Linear Approximation

• A tangent line to $y=f(x)$ at $x=a$ is
• $y=f(a)+f^{\prime }(a)(x-a)$
• Derivation:

• You can use this to approximate a difficult function at an x value close to an easy number (eg. $(0.97{)}^{\frac{1}{3}}$)
  • Simply set a to the nice number
• Of course, this is extremely bald so use Taylor Series

Planar Approximation

• It's the same, but a plane
• So $y-f(a)=f^{\prime }(a)(x-a)$ becomes $z-f(a,b)=\frac{\partial f}{\partial x}(a,b)(x-a)+\frac{\partial f}{\partial y}(a,b)(y-b)$
  • This can also be written as $\mathrm{\Delta }f\approx f_x(a,b)\mathrm{\Delta }x+f_y(a,b)\mathrm{\Delta }y$
  • $\mathrm{d}f=f_x(a,b)\mathrm{d}x+f_y(a,b)\mathrm{d}y=\frac{\partial f}{\partial x}\mathrm{d}x+\frac{\partial f}{\partial y}\mathrm{d}y$
• very natural
• The linearization of $f(x,y)$ at the point $(a,b)$ is $f(x,y)=f(a,b)+\frac{\partial f}{\partial x}f(a,b)(x-a)+\frac{\partial f}{\partial y}(a,b)(y-b)=f(a,b)+\overrightarrow{\mathrm{\nabla }}f(a,b)\cdot (x-a,y-b)$