Consider the curve C = {(x, y) E R?: x - y3 = 0}. Let o: R → R be defined by o(t) = t3, then o is a bijection from R to R and so it has an inverse o-1. The equation that defines C can be written as a = ¢(y), and by applying o-1 on both sides we obtain the equivalent equation y = 0-(x). Therefore y is a function of r. Taking this into account, differentiating the equation y3 with respect to a on both sides yields 1 = 3y?. %3D 2 dy Since (0, 0) E C, plugging in r = 0 da %3D and y = 0 into the last equation yields 1 = 0. (a) Explain the mistake(s) in the above argument. (b) Explain why the Implicit. Function Theorem cannot be used to fix the above argument.
Consider the curve C = {(x, y) E R?: x - y3 = 0}. Let o: R → R be defined by o(t) = t3, then o is a bijection from R to R and so it has an inverse o-1. The equation that defines C can be written as a = ¢(y), and by applying o-1 on both sides we obtain the equivalent equation y = 0-(x). Therefore y is a function of r. Taking this into account, differentiating the equation y3 with respect to a on both sides yields 1 = 3y?. %3D 2 dy Since (0, 0) E C, plugging in r = 0 da %3D and y = 0 into the last equation yields 1 = 0. (a) Explain the mistake(s) in the above argument. (b) Explain why the Implicit. Function Theorem cannot be used to fix the above argument.
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
Please help with 1a and b
Transcribed Image Text:1. Consider the curve C = {(x, y) E R²: x – y3 = 0}. Let o: R → R be defined by ø(t) = t³,
then o is a bijection from R to R and so it has an inverse o-1. The equation that defines C can
be written as x = ¢(y), and by applying o-1 on both sides we obtain the equivalent equation
y = 0-1(x). Therefore y is a function of x. Taking this into account, differentiating the equation
y with respect to x on both sides yields 1 = 3y
da
dy
Since (0,0) e C, plugging in x = 0
and y = 0 into the last equation yields 1 = 0.
(a) Explain the mistake(s) in the above argument.
(b) Explain why the Implicit Function Theorem cannot be used to fix the above argument.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
