Question

Need help with gradient vectors (Calculus)

Let f(x,y)= x^2+y^2+4x
1. Find the gradient vector f(2,1)
2. Sketch the level set which contains the point (2,1) together with the gradient vector
3. In what direction does f decrease most rapidly
4. In what direction does f remain constant?

Answer 1

find the partial and plug in the given points, x=2, y=1

Answer 2

I've taken the derivative and found out that the gradient vector is <8.2>, but, remain to be clueless on how to sketch

& As for the 3rd question, is it (-8,2)?

Answer 3

@abb0t Can you please check for me? (whether 1 and 3 is correct), and perhaps give me a clue on how to sketch?

Answer 4

gradient vector is perpendicular to the tangent line at the point, am I right?

Answer 5

take the dot product.

Answer 6

you'll get an equation of the line.

Answer 7

@abb0t Could you please elaborate a bit more? (As of how to take the dot product with <8,2>) & is my assumption of (3) correct?

Answer 8

\(<8,2> \cdot < x-2, y-1> \)