8.1 Exponential Functions & Differentiation “The biggest challenge in life is being yourself… In a world trying to make you like everyone else.” CoolNSmart.com Warm – up: Graph y = 2ex. Identify each: Domain, Range, asymptote(s), intercept(s).

Review and Refresh How do you determine the slope of a function at a point, using Calculus? Describe the chain rule.

The derivative of y = ex The slope at (0, 1) is 1. The slope at (1, e) is e. m = 1 m ≈ 2.718

New Derivative Rules Derivative of Natural Base Function: Let u be a differentiable function of x. 1. 2.

Example 1 Find the slope of the tangent line to y = e-2x at the point (0, 1). (0, 1)

Example 2 Find the derivative of each function: (a) (b) (c)

Example 2 Continued… Practice with a partner: (d) (e)

Example 3 Use implicit differentiation to find dy/dx.

See if you can example…. Find the second derivative of the function:

Exit Pass Write your name and find the derivative of each function: 1. 2. HW: 8.1 Day 1 Assignment

8.1 Day 2 Curve Sketching & Optimization WARM – UP: Determine the second derivative of: A SPECIAL friend is hard to FIND, Hard to LOSE And impossible to FORGET. TRUE friends are never APART maybe in DISTANCE BUT NOT IN THE HEART!! CoolNSmart.com

Review of the day Answer these independently and then we will discuss: How do you determine relative extrema? How do you determine points of inflection, and what are they?

Graph me!! Example 1 Find the extrema and the points of inflection (if any exist) of the function.

Try one with a partner: Find extrema (justify), points of inflection (justify):

Last Example / Exit Pass? The value V of an item t years after it is purchased is given by Find the rates of change of V with respect to t when t = 1 and t = 5.