Section 3.6 A Summary of Curve Sketching AP Calculus BC

Warm-up:

Warm-up:

Warm-up -- Solution:

Learning Objective: 1.) Analyze and sketch the graph of a function.

Tools/concepts for curve sketching All the different components of a graph that would help us determine its behavior are listed below. Some of these will be names of specific points on a graph, while some will be graph qualities or characteristics. x- and y-intercepts vertical & horizontal asymptotes relative and absolute max/min domain & range increasing & decreasing intervals continuity points of inflection differentiability concavity symmetry (even/odd functions)  

Steps for Curve Sketching: Use Algebra – Find Asymptotes (use this to determine the domain/range) Find x-intercepts Find y-intercepts Use Calculus Find 𝑓 ′ (𝑥) and critical value(s) Find where 𝑓 𝑥 is increasing/decreasing Find 𝑓 ′′ (𝑥) and point(s) of inflection(s) Find where 𝑓 𝑥 is concave up/concave down

Example 1: Analyze and sketch the graph of: 𝑓 𝑥 = 2( 𝑥 2 −9) 𝑥 2 −4 Example on p. 207

Example 2: Analyze and sketch the graph of: 𝑓 𝑥 = 𝑥 𝑥 2 +2 Example on p. 207

Example 3: Analyze and sketch the graph of: 𝑓 𝑥 =2 𝑥 5 3 −5 𝑥 4 3 Example on p. 207

Example 4: Analyze and sketch the graph of: 𝑓 𝑥 = 𝑥 4 −12 𝑥 3 +48 𝑥 2 −64𝑥 **Note: A polynomial function of even degree must have at least one relative extrema. Example on p. 207

Example 5: Analyze and sketch the graph of: 𝑓 𝑥 = cos⁡(𝑥) 1+sin⁡(𝑥) . Example on p. 207

Homework: pg. 212-213: #1-4 all, 5-23 odd, 37, 38, 49, 51