aim: what concepts have we available to aide us in sketching functions?

Aim: Curve Sketching Course: Calculus

Do Now:

Aim: What concepts have we available to aide us in sketching functions?

2

2

2 9

4

xf x

x

Find the domain of


Concepts used in Sketching

• x- and y-intercepts• symmetry• domain & range• continuity• vertical asymptotes• differentiability• relative extrema• concavity• points of inflection• horizontal asymptotes

Use them all? If not all, which are best?


Guidelines for Analyzing Graph

1. Determine the domain and range of the function.

2. Determine the intercepts and asymptotes of the graph.

3. Locate the x-values for which f’(x) and f’’(x) are either zero or undefined. Use the results to determine relative extrema and points of inflection.

Also helpful: symmetry; end behavior


Abridged Guidelines – the 4 Tees

T1 Test the function

T2 Test the 1st Derivative

T3 Test the 2nd Derivative

T4 Test End Behavior


Model Problem 1

Analyze the graph of 2

2

2 9

4

xf x

x

1. find domain & range

exclusions at zeros of denominator

domain: all reals except ±24

2

-2

-4

-5 5


Model Problem 1


2

2 9

4

xf x

x

2. find intercepts & asymptotes

2

2

2 90

4

xf

x

y-intercept

2

2

2 0 9 18 9

0 4 4 2

2

2

2 90

4

xf x

x

x-intercept

2

2

2 90 , 3

4

xx

x

4

2

-2

-4

-5 5


horizontal asymptote

verticals asymptotes found at zeros of denominator

Model Problem 1


2

2 9

4

xf x

x


x = ±2

If degree of p = degree of q, then the line y = an/bm is a horizontal asymptote.

2 2

2 2

2 9 2 18

4 4

x x

x x

y = 2

4

2

-2

-4

-5 5


Model Problem 1


2

2 9

4

xf x

x

3. find f’(x) = 0 and f’’(x) = 0 or undefined

2 2

22

4 4 2 18 2'

4

x x x xf x

x

22

20'

4

xf x

x

0 x = 0

(x2 – 4)2 = 0

undefined at zeros of denominator

x = ±2 2 2

2 2

2 9 2 18

4 4

x x

x x

4

2

-2

-4

-5 5


Model Problem 1


2

2 9

4

xf x

x

3. find f’(x) = 0 and f’’(x) = 0 or undefined

22 2 1

222

4 20 20 2( 4) 2''

4

x x x xf x

x

22

20'

4

xf x

x

2

32

20 3 4''

4

xf x

x

0 no real

solution

no possible points of inflection


Model Problem 1

3. test intervals

f(x) f’(x) f’’(x)characteristic of

Graph

- < x < -2

x = -2 Undef Undef Undef

-2 < x < 0

x = 0 9/2

0 < x < 2

x = 2 Undef Undef Undef

2 < x <

decreasing, concave down

decreasing, concave up

relative minimum

increasing, concave up

vertical asymptote

vertical asymptote

increasing, concave down

+

+

++

+

0


6

4

2

-2

-4

-6

-5 5

q x = 2x2-9

x2-4

Model Problem 1

(0, 9/2) relative minimum

increasing, concave down 2 < x <

decreasing, concave up -2 < x < 0

increasing, concave down - < x < -2

increasing, concave up 0 < x < 2


Model Problem 2 – What the cusp!!


32y x

Find Domain

no vertical or horizontal asymptotes

Find intercepts & asymptotes

all reals

-intercepts at 2 2x

3

32 2 23 3 22 2x x

-intercepts at (0,2)y

T1

230 2 x


3

2

1

-1

-2

-2 2

h x = 2-x

2

33

2

1

-1

-2

-2 2

g x = -2

3 x

-1

3



32y x

1st Derivative Test T21

32

3

dyx

dx

1

32

03

x

x at 0 is undefined

BUT . . . x = 0 is defined for original function

f’ > 0 inc

f’ < 0 dec

a cusp!!!




32y x

2nd Derivative Test T3423

2

2

9

d yx

dx

x at 0 is

undefined

4

32

09

x

3

2

1

-1

-2

-2 2

h x = 2-x

2

3

f’’ > 0 con up

f’’ > 0 con up

cusp


3

2

1

-1

-2

-3

-4 -2 2 4

f x = 3x5-20x3

32

Model Problem 35 33 20

Sketch the graph ( )32

x xf x

3

2

1

-1

-2

-3

-4 -2 2 4

f x = 3x5-20x3

32

f’

f’’

> 0 inc

< 0 dec

< 0 dec

> 0 inc

< 0 c.d.

> 0 c.u.

< 0 c.d.

> 0 c.u.

(-2,2)

(2,-2)

inflection points: (-1.4,1.2), (0,0), (1.4,-1.2)

relative max.

relative min.


Model Problem 4

Analyze the graph of cos

1 sin

xf x

x

1. find Domain

verticals asymptotes found a zeros of denominator

x = ±2

1 + sin x = 0; sin x = -1


2x

aim: what concepts have we available to aide us in sketching functions?

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