How Do You Calculate the Gradient at a Point in a Function?

[tinkus]

Homework Statement

Consider the function f (x,y). if you start at the point (4,5) and move to the point (5,6) . the directional derivative is 2. Starting at the point (4,5) and moving toward the point (6,6)gives a directional derivative of 3.Find grad f at the point (4,5) .

Homework Equations

The Attempt at a Solution

I don't really know how to go about this question. All I can do so far is find the unit vector.
PQ = (5-4) i + (6-5) j = i+j ; u = 1/sqrt 2 i + 1/sqrt 2 j

PR = (6-4 i +( 6-5) j = 2i+j ; u = 2/sqrt 5 i + 1/sqrt 5 j

 

[Dick]

If the gradient G=ai+bj, then the given information tells you PQ.G=2 and PR.G=3. That's two equations in two unknowns.

 

[phreak]

Use the equation $$f_u = \nabla f \cdot u$$. You'll set yourself up with a system of equations, solve them, and you're done.

 

[tinkus]

ok i set the system of equation and I'm getting nowhere.
grad f1 = .5i + .5j
grad f2 = .2981i + .1491j
how do i set up the system of equation.

 

[Dick]

Those aren't the right equations. E.g. PQ.G doesn't have i or j in it.

 

[HallsofIvy]

Write the gradient as $f\vec{i}+ g\vec{j}$.

1. What is the unit vector in the direction from (4, 5) to (5, 6)? What is the dot product of that vector with $f\vec{i}+ g\vec{j}$? Set that equal to 2.

2. What is the unit vector in the direction from (4, 5) to (6, 6)? What is the dot product of that vector with $f\vec{i}+ g\vec{j}$? Set that equal to 3.

You now have two equations to solve for f and g.

 

[tinkus]

Write the gradient as $f\vec{i}+ g\vec{j}$.

1. What is the unit vector in the direction from (4, 5) to (5, 6)? What is the dot product of that vector with $f\vec{i}+ g\vec{j}$? Set that equal to 2.

2. What is the unit vector in the direction from (4, 5) to (6, 6)? What is the dot product of that vector with $f\vec{i}+ g\vec{j}$? Set that equal to 3.

You now have two equations to solve for f and g.

I got it now, thank you so much.