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# Curve Sketching

Calculus Unit 4 Outline

by

Tweet## Andrea Hanke

on 3 April 2013#### Transcript of Curve Sketching

Calculus- Unit 4 Curve Sketching Find x & y

Intercepts x-Intercept -set y = 0

-solve for x y= x/(x^2-9)

0=x/(x^2-9)

0=x y-Intercept -set x = 0

-solve for y y= x/(x^2-9)

y=0/(〖(0)〗^2-9)

y=0/9

y=0 Check for

Discontinuities Vertical Asymptotes y= x/(x^2-9)

y=x/((x-3)(x+3))

va= ∓-3, +3 Holes y= (x-3)/(x^2-9)

y=(x-3)/(x-3)(x+3)

y=1/(x+3)

hole at x=+3 VA Limits 〖 lim of +3 〖behaviour=lim┬(x→->3-) 〗〖x/(x^2-9)〗

= (+ve)/(-ve)

= -ve

∴y→ -> - infinity 〖behaviour=lim┬(x→->3-) 〗〖x/(x^2-9)〗

= (+ve)/(-ve)

= -ve

∴y→ -> - infinity∞ lim of -3 Horizontal &

Oblique

Asymptotes Tester- Above & Below Critical Points Sign Table 1 Inflection Points Sign Table 2 Have Fun

& Graph y= x/(x^2-9) - divide through

by the highest

degree Horizontal Asymptotes Positive Infinity 〖=lim(x->infinity) (x/x^2 )/((x^2-9)/x^2 )

=lim(x->infinity) (1/x)/(1-(9/x^2 ))

= (0)/(1 -(0))

= 0/1

= 0

∞ behaviour =lim(x-3-) x/(x^2-9)

= (-ve)/(+ve)

= -ve

y -> - infinity 〖behaviour =lim┬(x->→-3+)〗 〖x/(x^2-9)〗

= (-ve)/(-ve)

= +ve

∴y→ -> +infinity∞ Negative Infinity =lim┬(x→->-infinity) (x/x^2 )/((x^2-9)/x^2 )

=lim┬(x→->-infinity) (1/x)/(1-9/x^2 ))

= 0/(1-(0))

=0/1

=0 Oblique Asymptotes x +5 R6

(x-3)/(x^2+2x-9)

-〖(x〗^2-3x)

-(0 +5x)

-(5x-15)

0 +6 y=(x^2+2x-9)/(x-3) HA tester =(x^2+2x-9)/(x-3) -0

=(x^2+2x-9)/(x-3) as x-> +infinity, y= above

tester=(x^2+2x-9)/(x-3)

=(+ve)+(+ve)/(+ve)

=+ve as x-> -infinity, y= below

tester =(x^2+2x-9)/(x-3)

=(+ve^2+-ve)(-ve)

=(+ve)/(-ve)

=-ve "The First Derivative Test" -take the first derviative

-set to zero

-solve for x

-x= max & min dy/dx=(1(x^2-9)-2x(x))/(〖(x〗^2-9)^2)

dy/dx= (x^2-9-2x^2)/((x^2-9)^2)

0=-x^2-9

x^2=-9

x=√(-9)^1/2

=DNE x<-3 -3<x<3 x>3

-x^2-9 - - -

x^2-9 + - +

x^2-9 + - +

dy/dx - - -

y "The Second Derivative Test" -take the second derivative

-set to zero

-solve for x

-x=inflection points x=-3 x=3 y=0 x<1/6 x>1/6

12x-2 - +

f''(x) - +

f'(x) concave

down concave

up Andrea

Hanke OA tester= remainder

= 6/(x-3) as x-> infinity, y= above

tester =6/(x-3)

= (+ve)/(+ve)

= +ve as x -> -infinity, y= below

tester= 6/(x-3)

= (+ve)/ (-ve)

= -ve -long divide By: No energy drinks were used in the creation of this prezi: Polynomial Rational f(x) = 3x^3-x^2+7

f'(x) = 6x^2-2x

f''(x) = 12x-2

0 = 12x-2

2/12 =x

1/6 =x f(x) = x/(x^2-9)

f'(x)= (x^2-9-2x^2)/(x^2-9)^2

=(-x^2-9)/(x^2-9)^2

f''(x)= (-2x(x^2-9)^2-(x^2-9)(2x)(-x^2-9))/(x^2-9)^2

0 = (-2(x^2-9)-2(2x)(-x^2-9))/(x^2-9)^3

= -2x^3+18x-4x(-x^2-9)

= -2x^3+18x+4x^3+36x

= 2x^3+54x

= 2x(x^2+27)

x = 0, DNE Polynomial Rational x<0 x>0

2x - +

(x^2+27) + +

(x^2-9)^3 - -

f''(x) + -

f'(x) concave

up concave

down

Full transcriptIntercepts x-Intercept -set y = 0

-solve for x y= x/(x^2-9)

0=x/(x^2-9)

0=x y-Intercept -set x = 0

-solve for y y= x/(x^2-9)

y=0/(〖(0)〗^2-9)

y=0/9

y=0 Check for

Discontinuities Vertical Asymptotes y= x/(x^2-9)

y=x/((x-3)(x+3))

va= ∓-3, +3 Holes y= (x-3)/(x^2-9)

y=(x-3)/(x-3)(x+3)

y=1/(x+3)

hole at x=+3 VA Limits 〖 lim of +3 〖behaviour=lim┬(x→->3-) 〗〖x/(x^2-9)〗

= (+ve)/(-ve)

= -ve

∴y→ -> - infinity 〖behaviour=lim┬(x→->3-) 〗〖x/(x^2-9)〗

= (+ve)/(-ve)

= -ve

∴y→ -> - infinity∞ lim of -3 Horizontal &

Oblique

Asymptotes Tester- Above & Below Critical Points Sign Table 1 Inflection Points Sign Table 2 Have Fun

& Graph y= x/(x^2-9) - divide through

by the highest

degree Horizontal Asymptotes Positive Infinity 〖=lim(x->infinity) (x/x^2 )/((x^2-9)/x^2 )

=lim(x->infinity) (1/x)/(1-(9/x^2 ))

= (0)/(1 -(0))

= 0/1

= 0

∞ behaviour =lim(x-3-) x/(x^2-9)

= (-ve)/(+ve)

= -ve

y -> - infinity 〖behaviour =lim┬(x->→-3+)〗 〖x/(x^2-9)〗

= (-ve)/(-ve)

= +ve

∴y→ -> +infinity∞ Negative Infinity =lim┬(x→->-infinity) (x/x^2 )/((x^2-9)/x^2 )

=lim┬(x→->-infinity) (1/x)/(1-9/x^2 ))

= 0/(1-(0))

=0/1

=0 Oblique Asymptotes x +5 R6

(x-3)/(x^2+2x-9)

-〖(x〗^2-3x)

-(0 +5x)

-(5x-15)

0 +6 y=(x^2+2x-9)/(x-3) HA tester =(x^2+2x-9)/(x-3) -0

=(x^2+2x-9)/(x-3) as x-> +infinity, y= above

tester=(x^2+2x-9)/(x-3)

=(+ve)+(+ve)/(+ve)

=+ve as x-> -infinity, y= below

tester =(x^2+2x-9)/(x-3)

=(+ve^2+-ve)(-ve)

=(+ve)/(-ve)

=-ve "The First Derivative Test" -take the first derviative

-set to zero

-solve for x

-x= max & min dy/dx=(1(x^2-9)-2x(x))/(〖(x〗^2-9)^2)

dy/dx= (x^2-9-2x^2)/((x^2-9)^2)

0=-x^2-9

x^2=-9

x=√(-9)^1/2

=DNE x<-3 -3<x<3 x>3

-x^2-9 - - -

x^2-9 + - +

x^2-9 + - +

dy/dx - - -

y "The Second Derivative Test" -take the second derivative

-set to zero

-solve for x

-x=inflection points x=-3 x=3 y=0 x<1/6 x>1/6

12x-2 - +

f''(x) - +

f'(x) concave

down concave

up Andrea

Hanke OA tester= remainder

= 6/(x-3) as x-> infinity, y= above

tester =6/(x-3)

= (+ve)/(+ve)

= +ve as x -> -infinity, y= below

tester= 6/(x-3)

= (+ve)/ (-ve)

= -ve -long divide By: No energy drinks were used in the creation of this prezi: Polynomial Rational f(x) = 3x^3-x^2+7

f'(x) = 6x^2-2x

f''(x) = 12x-2

0 = 12x-2

2/12 =x

1/6 =x f(x) = x/(x^2-9)

f'(x)= (x^2-9-2x^2)/(x^2-9)^2

=(-x^2-9)/(x^2-9)^2

f''(x)= (-2x(x^2-9)^2-(x^2-9)(2x)(-x^2-9))/(x^2-9)^2

0 = (-2(x^2-9)-2(2x)(-x^2-9))/(x^2-9)^3

= -2x^3+18x-4x(-x^2-9)

= -2x^3+18x+4x^3+36x

= 2x^3+54x

= 2x(x^2+27)

x = 0, DNE Polynomial Rational x<0 x>0

2x - +

(x^2+27) + +

(x^2-9)^3 - -

f''(x) + -

f'(x) concave

up concave

down