Curve Sketching
Lesson 5.4
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
2
Tools for Curve Sketching
Test for concavity
Test for increasing/decreasing functions
Critical points
Zeros
Maximums and Minimums
3
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 4
Using First, Second Derivatives
Note the four possibilities for a function to be … • Increasing or decreasing• Concave up or concave down
5
Positive(increasing function)
Negative (decreasing
function)
Positive (concave up)
Negative (concave
down)
f '(x)
f ''(x)
Try It Out
Find as much as you can about the function without graphing it on the calculator
6
3 215( ) 18 1
2f x x x x
2 1
xy
x
( ) lnf x x x
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)
7
'( ) 0 on (- , -6) and (1, 3)f x '( ) 0 on (-6, 1) and (3, )f x ''( ) 0 on (- , -6) and (3, )f x ''( ) < 0 on interval (-6, 3)f x
Assignment
Lesson 5.4
Page 354
Exercises 1 – 39 odd
8