Let f(x, y) = 1 + x ln(xy — 5). (a) Explain why the function is differentiable at the point (2, 3) (b) Find the equation of the tangent plane at the point (2,3).
Let f(x, y) = 1 + x ln(xy — 5). (a) Explain why the function is differentiable at the point (2, 3) (b) Find the equation of the tangent plane at the point (2,3).
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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![### Differentiability and Tangent Plane of a Given Function
#### Problem Statement
Consider the function \( f(x, y) = 1 + x \ln(xy - 5) \).
**Tasks:**
**(a)** Explain why the function is differentiable at the point (2, 3).
**(b)** Find the equation of the tangent plane at the point (2, 3).
#### Solutions
**(a) Differentiability at the Point (2, 3):**
To determine whether the function \( f(x, y) \) is differentiable at the point (2, 3), we need to check the existence and continuity of the partial derivatives of \( f \) at that point. Specifically, the partial derivatives with respect to \( x \) and \( y \) must exist and be continuous at (2, 3).
**(b) Equation of the Tangent Plane:**
To find the equation of the tangent plane to the surface defined by \( f(x, y) \) at the point (2, 3), we use the formula for the tangent plane of a function of two variables. The formula is:
\[ z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) \]
where \( (a, b) \) is the point of tangency, \( f_x \) and \( f_y \) are the partial derivatives of \( f \) with respect to \( x \) and \( y \), respectively.
1. Calculate \( f_x \) and \( f_y \) at the point (2, 3).
2. Evaluate \( f(2, 3) \).
3. Plug in these values into the tangent plane formula to get the equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa40558bc-afda-4e2b-be1a-c4211c27f73c%2Fb374c1af-5b27-4416-8fbd-8fba1ed00344%2Fidnn5qj_processed.png&w=3840&q=75)
Transcribed Image Text:### Differentiability and Tangent Plane of a Given Function
#### Problem Statement
Consider the function \( f(x, y) = 1 + x \ln(xy - 5) \).
**Tasks:**
**(a)** Explain why the function is differentiable at the point (2, 3).
**(b)** Find the equation of the tangent plane at the point (2, 3).
#### Solutions
**(a) Differentiability at the Point (2, 3):**
To determine whether the function \( f(x, y) \) is differentiable at the point (2, 3), we need to check the existence and continuity of the partial derivatives of \( f \) at that point. Specifically, the partial derivatives with respect to \( x \) and \( y \) must exist and be continuous at (2, 3).
**(b) Equation of the Tangent Plane:**
To find the equation of the tangent plane to the surface defined by \( f(x, y) \) at the point (2, 3), we use the formula for the tangent plane of a function of two variables. The formula is:
\[ z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) \]
where \( (a, b) \) is the point of tangency, \( f_x \) and \( f_y \) are the partial derivatives of \( f \) with respect to \( x \) and \( y \), respectively.
1. Calculate \( f_x \) and \( f_y \) at the point (2, 3).
2. Evaluate \( f(2, 3) \).
3. Plug in these values into the tangent plane formula to get the equation.
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