Download Sect 4.5: Sketch The Graph

Math 221-16: Calculus I
Sect 4.5: Sketch The Graph
Sketch the graph of the following functions using information about where the function is increasing/decreasing,
local max/min, intervals of concavity, points of inflection and asymptotes.
We Have done some/most of the work in class before.
1. f (x) = 21 x −
√
x
• Domain: [0, ∞)
• x and y Intercepts: (4, 0), (0, 0)
• Asymptotes: Find These.....
• Intervals of Inc/Dec, Local Max/Min: f ′ (x) =
Inc: (1, ∞) , Dec: (0, 1), (1, −1
2 ) is a local min
1
2
−
1
√
2 x
• Intervals of CU/CD, Inflection Points: f ′′ (x) = 0 −
=
− √1x
√
(2 x)2
√
x−1
√
2 x
=
1√
4x x
CU: (0, ∞), CD: nowhere, POI: none
2. f (x) =
x−1
x+1
• Domain: (−∞, −1) ∪ (−1, ∞)
• x and y Intercepts: (1, 0) and (0, −1)
• Asymptotes: Find These....
• Intervals of Inc/Dec, Local Max/Min:
f ′ (x) =
(x+1)1−(x−1)1
(x+1)2
=
x+1−x+1
(x+1)2
=
2
(x+1)2
Inc: (−∞, −1) and (−1, ∞), Dec: nowhere
No local minima or maxima
• Intervals of CU/CD, Inflection Points:
f ′′ (x) =
x+1)2 (0)−2(2(x+1)1 (1))
x+1)4
=
−4(x+1)
x+1)4
(optional simplification
CU: (−∞, −1) and CD: (−1, ∞)
No points of inflection.
1
−4
)
(x+1)3
Sketch the graph of the following functions using information about where the function is increasing/decreasing, local max/min, intervals of concavity, points of inflection and asymptotes.
Most of the work has been done for you
3. f (x) =
x2
x2 + 1
• Domain: R
• x and y Intercepts: (0, 0)
• Asymptotes:
Vertical: none
Horizontal:
2
limx→+∞ x2x+1 = 1
limx→−∞
x2
x2 +1
=1
• Intervals of Inc/Dec, Local Max/Min: Find These....
• Intervals of CU/CD, Inflection Points:
q
q
q
q
q
q
q q
CU: (− 13 , 13 ) CD: (−∞, 13 ) and ( 13 , ∞). Point of Inflection: ( 13 , f ( 13 )) and (− 13 , f (− 13 ))
4. f (x) = −(x − 9)3
• Domain: R
• x and y Intercepts: (9 , 0) and (0, 729)
• Asymptotes: none
• Intervals of Inc/Dec, Local Max/Min:
f ′ (x) = −3(x − 9)2
Inc: nowhere, Dec: (−∞, 9) and (9, ∞)
• Intervals of CU/CD, Inflection Points: Find These...
2
Sketch the graph of f (x) with the following properties:
5.
• Natural Domain of f (x): (−∞, −1) ∪ (−1, ∞)
• Points on the graph: (−5, 0), (−7, 0), (7, 0)
• Some Relevant Limits:
limx→−1− f (x) = +∞, limx→−1+ f (x) = +∞
limx→+∞ f (x) = −∞, limx→−∞ f (x) = +∞
• Intervals of Inc/Dec, Local Max/Min:
f (x) is increasing on (−6, −1) and decreasing on (−∞, −6) and (−1, ∞)
(−6, −1
2 ) is a point on the graph
• Intervals of CU/CD, Inflection Points:
f (x) is concave up on (−∞, −1) and (−1, ∞)
6.
• Natural Domain of f (x): R
• Points on the graph: (−8, 0) and (0, 3)
• Some Relevant Limits:
limx→+∞ f (x) = −8, limx→−∞ f (x) = −8
• Intervals of Inc/Dec, Local Max/Min:
f (x) is increasing on (−∞, −4), (2, 7) and decreasing on (−4, 2) and (7, ∞)
(−4, 4), (2, 1) and (7, 7) are points on the graph
• Intervals of CU/CD, Inflection Points:
f (x) is concave up on (−∞, −10), (1, 3) and (9, ∞) and concave down on (−10, 1) and (3, 9)
and (−10, −7), (1, 2), (3, 5) and (9, 5) are points of inflection.
7.
• Natural Domain of f (x): (−∞, 5) ∪ (5, ∞)
• Points on the graph: (− 32 , 0), (4, 0), (0, 3)
• Some Relevant Limits:
limx→5− f (x) = −∞, limx→5+ f (x) = +∞
limx→+∞ f (x) = −3, limx→−∞ f (x) = −3
• Intervals of Inc/Dec, Local Max/Min:
f (x) is increasing on (−∞, 1) and decreasing on (1, 5) and (5, ∞)
(1, 7) is a point on the graph
• Intervals of CU/CD, Inflection Points:
f (x) is concave up on (−∞, −1) and (5, ∞) and concave down on (−1, 5)
3
8.
• Natural Domain of f (x): [−4, ∞)
• Points on the graph: (−4, 0), (−1, 0), (7, 0), (0, −3)
• Some Relevant Limits:
limx→+∞ f (x) = +∞,
• Intervals of Inc/Dec, Local Max/Min:
f (x) is increasing on (−4, −3), (1, ∞) and decreasing on (−3, 1)
(−3, 2), and (1, −4) are points on the graph
• Intervals of CU/CD, Inflection Points:
f (x) is concave up on (−2, ∞), and concave down on (−4, −2)
and (−2, 1) is a points of inflection.
9.
• Natural Domain of f (x): (−∞, −6) ∪ (−6, 4) ∪ (4, ∞)
• Some Relevant Limits:
limx→−6− f (x) = −∞, limx→−6+ f (x) = −∞
limx→4− f (x) = +∞, limx→4+ f (x) = −∞
limx→−∞ f (x) = −3, limx→+∞ f (x) = +∞
• Info From the First Derivative:
f ′ (x) is positive on (−6, 4) and (4, ∞) and negative on (−∞, −6)
• Info from the Second Derivative:
f ′′ (x) is positive on (−2, 4) and negative on (−∞, −6), (−6, −2) and (4, ∞)
f ′′ (−2) = 0, and (−2, 3) is a point on the graph
4