Linear Approximations

Tangent Lines

• Each partial derivative forms a tangent line parallel to that coordinate plane

$$\begin{array}{r}{\vec{v}}_1=⟨1,0,f_x^{\prime }(x_0,y_0)⟩\\ {\vec{v}}_2=⟨0,1,f_y^{\prime }(x_0,y_0)⟩\end{array}$$

Relationship to Tangents

• $\tan {\theta }_x=f_x^{\prime }(x_0,y_0)$ where ${\theta }_x$ is the angle between the tangent line and a horizontal line
• Same applies for y

Tangent Plane

• The plane that contains both tangent lines
• $\vec{n}=⟨f_x^{\prime }(x_0,y_0),f_y^{\prime }(x_0,y_0),-1⟩$
• $\mathcal{P}:z=z_0+f_x^{\prime }(x_0,y_0)(x-x_0)+f_y^{\prime }(x_0,y_0)(y-y_0)$
• Serves the same purpose as tangent lines just for multiple variables
• The negative one in $\vec{n}$ doesn’t always have to be on the $z$, if there is a partial deriv for $z$ but not a different variable they would switch places

Tangent Plane Approximation

1. Round values of x and y to get $x=x_0+\Delta x$ and $y=y_0+\Delta y$
2. Find $f(x_0,y_0)$, $f_x^{\prime }(x_0,y_0)$, and $f_y^{\prime }(x_0,y_0)$
3. $L(x,y)=f(x_0,y_0)+f_x^{\prime }(x_0,y_0)\Delta x+f_y^{\prime }(x_0,y_0)\Delta y$

Multivariable Newton’s Method

• Approximates a $\Delta x$ and $\Delta y$ towards the $x$ and $y$ values that satisfy the following system from an initial point $P_0$
  • $f(x,y)=a,g(x,y)=b$
• Needs a solution near the point where you start
• To do so solve this system of equations for $\Delta x$ and $\Delta y$ :
  • $f_x^{\prime }(x_1,y_1)\Delta x_1+f_y^{\prime }(x_1,y_1)\Delta y_1=a-f(x_1,y_1)$
  • $g_x^{\prime }(x_1,y_1)\Delta x_1+g_y^{\prime }(x_1,y_1)\Delta y_1=b-g(x_1,y_1)$
• Can be iterated. Take the new point and treat it as a starting point and repeat