questions?

Logarithms Logarithms were invented by Scottish Baron John Napier in 1614

Logarithms AND by Swiss craftsman Joost Bürgi in 1620

Logarithms Napier defined logarithms as the algebraic ratio of two distances in a geometric form while Bürgi’s definition was purely geometric

Logarithms Now we use them totally differently: as an exponent

Logarithms Logarithmic Functions y = log b x is the same as b y = x if x>0, b>0 and b≠1

logarithmic function with base b ~or~ exponential function with base b Logarithms

log to what power gives 16? Logarithms

log 2 16 = 4 because 2 4 = 16 Logarithms

Exponentials IN-CLASS PROBLEMS

Logarithms The default logarithm has base 10: log 10 x = log x

Logarithms Some definitions: log b 1 = 0log 1 = 0 log b b = 1log 10 = 1 log b b x = xlog 10 x = x b logb x = x10 logx = x

Logarithms Logarithms are the inverse of exponential functions

Write in exponential form: 4 = log = log = log 9 x3 = log x 27 log = y Exponentials IN-CLASS PROBLEMS

Write in logarithmic form: 2 3 = = = x8 y = = 1/125 Exponentials IN-CLASS PROBLEMS

Evaluate (w/o calc): log 4 16log 7 49 log 6 1/6log Exponentials IN-CLASS PROBLEMS

Logarithms Natural logarithms: logs with base e written "ln"

Logarithms log b 1 = 0ln 1 = 0 log b b = 1ln e = 1 log b b x = xln e x = x b logb x = xe lnx = x

Use a calculator to find: ln (7)ln(2.72) ln(-3)ln(896.5) ln e 7 e ln 125 Exponentials IN-CLASS PROBLEMS

Logarithms vs e Exponents Graph demo

Percentage of college graduates Exponentials IN-CLASS PROBLEMS

Change of temperature in an enclosed vehicle (∆ means “change”) Exponentials IN-CLASS PROBLEMS

Liberation! Be sure to turn in your assignments from last week to me before you leave Don’t forget your homework due next week! Have a great rest of the week!