04 Gradients and Optimization Free Online FlashCards

What is the gradient formula for a directional derivative?

For a differentiable function, the directional derivative in unit direction $\mathbf{u}$ is $D_{\mathbf{u}}f=\nabla f\cdot\mathbf{u}$.

How do you convert a direction vector into a unit vector?

Normalize a nonzero direction vector $\mathbf{v}$ by dividing it by its magnitude: $\mathbf{u}=\frac{\mathbf{v}}{\|\mathbf{v}\|}$.

How is the gradient of $f(x,y,z)$ defined?

For $f(x,y,z)$, the gradient is $\nabla f=\left\langle\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right\rangle$.

What do the direction and magnitude of the gradient represent?

The gradient points in the direction of steepest ascent, and its magnitude $\|\nabla f\|$ is the maximum rate of increase.

When is a directional derivative zero?

A direction perpendicular to $\nabla f$ has directional derivative zero, because $D_{\mathbf{u}}f=\nabla f\cdot\mathbf{u}=0$.

What is the geometric relation between a gradient and a level curve?

The gradient is normal, or perpendicular, to a level curve $f(x,y)=c$ at a regular point.

What equation gives the tangent plane to $F(x,y,z)=c$?

For $F(x,y,z)=c$ at $P=(x_0,y_0,z_0)$, the tangent plane is $\nabla F(P)\cdot\langle x-x_0,y-y_0,z-z_0\rangle=0$.

What is the tangent-plane formula for $z=f(x,y)$?

For $z=f(x,y)$, the tangent plane at $(x_0,y_0,f(x_0,y_0))$ is $z-f(x_0,y_0)=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$.

What makes an interior point a critical point?

An interior point is critical if $\nabla f(\mathbf{a})=\mathbf{0}$ or if a relevant partial derivative does not exist.

How does the second-derivative test classify a critical point?

For a critical point, let $D=f_{xx}f_{yy}-(f_{xy})^2$. If $D>0$ and $f_{xx}>0$, it is a local minimum; if $D>0$ and $f_{xx}<0$, a local maximum; if $D<0$, a saddle.

What equations define Lagrange multipliers for one constraint?

To optimize $f$ subject to $g=c$, solve $\nabla f=\lambda\nabla g$ together with $g=c$, then compare the objective values at all candidates.

What are the extrema of $xy$ on the unit circle?

For $f(x,y)=xy$ on $x^2+y^2=1$, the maximum is $\frac{1}{2}$ when $y=x$, and the minimum is $-\frac{1}{2}$ when $y=-x$.