Chapter 12: Q. 12. (page 963)

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

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

Step by step solution

Given information

Let, $f (x, y)$ is a function of two variables defined on an open set containing the point $(a, b)$

Let $Δ z = f (a + Δ x, b + Δ y) - f (a, b)$

If both $f_{x} (a, b)$ and $f_{y} (a, b)$ exist, the function $f$ is said to be differentiable and

$Δ z = f_{x} (a, b) Δ x + f_{y} (a, b) Δ y + ε_{1} Δ x + ε_{2} Δ y \dots \dots (1)$ .where $ε_{1}$ and $ε_{2}$ are functions of $Δ x$ and $Δ y$ and

both are zero when $(Δ x, Δ y) \to (0, 0)$

The objective is to find the equation of the plane tangent to the graph of f(x, y) at (a, b)

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

$z - f (a, b) = f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b) \Rightarrow ∆ z = (f_{x} (a, b), f_{y} (a, b)) \cdot ((x - a), (y - b)) = \nabla f (a, b) \cdot ((x, y) - (a, b))$

Hence, the equation of tangent plane to the surface $f (x, y)$ at $(a, b)$ is:

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

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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

Given information

The objective is to find the equation of the plane tangent to the graph of f(x, y) at (a, b)

One App. One Place for Learning.

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If a function f(x,y)is differentiable at (a,b), explain how touse the gradient ∇f(a,b)to find the equation of the planetangent to the graph of fat (a,b).

Short Answer

Step by step solution

Given information

The objective is to find the equation of the plane tangent to the graph of f(x, y) at (a, b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Pure Maths

Decision Maths

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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 $(a, b)$ .