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introducing WT COUNCIL a tour of DERIVATIVES introducing WT COUNCIL This presentation demonstrates the new capabilities of PowerPoint and it is best viewed in Slide Show. These slides are designed to give you great ideas for the presentations you’ll create in PowerPoint 2011! For more sample templates, click the File menu, and then click New From Template. Under Templates, click Presentations.

Today’s Lesson Derivatives of e & Logarithmic Functions

RECALL Formula for CONTINUOUSLY COMPOUNDED Interest: To solve A = Pert problems, list the variables known to find the NEED to know. Natural Log is the INVERSE of exponential: Solving A = Pert for rate or time requires using the inverse of ex (use of ln)

Derivative of e The derivative of the exponential function is the exponential function.

Derivative of ln The derivative of the natural logarithm function is the inverse of the function times the derivative of the function.

Derivatives of Other Forms

EXAMPLES Find the derivative

EXAMPLES Find the derivative

EXAMPLES Find the derivative

EXAMPLES Find the derivative

EXAMPLES Find the derivative

EXAMPLES A single levitrus bacterium divides every 2 hours to produce two complete levitrae bacteria. If we start with a colony of 500 bacteria, then after t hours there will be: A(t) = 500(2)4t

EXAMPLES A single levitrus bacterium divides every 2 hours to produce two complete levitrae bacteria. If we start with a colony of 500 bacteria, then after t hours there will be: A(t) = 500(2)4t A’(1): The bacteria grow at a rate of 22,180.7 per hour. A’(5): The bacteria grow at a rate of 1,453,634,996 per 5 hour period.

EXAMPLES A mathematical model for the average of a state’s newspapers’ circulations is C(t) = 10 – 0.6lnt (t > 0); where C(t) is newspaper circulation (in millions) in year t (t = 0 corresponds to 1990). Use this model to estimate the circulation and the rate of change of circulation in 2015 and 2020. Round all numbers to the nearest million. Interpret these results.

EXAMPLES Using: C(25): After 25 years, the circulation for the state’s newspapers = 8,068,675 subscribers. C’(25): After 25 years, the rate of circulation for the state’s newspapers decreases at a rate of 24,000 subscribers per year. C(30): After 30 years, the circulation for the state’s newspapers = 7,959,282 subscribers. C’(30): After 30 years, the rate of circulation for the state’s newspapers decreases at a rate of 20,000 subscribers per year.

SUMMARY The derivative of ex is the same function ex The derivative of lnx is The derivative of: