vertical
DESCRIPTION
Sec 4.4: Curve Sketching. Horizontal. Asymptotes. Vertical. 1)Vertical Asymptotes 2) Horizontal Asymptotes 3) Slant Asymptotes. Slant or Oblique. called a slant asymptote because the vertical distance between the curve and the line approaches 0. - PowerPoint PPT PresentationTRANSCRIPT
Vertical1)Vertical Asymptotes2) Horizontal Asymptotes3) Slant Asymptotes
Asymptotes
1)(
2
3
x
xxf
Horizontal
)(lim
),(lim
xf
xf
study
x
x
Slant or Oblique
0)()(lim
bmxxfstudyx
called a slant asymptote because the vertical distance between the curve and the line approaches 0.
For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following
:Example
Sec 4.4: Curve Sketching
1)(
2
3
x
xxf
Slant or Oblique
0)()(lim
bmxxfstudyx
called a slant asymptote because the vertical distance between the curve and the line approaches 0
For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following
:Example
Sec 4.4: Curve Sketching
3
1)Vertical Asymptotes2) Horizontal Asymptotes3) Slant Asymptotes
Asymptotes
Degree Example Horizontal Slant
Deg(num)<Deg(den)
Deg(num)=Deg(den)
Deg(num)=Deg(den)+1
0y NO
n
n
x
xy
of coeff
of coeff NO
NO
Special Case: (Rational function) Horizontal or Slant
onLongDivisi
4
2
4
1
x
xy
2
2
24
31
x
xy
1
12
3
xx
xy
Horizontal
Sec 4.4: Curve Sketching
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Sec 4.4: Curve Sketching
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Sec 4.4: Curve Sketching
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Sec 4.4: Curve Sketching
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Sec 4.4: Curve Sketching
1)(
2
3
x
xxf
Slant or Oblique
0)()(lim
bmxxfstudyx
called a slant asymptote because the vertical distance between the curve and the line approaches 0
For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following
:Example
xexf x )(
:Example
Sec 4.4: Curve Sketching
9
F101
Sec 4.4: Curve Sketching
A. InterceptsB. Asymptotes
SKETCHING A RATIONAL FUNCTION
)2()2(3
)4()(
2
2
xx
xxxf
:Example
Sec 4.4: Curve Sketching
A. DomainB. InterceptsC. SymmetryD. AsymptotesE. Intervals of Increase or DecreaseF. Local Maximum and Minimum ValuesG. Concavity and Points of InflectionH. Sketch the Curve
GUIDELINES FOR SKETCHING A CURVE
Symmetry
)()( :functioneven xfxf
)()( :function odd xfxf
symmetric aboutthe y-axis
symmetric aboutthe origin
Sec 4.4: Curve Sketching
12
Example
1
22
2
x
xy
A. DomainB. InterceptsC. SymmetryD. AsymptotesE. Intervals of Increase or DecreaseF. Local Maximum and Minimum ValuesG. Concavity and Points of InflectionH. Sketch the Curve
A. Domain: R-{1,-1}B. Intercepts : x=0C. Symmetry: y-axisD. Asymptotes: V:x=1,-1 H:y=2E. Intervals of Increase or Decrease: inc (-
inf,-1) and (-1,0) dec (0,1) and (1,-inf)F. Local Maximum and Minimum Values:
max at (0,0)G. Concavity and Points of Inflection down
in (-1,1) UP in (-inf,-1) and (1,inf)H. Sketch the Curve
Sec 4.4: Curve Sketching
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Sec 4.4: Curve Sketching
Easy to sketch:
2)( xxf
xxf )(
21 xy
2410 xy
Study the limit at inf
criticals all find 1)
lim lim :study 2)xx
asymptotes verticalall find 3)
Sec 4.4: Curve Sketching
Study the limit at inf
criticals all find 1)
lim lim :study 2)xx
asymptotes verticalall find 3)
Sec 4.4: Curve Sketching
Study the limit at inf
degeven poly with 1)
or )(lim x
xf
deg oddpoly with 2)
second the is one)(lim x
xf
Sec 4.4: Curve Sketching
F083
Sec 4.4: Curve Sketching
18
F091
Sec 4.4: Curve Sketching
19
F091
Sec 4.4: Curve Sketching