Chapter 12: Q. 12 (page 963)

12. If a function $f (x, y)$ is differentiable at $(a, b)$ , explain how to use the gradient $\nabla f (a, b)$ to find the equation of the plane tangent to the graph of $f$ at $(a, b)$ .

Short Answer

Expert verified

The equation of the plan tangent to the graph of the function at $(a, b)$ is $z - f (a, b) = \nabla f (a, b) \cdot ⟨ (x, y) - (a, b) ⟩$

Step by step solution

Introduction

The given is the function is $f (x, y)$ is differentiable at the point $(a, b)$

The objective is to find the equation of the plane tangent

Step 1

Let $f (x, y)$ be defined as a function of two variables on an open set including the point $(a, b)$ and let $Δ z = f (a + Δ x, b + Δ y) - f (a, b)$ . The function $f$ is said to be differentiable if $f_{x} (a, b) a n d f_{y} (a, b)$ both exists and

$Δ z = f_{x} (a, b) Δ x + f_{y} (a, b) Δ y + ε_{1} Δ x + ε_{2} Δ y$

where $ε_{1}$ and $ε_{2}$ are functions of $Δ x$ and $Δ y$ , and they're both zero when $(Δ x, Δ y) \to (0, 0)$ .

Step 2

Substitute $Δ x = x - a, Δ y = y - b$ and $Δ z = z - f (a, b)$ in $(1)$ and eliminate $ε_{1} ∆ x a n d ε_{2} ∆ y$ terms.

$z - f (a, b) = f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$

$Δ z = <f_{x} (a, b), f_{y} (a, b)> \cdot ⟨ (x - a), (y - b) ⟩$

$= \nabla f_{+} (a, b) \cdot ⟨ (x, y) - (a, b) ⟩$

As a result, the tangent plane to the surface equation $f (x, y)$ at $(a, b)$ is

z - f (a, b) = \nabla f (a, b) \cdot ⟨ (x, y) - (a, b) ⟩

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

12. If a function $f (x, y)$ is differentiable at $(a, b)$ , explain how to use the gradient $\nabla f (a, b)$ to find the equation of the plane tangent to the graph of $f$ at $(a, b)$ .

Short Answer

Step by step solution

Introduction

Step 1

Step 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Applied Mathematics

Mechanics Maths

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

12. If a function f(x,y) is differentiable at (a,b), explain how to use the gradient ∇f(a,b) to find the equation of the plane tangent to the graph of f at (a,b).

Short Answer

Step by step solution

Introduction

Step 1

Step 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Applied Mathematics

Mechanics Maths

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

12. If a function $f (x, y)$ is differentiable at $(a, b)$ , explain how to use the gradient $\nabla f (a, b)$ to find the equation of the plane tangent to the graph of $f$ at $(a, b)$ .