A point ( (a, b) ) is a critical point if ( \nabla f(a, b) = \langle 0, 0 \rangle ) (i.e., both partial derivatives are zero). At such points, the tangent plane is horizontal.
Measures rate of change in direction ( \mathbfu ): [ D_\mathbfu f(\mathbfa) = \lim_t \to 0 \fracf(\mathbfa + t\mathbfu) - f(\mathbfa)t ] If ( f ) is differentiable: [ D_\mathbfu f = \nabla f(\mathbfa) \cdot \mathbfu ] multivariable differential calculus
Given: Maximize ( f(x, y) ) subject to ( g(x, y) = k ). A point ( (a, b) ) is a
In single-variable calculus, a limit approaches a point from only two directions: left or right. In multivariable calculus, a limit must hold true from infinite directions. A point ( (a
𝜕2f𝜕y2partial squared f over partial y squared end-fraction