3.6b Summary of Curve Sketching

Intercepts, symmetry, domain & range, continuity, asymptotes, differentiability, extrema, concavity,infinite

limits at infinity, WHEW!

Ex 4 p.211 Radical FunctionAnalyze and graph

1st Derivative:

2nd Derivative:

Domain:

x-intercepts:

y-intercept:

Vertical asymptote(s):

Horizontal asymptote(s):

Critical number(s):

Possible points of inflection:

Symmetry:

f x x x( ) 2 55

34

3

All realsf x x x' ( ) ( ) 10

3 21

31

3

f xx

x' ' ( )

( )

20 1

9

13

23

(0, 0), (125/8, 0)

(0, 0)

None

None

x = 0, x = 8

x = 0, x = 1

None

Ex 4 continued

f(x) f’(x) f”(x) Characteristics of Graph-∞ < x < 0 Pos Neg Increasing, concave downx = 0 0 0 Undef Relative maximum0 < x < 1 Neg Neg Decreasing, concave downx = 1 -3 Neg 0 Point of inflection1 < x < 8 Neg Pos Decreasing, concave upx = 8 -16 0 Pos Relative Minimum8 < x < ∞ Pos Pos Increasing, concave up

Ex 5 p.213 Polynomial FunctionAnalyze and graph

1st Derivative:

2nd Derivative:

Domain:

x-intercepts:

y-intercept:

Vertical asymptote(s):Horizontal asymptote(s):

Critical number(s):

Possible points of inflection:

Symmetry:

None

x = 1, x = 4

At x = 2, x = 4

None

None

(0, 0)

(0, 0), (4, 0)

All real #’s

4 3 2( ) 12 48 64f x x x x x 3( 4)x x

2'( ) 4( 1)( 4)f x x x

''( ) 12( 4)( 2)f x x x

f(x) f’(x) f”(x) Characteristics of Graph-∞ < x < 1 Neg Pos Decreasing, concave upx = 1 -27 0 Pos Relative minimum1 < x < 2 Pos Pos Increasing, concave upx = 2 -16 Pos 0 Point of inflection2 < x < 4 Pos Neg Increasing, concave downx = 4 0 0 0 Point of Inflection4 < x < ∞ Pos Pos Increasing, concave up

Ex 5 continued

Ex 6 p.214 Trigonometric functionAnalyze and graph

1st Derivative:

2nd Derivative:

Domain:

x-intercepts:

y-intercept:

Vertical asymptote(s):Horizontal asymptote(s):

Critical number(s):

Possible points of inflection:

Symmetry:

x = - π/2, x = 3π/2

None

At x = π/2

None

None

(0, 1)

(π/2, 0

Period is 2π. For convenience, choose (- π/2, 3π/2)1

'( )1 sin

f xx

2

cos''( )

(1 sin )

xf x

x

cos( )

1 sin

xf x

x

All real except x = (3+4n)π/2

f(x) f’(x) f”(x) Characteristics of Graphx = -π/2 Undef Undef Undef Vertical asymptote-π/2< x < π/2 Neg Pos Decreasing, concave up

x = π/2 0 -1/2 0 Point of inflectionπ/2< x < 3π/2 Neg Neg Decreasing, concave downx = 3π/2 Undef Undef Undef Vertical asymptote

Ex 5 continued

3.6b p. 215/ 45, 49-59 odd, 67-70