A Summary of Curve Sketching Lesson 4.6 How It Was Done BC (Before Calculators) • How can knowledge of a function and it's derivative help.

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A Summary of Curve
Sketching
Lesson 4.6
How It Was Done BC
(Before Calculators)
• How can knowledge of a function and it's
derivative help graph the function?
Regis might be calling
for this information!
• How much can you tell about the graph of a
function without using your calculator's
graphing?
Algorithm for Curve Sketching
• Determine domain, range of the function
• Determine critical points
 Places where f ‘(x) = 0
• Plot these points on f(x)
• Use second derivative f’’(x) = 0
 Determine concavity, inflection points
• Use x = 0 (y intercept)
• Find f(x) = 0 (x intercepts)
• Sketch
Recall … Rational Functions
an x n  ...
m
bm x  ...
• Leading terms dominate



m = n => limit = an/bm
m > n => limit = 0
m < n => asymptote linear diagonal
or higher power polynomial
Finding Other Asymptotes
• Use PropFrac to get
r
y  m( x)  b 
d ( x)
• If power of numerator is larger by two
 result of PropFrac is quadratic
 asymptote is a parabola
Example
• Consider
• Propfrac gives
x  2x  7x
3
2
x  5 x  3x  3
5
4
Example
• Note the
parabolic asymptote
Other Kinds of Functions
• Logistic functions
• Radical functions
• Trig functions
10
h( x ) 
2  3e  x / 2
y  x  16  x 2
1
f ( x)  cos x  cos 2 x
2
Assignment
• Lesson 4.6
• Page 255
• Exercises 1 – 61 EOO

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