Let A CR2 be the closed unit disk (so A = {(x, y) = R²: x² + y² ≤ 1}). Let f: A→R where (a) (b) f(x, y) = √1-x² - y². Find and simplify Df(c) for c = Let c = (2,0). Find and simplify f(c) + L((x, y) — c). Reminder: L = Df(c). =(√2,0). (c) What is the equation of the tangent plane to the graph of z =√1-x² - y² through the point (2,0, f(c))?
Let A CR2 be the closed unit disk (so A = {(x, y) = R²: x² + y² ≤ 1}). Let f: A→R where (a) (b) f(x, y) = √1-x² - y². Find and simplify Df(c) for c = Let c = (2,0). Find and simplify f(c) + L((x, y) — c). Reminder: L = Df(c). =(√2,0). (c) What is the equation of the tangent plane to the graph of z =√1-x² - y² through the point (2,0, f(c))?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![4. Let \( A \subseteq \mathbb{R}^2 \) be the closed unit disk (so \( A = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1 \} \)).
Let \( f : A \to \mathbb{R} \) where
\[
f(x,y) = \sqrt{1 - x^2 - y^2}.
\]
(a) Find and simplify \( Df(c) \) for \( c = \left( \frac{\sqrt{2}}{2}, 0 \right) \).
(b) Let \( c = \left( \frac{\sqrt{2}}{2}, 0 \right) \). Find and simplify \( f(c) + L((x,y) - c) \).
Reminder: \( L = Df(c) \).
(c) What is the equation of the tangent plane to the graph of \( z = \sqrt{1 - x^2 - y^2} \) through the point \( \left( \frac{\sqrt{2}}{2}, 0, f(c) \right) \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2F5513556a-ca30-449f-a262-a3d0d7bc7553%2Friuw2z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Let \( A \subseteq \mathbb{R}^2 \) be the closed unit disk (so \( A = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1 \} \)).
Let \( f : A \to \mathbb{R} \) where
\[
f(x,y) = \sqrt{1 - x^2 - y^2}.
\]
(a) Find and simplify \( Df(c) \) for \( c = \left( \frac{\sqrt{2}}{2}, 0 \right) \).
(b) Let \( c = \left( \frac{\sqrt{2}}{2}, 0 \right) \). Find and simplify \( f(c) + L((x,y) - c) \).
Reminder: \( L = Df(c) \).
(c) What is the equation of the tangent plane to the graph of \( z = \sqrt{1 - x^2 - y^2} \) through the point \( \left( \frac{\sqrt{2}}{2}, 0, f(c) \right) \)?
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