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
icon
Related questions
Question

f(x) = x^3/x^2+7

 
 
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: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
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, ∞)
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, ∞)
Expert Solution
Step 1: Given data and guidelines

Advanced Math homework question answer, step 1, image 1

steps

Step by step

Solved in 5 steps with 4 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,