14.2 Limits and Continuity

Limit of $f(x,y)$ as $(x,y)$ approach $(a,b)$:

$\lim_{(x,y) \to (a,b)} f(x,y) = L$ if $\forall \epsilon > 0, \exists \delta > 0, (x,y) \in D \land 0 < \sqrt{(x-a)^2+(y-b)^2}<\delta \implies |f(x,y) - L| \le \epsilon$

Continuity: $f(x,y)$ is continuous at $(a,b)$ if

• $\lim_{(x,y)\to(a,b)} f(x,y)=f(a,b)$

Continuity on Interval: $f$ is continuous on $D$ if

• $\forall (a,b)\in D, \lim_{(x,y)\to (a,b)} f(x,y) = f(a,b)$

14.3 Partial Derivatives

Notations: $f_x(x,y)=f_x=\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} f(x,y) = \frac{\partial z}{\partial x} = f_1 = D_1f = D_xf$

Partial derivative of $f$ with respect to $x$ at $(a,b)$

• $f_x(a,b)=g'(a) \text{ where } g(x)=f(x,b)$
• $f_x(a,b)=\lim_{h\to0}\frac{f(a+h,b)-f(a,b)}{h}$
• $f_y(a,b)=\lim_{h\to0}\frac{f(a,b+h)-f(a,b)}{h}$

Integrating Partial Derivatives: When integrating $\frac{\partial f}{\partial x}$, treat variables other than $x$ as constants

14.4 Tangent Planes

If $f$ has continuous partial derivatives, a tangent plane to $z=f(x,y)$ at $(x_0, y_0, z_0)$ is:

$z-z_0=f_x(x_0,y_0)\cdot(x-x_0)+f_y(x_0, y_0)\cdot(y-y_0)$

Diffable: $f$ is diffable at $(a,b)$ if $\Delta z = f_x(a,b) \Delta x + f_y(a,b)\Delta y + \epsilon_1 \Delta x + \epsilon_2 \Delta y$

• Where $\epsilon_1,\epsilon_2 \rightarrow 0$ as $(\Delta x, \Delta y) \rightarrow (0,0)$
• If $f_x,f_y$ exists near $(a,b)$ and are continuous at $(a,b)$, then $f$ is diffable at $(a,b)$