Lecture 09     

Linearization

APMA E2000

Drew Youngren dcy2@columbia.edu

$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$ $\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$ $\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$

Announcements

  • Midterm 1 Thursday, 2/20.
    • No outside notes/calculators
    • Review Wednesday morning
    • Seat bonus (details to follow)
  • Recitation optional (office hours)
  • No quiz this week
  • (Short) HW5 due next Tues

1-minute review

Partial $\partial$erivatives

The partial derivative of a function $f(x_1,\ldots,x_n)$ with respect to $x_i$ is \[\frac{\partial f}{\partial x_i} = \lim_{h\to 0} \frac{f(x_1,\ldots,x_i+h,\ldots,x_n) - f(x_1,\ldots,x_n)}{h}.\]

Linear Functions & Differentiability

A linear function on $\RR^n$ has the form \[L(x_1, \ldots, x_n) = a_0 + \sum_{i=1}^n a_i x_i\] where the $a_i$'s are constants.

*really affine linear

Key Example

In one variable, a linear function $L(x)$ is one whose graph is a line\[y = ax + b.\]

In two variables, a linear function $L(x,y)$ is one whose graph is a plane \[z = ax + by + c. \]

Differentiability

In one dimension, a function $f(x)$ is differentiable at $a$ if $f'(a)$ exists.

Graphically, this means the graph resembles a line at small scales.

Differentiability

In 2-D, a function $f(x,y)$ is differentiable at a point if its graph looks like a plane close-up.

Smooth example
Nonsmooth example
Definition

A scalar field $f(x_1, \ldots, x_n)$ is differentiable at $(p_1, \ldots, p_n)$ provided there is a linear function $L(x_1, \ldots, x_n)$ such that \[f(x_1, \ldots, x_n) - L(x_1, \ldots, x_n) = \sum_{i=1}^n \epsilon_i (x_i - p_i)\] where each $\epsilon_i \to 0$ as $\vec x \to \vec p$.

Example

Show $f(x) = x^3$ is differentiable at $x=1$.

Solution. Let $L(x) = 3x - 2$. Then, \[ f(x) - L(x) = x^3 - 3x + 2 \] \[ = (x^2 + x - 2)(x - 1) \]

Tangent Plane

Key example

We start with the graph of a smooth 2-D function \[ z = f(x,y) = \ln(x + 2y + 1) \] and calculate the tangent plane at $(1,2, \ln 6)$.

Program for Tangent Plane
  1. Construct curve in $x$-direction $\langle t, 2, \ln(t + 5) \rangle$
  2. Take derivative at $t=1$, $\langle 1, 0, \frac{1}{6} \rangle$.
  3. Construct curve in $y$-direction $\langle 1, t, \ln(2 + 2t) \rangle$
  4. Take derivative at $t=2$, $\langle 0, 1, \frac{1}{3} \rangle$.
  5. Cross tangent vectors to get normal.
  6. Form plane equation $-\frac{x}{6} -\frac{y}{3} + z = \ln 6 - \frac56$.
  7. Profit.
Tangent Plane at $(a, b, f(a,b))$.
  1. Construct curve in $x$-direction $\langle t, b, f(t,b) \rangle$
  2. Take derivative at $t=a$, $\langle 1, 0, f_x(a,b) \rangle$.
  3. Construct curve in $y$-direction $\langle a, t, f(a,t) \rangle$
  4. Take derivative at $t=b$, $\langle 0, 1, f_y(a,b) \rangle$.
  5. Cross tangent vectors to get normal $\vec n = \langle -f_x(a,b), -f_y(a,b), 1 \rangle$.
  6. Form plane equation.
  7. Profit. \[ z = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b). \]
Normal vector
\[ \vec n = \left\langle -f_x(a,b), -f_y(a,b), 1 \right\rangle \]
Tangent Plane
\[ z = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b) \]
\[ = L(x,y) \] This is the linearization of $f$ at the point $(a,b)$.

Linearization

Definition
The linearization of a function $f$ at $\vec p = \langle p_1, \ldots, p_n \rangle$ is a linear function \[L(\vec x) = f(\vec p) + \sum_{i=1}^n \frac{\partial f}{\partial x_i}(\vec p)(x_i - p_i) \]

Use linearization to estimate values of a function nearby known points.

Example

\[f(x,y,z) = e^{-x^2 + 3y + 3z}\] Estimate $f(3.01, 2.02, 0.99)$.

Example

Estimate $\sqrt{2/7}$.

  1. Use $f(x)= \sqrt{\frac{2}{x}}$ around $x = 8$.
Example

Estimate $\sqrt{2/7}$.

  1. Use $g(x,y) = \sqrt{\frac{x}{y}}$ around $(9, 36)$.
Example

Estimate $\sqrt{2/7}$.

  1. Use $f(x)= \sqrt{\frac{2}{x}}$ around $x = 8$. \[\sqrt{\frac27} = f(7) \approx f(8) + f'(8)(-1) = \frac{17}{32}\]
  2. Use $g(x,y) = \sqrt{\frac{x}{y}}$ around $(9, 36)$.\[\sqrt{\frac{10}{35}} \approx \frac12 + g_x(9,36)(1) + g_y(9,36)(-1) = \frac{77}{144}\]

Differentials

An alternative, compact approach to the same concept is to estimate changes in functions using differentials.

We write $\Delta f = f(x,y) - f(a,b) \approx $ \[df = f_x(a,b)dx + f_y(a,b)dy\]

Example

A cylindrical aluminum can is is 5cm high with a diameter of 6cm and is 0.4mm thick. Estimate the volume of the aluminum.

Solution. Let $V = \pi r^2 h$. The volume of the aluminum is approximately \[ dV = 2\pi r h\, dr + \pi r^2 \,dh = \pi(30\times0.04 + 9\times 0.08) \]

Learning Outcomes

You should be able to...
  • Determine (quickly) whether a functions of several variables is linear.
  • Identify the correct linearization formula for a given function and connect it to tangent lines/planes.
  • Identify an appropriate function and point to use linearization in estimating various quantities.
  • Use linearization and/or differentials to estimate error (and relative error.)