Sketch the graph of f(x)= A. The domain is R. B. C. D. The x- and y-intercepts are both Since f(-x) = -f(x), fis ---Select--- and its graph is symmetric about the origin. f(x) = Since x² + 7 is never 0, there is no vertical asymptote. Since f(x) → ∞o as x → ∞o and f(x)→-coas x→∞o, there is no horizontal asymptote. But long division gives us the following. f(x) - = - X = = x² + 7 x³ +7 So the line y = = X- x² + 7 1 + x² + 7 as x ±0o is a slant asymptote.
Sketch the graph of f(x)= A. The domain is R. B. C. D. The x- and y-intercepts are both Since f(-x) = -f(x), fis ---Select--- and its graph is symmetric about the origin. f(x) = Since x² + 7 is never 0, there is no vertical asymptote. Since f(x) → ∞o as x → ∞o and f(x)→-coas x→∞o, there is no horizontal asymptote. But long division gives us the following. f(x) - = - X = = x² + 7 x³ +7 So the line y = = X- x² + 7 1 + x² + 7 as x ±0o is a slant asymptote.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
f(x) = x^3/x^2+7
Transcribed Image Text:Sketch the graph of f(x)=
A. The domain is R.
B.
C.
D.
E.
F.
G.
The x- and y-intercepts are both
Since f(-x) = -f(x), fis--Select--- and its graph is symmetric about the origin.
f(x) =
Since x² + 7 is never 0, there is no vertical asymptote. Since f(x) → ∞o as x → ∞o and f(x) →-coas x→-co, there is
no horizontal asymptote. But long division gives us the following.
f(x) -
- X =
=
=
So the line y =
f'(x) =
x² + 7
x³
+7
f"(x) =
= X-
x² + 7
1 +
x² + 7
3x²(x² + 7) x³. 2x
(x² + 7)²
x²(x² +21)
(x² + 7)²
Since f'(x) > 0 for all x (except 0), f is increasing on (-00, 00).
Although f'(0) =
as x ±0o
is a slant asymptote.
=
,f'(x) does not change sign at 0, so there is no local maximum or minimum.
(4x³ + 42x)(x² + 7)² − (x4 + 21x²) · 2(x² + 7)2x
(+²+7)4
14x(21-x²)
(+²+7)³
Transcribed Image Text:H.
Since f"(x) = 0 when x = 0 or x = ± √/21, we set up the following chart.
Interval
x < -√21
-√21 < x < 0
0 < x < √21
√21 < x
and (x, y) =
7x 21 − x²(x² + 7)³
-
-
The points of inflection are (x, y):
-----
+
+
+
(larger x-value).
f"(x)
+
+
The graph of f is sketched in the following figure.
5
*
-5
5
-5
X
CU on (-∞, -√21)
@
CD on (-√21, 0)
(smaller x-value), (0, 0),
CU on (0, √21)
CD on (√21, ∞)
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