Aim: What concepts have we available to aide us in sketching functions?
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Transcript of 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
Aim: Curve Sketching Course: Calculus
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?
Aim: Curve Sketching Course: Calculus
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
Aim: Curve Sketching Course: Calculus
Abridged Guidelines – the 4 Tees
T1 Test the function
T2 Test the 1st Derivative
T3 Test the 2nd Derivative
T4 Test End Behavior
Aim: Curve Sketching Course: Calculus
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
Aim: Curve Sketching Course: Calculus
Model Problem 1
Analyze the graph of 2
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
Aim: Curve Sketching Course: Calculus
horizontal asymptote
verticals asymptotes found at zeros of denominator
Model Problem 1
Analyze the graph of 2
2
2 9
4
xf x
x
2. find intercepts & asymptotes
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
Aim: Curve Sketching Course: Calculus
Model Problem 1
Analyze the graph of 2
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
Aim: Curve Sketching Course: Calculus
Model Problem 1
Analyze the graph of 2
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
Aim: Curve Sketching Course: Calculus
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
Aim: Curve Sketching Course: Calculus
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
Aim: Curve Sketching Course: Calculus
Model Problem 2 – What the cusp!!
Analyze the graph of 2
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
Aim: Curve Sketching Course: Calculus
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
Model Problem 2 – What the cusp!!
Analyze the graph of 2
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!!!
Aim: Curve Sketching Course: Calculus
Model Problem 2 – What the cusp!!
Analyze the graph of 2
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
Aim: Curve Sketching Course: Calculus
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.
Aim: Curve Sketching Course: Calculus
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
2. find intercepts & asymptotes
2x