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# Calculus Notes ## Limits and Continuity * Use limit tests (L'Hôpital's Rule, Two-Path Test, Substitution) to determine if the limit exists. * Boundary points lie on the edge of the domain. * Interior points lie within the domain. * Two-path test: if the limit depends on what path is taken, the...

# Calculus Notes ## Limits and Continuity * Use limit tests (L'Hôpital's Rule, Two-Path Test, Substitution) to determine if the limit exists. * Boundary points lie on the edge of the domain. * Interior points lie within the domain. * Two-path test: if the limit depends on what path is taken, the limit may not exist. * Continuity: A function `f(x, y)` is continuous if `lim_(x,y)→(a,b) f(x,y) = f(a,b)`. ## Partial Derivatives * **First partial derivatives:** * `fx = ∂f/∂x` (derivative with respect to x, holding y constant) * `fy = ∂f/∂y` (derivative with respect to y, holding x constant) * **Second partial derivatives:** * `fxy = ∂²f/∂x∂y` (repeat differentiation) * `fyx = ∂²f/∂y∂x` (repeat differentiation) ## Chain Rule * `dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)` * Used when a function depends on more than one variable. ## Implicit Differentiation * Differentate both sides of the equation and then solve for a desired derivative. ## Directional Derivatives and the Gradient * **Gradient vector:** `∇f = (∂f/∂x)i + (∂f/∂y)j` (vector of partial derivatives). * **Directional derivative:** The rate of change of `f(x,y)` in the direction of a unit vector `u`. `Duf = ∇f ⋅ u = ||∇f|| cos θ` ## Tangent Plane and Linear Approximation * **Tangent plane:** * `∇f(a,b,c)`⋅ where *(a,b,c)* is a point on the surface * **Tangent line of a curve:** `y = f'(a)(x-a) + f(a)` * **Linear approximation:** `dz = fx(a,b)(x-a) + fy(a,b)(y-b)` ## Maximum and Minimum Problems * Critical points: Find points where the gradient `∇f = 0` or is undefined. * Second derivative test: `D = fxxfyy - (fyx)²` * If `D>0` and `fxx>0` → local minimum * If `D>0` and `fxx