Gradient vectors and tangent lines

[jcook735]

gradient vectors and tangent lines!

If f(x, y) = xy, find the gradient vector f(3, 7) and use it to find the tangent line to the level curve f(x, y) = 21 at the point (3, 7).


I already found the gradient vector to be <7, 3>, Maybe I am missing something obvious, but I have no clue how to find the tangent line.

 

[cronxeh]

grad(xy)*(x-3,y-7) = 0
<7,3>*(x-3,y-7) = 0
7*(x-3) + 3*(y-7) = 0
7x - 21 + 3y - 21 = 0
7x + 3y - 42 = 0
3y = -7x + 42
y = -7/3*(x) + 42/3

y= -7/3*(x) + 14 is the equation of a tangent line to f(x,y)=xy at point (3,7)

And by the way, the normal to this tangent will have a -1/m slope, so your normal to this tangent is y-7=3/7*(x-3) -> y=3/7*x+40/7

Review the attached graph