Q. 11 - Calculus

Question

Continue with the function $$f(x, y) = ax + by$$ from Exercise 10.

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

(b) Show that every gradient vector, $$\bigtriangledown f(x,y)$$ , is orthogo-

nal to every level curve of $$f$$ .

Step-by-Step Solution

Answer

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

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

Step 1. Given Information

$$f(x, y) = ax + by$$ 

Step 2. Explanation

We have the function, $$f(x, y) = ax + by$$ 

Evaluating, we get, $$y=-\frac{a}{b}x+C$$

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

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 $$f$$

Step 4. Explanation

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

This can be shown using the direction vector, $$(b,-a)$$ for every level curve.

Hence, we get, $$\bigtriangledown f(x,y)= \langle a,b\}$$, and $$\langle b,-a \rangle \cdot \langle a,b \rangle =0$$