ObsidianVault/.trash/Class 11-13.md

Calculus 3

• Chain rule
  • If there is a function z(x(t), y(t)) then \frac{dz}{dt} = \frac{dz}{dx} \frac{dx}{dt} + \frac{dz}{dy} \frac{dy}{dt}
  • If z=f(x(u,v), y(u,v)) then \frac{{\partial z}}{\partial u} = \frac{{\partial z}}{\partial x} \frac{{\partial x}}{\partial u} + \frac{{\partial z}}{\partial y} \frac{{\partial y}}{\partial u} \frac{{\partial z}}{\partial v} = \frac{{\partial z}}{\partial x} \frac{{\partial x}}{\partial v} + \frac{{\partial z}}{\partial y} \frac{{\partial y}}{\partial v}
• Implicit differentiation
  • If z is defined implicitly as a function of x and y, then \frac{dz}{dx} = - \frac{{\frac{{\partial f}}{\partial x}}}{\frac{{\partial f}}{\partial z}} \frac{dz}{dx} = - \frac{{\frac{{\partial f}}{\partial y}}}{\frac{{\partial f}}{\partial z}}
• Critical points
  • For functions of two variables, this is when they both equal 0 or when one is undefined
  • Second derivative test: D = f_{x x}(x_{0}, y_{0})f_{y y}(x_{0}, y_{0}) - (f_{x y}(x_{0}, y_{0}))^2
    • if D>0 and f_{x x}(x_{0}, y_{0})>0 then f has a local minimum at (x_{0}, y_{0})
    • if D>0 and f_{x x}(x_{0}, y_{0})<0 then f has a local maximum at (x_{0}, y_{0})
    • if D<0 then f has a saddle point at (x_{0}, y_{0})
    • if D=0 then the test is inconclusive