1. Curve Sketching
Lesson 5.4

2. Motivation
Graphing calculators decrease the importance of curve sketching
So why a lesson on curve sketching?
A calculator graph may be misleading
What happens outside specified window?
Calculator plots, connects points without showing what happens between points
False asymptotes
Curve sketching is a good way to reinforce concepts of lessons in this chapter

3. Tools for Curve Sketching
Test for concavity
Test for increasing/decreasing functions
Critical points
Zeros
Maximums and Minimums

4. Strategy Determine domain of function
Find y-intercepts, x-intercepts (zeros)
Check for vertical, horizontal asymptotes
Determine values for f '(x) = 0, critical points
Determine f ''(x)
Gives inflection points
Test for intervals of concave up, down
Plot intercepts, critical points, inflection points
Connect points with smooth curve
Check sketch with graphing calculator

5. Using First, Second Derivatives
Note the four possibilities for a function to be …
Increasing or decreasing
Concave up or concave down
Positive (increasing function)
Negative (decreasing function)
Positive (concave up)
Negative (concave down)
f '(x)
f ''(x)

6. Try It Out
Find as much as you can about the function without graphing it on the calculator

7. Graphing Without the Formula
Consider a function of this description
Can you graph it?
This function is continuous for all reals
A y-intercept at (0, 2)

8. Assignment
Lesson 5.4
Page 354
Exercises 1 – 39 odd