The given function is: VIXI The given function can be written as: y = y = √x, x ≥ 0 S-√√x, x < 0 √x, x ≥ 0 X <0 To find the Left hand derivative, use the differentiable function formula. f(x)-f(a) x-a Left hand derivative = lim x->0 Substitute for f (x) and 0 for a in the above formula. Left hand derivative = lim X-0 -0-x = lim x-0 = lim x-0 = lim [] lim x2/3 x->0 Simplify the above limit further. =18 x175 1 --0 X Again, to find the Right hand derivative, use the differentiable formula Right hand derivative = lim f(x)-f(a) x-a x-0+ Substitute x forf (x) and 0 for a in the above formula. = lim x-0+ Right hand derivative = lim = = lim x-0+ [*] lim [] x-0+ Simplify the above limit further. = ∞ -x-f(0) x-0 VX-0 x +0+x x-f(0) x-0 lim [] x->0+ Thus, the results of the limit indicate that the given function is a cusp. The graph of the function y = x has a cusp at x = 0
The given function is: VIXI The given function can be written as: y = y = √x, x ≥ 0 S-√√x, x < 0 √x, x ≥ 0 X <0 To find the Left hand derivative, use the differentiable function formula. f(x)-f(a) x-a Left hand derivative = lim x->0 Substitute for f (x) and 0 for a in the above formula. Left hand derivative = lim X-0 -0-x = lim x-0 = lim x-0 = lim [] lim x2/3 x->0 Simplify the above limit further. =18 x175 1 --0 X Again, to find the Right hand derivative, use the differentiable formula Right hand derivative = lim f(x)-f(a) x-a x-0+ Substitute x forf (x) and 0 for a in the above formula. = lim x-0+ Right hand derivative = lim = = lim x-0+ [*] lim [] x-0+ Simplify the above limit further. = ∞ -x-f(0) x-0 VX-0 x +0+x x-f(0) x-0 lim [] x->0+ Thus, the results of the limit indicate that the given function is a cusp. The graph of the function y = x has a cusp at x = 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(see attached image) How to convert the original function into the form of y={ ... ? Especially in the absolute value scenarios. In this case, root 3 -x is matched with x < 0, and root 3 x is x>=0, how do i know which goes by which? Later in this question, it says limx->0- [-1/x^2/3] is negative infinity and limx->0+ of it is positive infinity, how do we know that? Thank you very much!
![RT
UTION &
WER
WLEDGE
STER
The given function is:
y = √|x|
The given function can be written as:
√x, x < 0
=
{
√√x, x ≥ 0
S-√√x, x < 0
√√x, x ≥ 0
To find the Left hand derivative, use the differentiable function formula.
f(x)-f(a)
Left hand derivative
x-a
lim
x→0¯
Substitute -x for f (x) and 0 for a in the above formula.
-√√x-f(0)
x-0
Left hand derivative = lim
x→0
_0+x
= lim
=
1
lim [-]
1/3
= lim [-]
x→0-
-0-x
= lim [-
-0-x
Simplify the above limit further.
=
-3√x-0
X
-8
Again, to find the Right hand derivative, use the differentiable formula
f(x)-f(a)
Right hand derivative
x-a
=
= lim
x→0+
Substitute x forf (x) and 0 for a in the above formula.
√x-f(0)
x-0
Right hand derivative = lim
= lim
x→0+
x 2/3
+0+x
lim X
+0+x
3x-0
lim [
= ∞
**
1/3
x2/3
+0+x
Simplify the above limit further.
lim []
x→0+
Thus, the results of the limit indicate that the given function is a cusp.
The graph of the function y = x has a cusp at x = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff34a8ef3-a700-4c73-a9ad-1c57a30c8083%2Fada1c0b9-2dd6-432f-aa74-0941a86940ed%2Ftnibgft_processed.png&w=3840&q=75)
Transcribed Image Text:RT
UTION &
WER
WLEDGE
STER
The given function is:
y = √|x|
The given function can be written as:
√x, x < 0
=
{
√√x, x ≥ 0
S-√√x, x < 0
√√x, x ≥ 0
To find the Left hand derivative, use the differentiable function formula.
f(x)-f(a)
Left hand derivative
x-a
lim
x→0¯
Substitute -x for f (x) and 0 for a in the above formula.
-√√x-f(0)
x-0
Left hand derivative = lim
x→0
_0+x
= lim
=
1
lim [-]
1/3
= lim [-]
x→0-
-0-x
= lim [-
-0-x
Simplify the above limit further.
=
-3√x-0
X
-8
Again, to find the Right hand derivative, use the differentiable formula
f(x)-f(a)
Right hand derivative
x-a
=
= lim
x→0+
Substitute x forf (x) and 0 for a in the above formula.
√x-f(0)
x-0
Right hand derivative = lim
= lim
x→0+
x 2/3
+0+x
lim X
+0+x
3x-0
lim [
= ∞
**
1/3
x2/3
+0+x
Simplify the above limit further.
lim []
x→0+
Thus, the results of the limit indicate that the given function is a cusp.
The graph of the function y = x has a cusp at x = 0
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