Find a function f(x, y) such thatf is continuous everywhere. the contour map of f consists of ellipses centered at the origin. (0,0, f(0, 0)) is the highest point on the surface z = f(x, y). Then find the range of f(x, y). Justify your answer using algebraic work.

Answered: Find a function f(x, y) such that f is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Find a function f(x, y) such that

f is continuous everywhere.
the contour map of f consists of ellipses centered at the origin.
(0,0, f(0, 0)) is the highest point on the surface z = f(x, y).

Then find the range of f(x, y). Justify your answer using algebraic work.

Expert Solution

Step 1

The two variable function which is continuous everywhere with contours being ellipses centered at the origin and $(0, 0, f (0, 0))$ being the highest point will be upper ellipsoid.

The general equation of the ellipsoid is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$ , where $a, b, c$ are positive non-zero real numbers.

Thus, to get the function we can solve the equation for $z$ as follows:

$\frac{z^{2}}{c^{2}} = 1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} \Rightarrow \frac{z}{c} = \pm \sqrt{1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}} \Rightarrow z = \pm c \sqrt{1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}}$

so, we get two functions $f (x, y) = c \sqrt{1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}} and f (x, y) = - c \sqrt{1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}}$ .

The function with highest point $(0, 0, f (0, 0))$ is $f (x, y) = c \sqrt{1 - \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}}$ .

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Double Integration$

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.

Similar questions