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Chapter 12: Q. 9 (page 963)

Continue with the function $$f(x, y) = 2x + 3y$$ from Exercise 8.

(a) What are the level curves of $$f$$ ?

(b) Show that every gradient vector, $$\bigtriangledown f(x,y)$$, is orthogonal to every level curve of $$f$$ .

Short Answer

Expert verified

(a) The level curves of $$f$$ are the lines satisfying the equation, $$y=-\frac{2}{3}x+C$$, where $$C\epsilon R$$

(b) Using the vector, $$\langle3,-2 \rangle$$, it is shown that gradient vector, $$\bigtriangledown f(x,y)$$, is orthogonal to every level curve of $$f$$.

Step by step solution

01

Step 1. Given Information

$$f(x, y) = 2x + 3y$$

02

Step 2. Explanation

We have the function, $$f(x, y) = 2x + 3y$$

From the above function, we get, $$y=-\frac{2}{3}x+C$$

Hence, the level curves of the given function, $$f$$ are lines satisfying the equation, $$y=-\frac{2}{3}x+C$$, where $$C\epsilon R$$.

03

Step 3. Given information

We have the gradient vector $$\bigtriangledown f(x,y)$$we need to show that it is orthogonal to every level curve of $$f$$

04

Step 4. Explanation

Every gradient vector, $$\bigtriangledown f(x,y)$$, is orthogonal to every level curve of $$f$$.

This can be shown using the vector, $$\langle3,-2 \rangle$$ and using this as direction vector for every level curve.

Hence, we get, $$\bigtriangledown f(x,y)= \langle 2,3)$$, and $$\{3,-2 \rangle \cdot (2,3 \rangle =0$$

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