Aim: Other Bases Course: Calculus Do Now: Aim: How do we differentiate and integrate the exponential function with other bases? Find the extrema for w/o calculator

Aim: Other Bases Course: Calculus Definition of Expo 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. Familiar properties

Aim: Other Bases Course: Calculus Definition of Log Function to Base a If a is a positive real number (a  1) and x is any real number, then the logarithmic function to the base a is denoted by log a x and is defined by Familiar properties

Aim: Other Bases Course: Calculus Properties of Inverses Solve for x:

Aim: Other Bases Course: Calculus More Rules Derivatives for Bases Other than e

Aim: Other Bases Course: Calculus Model Problems Find the derivative of each: 1. y = 2 x 2. y = 2 3x 3. y = log 10 cosx

Aim: Other Bases Course: Calculus Opt 1 Integrating non-e Expo Functions 1.Convert to base e using 2. Integrate directly using Two Options u = (ln2)x du = (ln2)dx Opt 2

Aim: Other Bases Course: Calculus Exponential Growth: Interest Formulas Exponential growth Compound Interest y = a b x Exponential function Exponential growth in general terms y = A(1 + r) t e n  n   Exponential growth Continuous compounding Continuous growth/decay k is a constant (±) P = amount of deposit, t = number of years, A = balance after t years, r = annual interest rate, n = number of compoundings per year.

Aim: Other Bases Course: Calculus Model Problem 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 compounding a) quarterly, b) monthly, and c) continuously

Aim: Other Bases Course: Calculus Model Problem 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) t = 0, b) t = 1,

Aim: Other Bases Course: Calculus Model Problem 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? c) t = 10, d) t  ,

Aim: Other Bases Course: Calculus The Product Rule