Question

Consider the function f(x,y)=3x2+3y2.
Find the the directional derivative of f at the point (1,−1) in the direction given by the angle θ=2π3.


Find the unit vector which describes the direction in which f is increasing most rapidly at (1,−1).

Answer 1

i got the first part as 6cos(2pi/3)-6sin(2pi/3), but shouldn't part b just be the gradient of f at (1,-1)?

Answer 2

the gradient in a given direction

Answer 3

if we take a plane and slice the surface we create a curve in a plane, a plane the is directed at a given angle.

the slope at a given point depends in the direction we have heading

Answer 4

would it be in the same direction as the first u? cos 2pi/3 and sin2pi/3?

Answer 5

reviewing some stuff to remember what i forgot

http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/grad/grad.html

Answer 6

gradient dotted with the direction vector

Answer 7

u = cos(2pi/3) , sin(2pi/3)

Answer 8

3x2+3y2

gradf = 6x, 6y

Answer 9

Duf = (cos(2pi/3) 6x , sin(2pi/3) 6y)

Answer 10

yes, that is the answer to part a. that's exactly what i got for part a.

Answer 11

so part b is: Find the unit vector which describes the direction in which f is increasing most rapidly at (1,−1).

Answer 12

refresh to get rid of the <?> marks

Answer 13

the gradient has a property that it actually points in the direction of fastest growth ... but i cnat figure out why

Answer 14

i think i figured it out.....it's the gradient over the magnitude of the gradient....

Answer 15

"the gradient vector points in the direction of greatest rate of increase of f(x,y)"

6x, 6y at (1,-1) is 6,-6

unit is yes, dividing a vector by its length

Answer 16

thanks. i got the right answer.

Answer 17

youre welcome