5-4: Differentiation and Integration of Exponentials Objectives: Solve logarithmic and exponential equations Differentiate and integrate exponential functions ©2002 Roy L. Gover (

“Wrong, Williamson! Wrong Wrong Wrong! You explained the way the ball bounces, when I specifically asked you to show me the way the cookie crumbles!”

Definition Natural Exponential Function:the inverse of the natural log function. y=ln x y=exy=ex A working definition:

Example Write as a natural logarithm:

Example Write as a natural exponential :

Example Solve for x :

Warm-Up Solve for x :

Definition The derivative of the natural exponential function Chain Rule Version e x is its own derivative

Example Find the derivative:

Try This Find the derivative:

Definition Chain Rule Version

Example

Assignment /35-55 odd odd, 103, 105

Try This Find the derivative:

Try This Find the derivative:

Try This The first permanent colony in America was established in Jamestown, Virginia, in For 1610 through 1780, the population, p ( in thousands), in colonial America can be modeled by...

where t =10 represents the year At what rate was the population of colonial America changing in 1650? Include the correct units. Hint:Find p ’ (t) and evaluate at t = people per year

Definition Chain Rule Version

Example

Try This

Example Find the area bounded by the x -axis and the lines x =-1 and x =0, 0

Lesson Close For, what is: The domain? The range? Over what part of the domain is f ( x ) concave up? Do not graph.

Assignment /35-55 odd odd, 103, 105