1. (10 pts) Find the critical points of the function
f (x, y) = 2x2
x2 y + y 2 .
For each critical point, classify it as a local minimum, local maximum, or saddle point.
2. (10 pts) Write at least two paragraphs that give a geometrical justification of wh
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3) m F"(f7f/<2c,os R, 1/ 231% Zt>
a. lm, W cfw_a MO) is m @ Val
Partial Derivatives 11.3
64
More partials
This works with more variables too.
@
@z
e xy ln z =
and
@
@x
e xy ln z =
We can also take higher derivatives.
@ @
@x @x f (x, y )
or
@2
f (x, y )
@x 2
or
fxx (x, y )
We might even decide to mix our partial deriva
Calculus with parametric curves 9.2
9
Finding slope on a parametric curve
When y is a function of x, what is the slope of the tangent line?
For a parametric curve cfw_x = f (t), y = g (t),
=
dy
dx
I Curve has a horizontal tangent where
dy
dt
= 0 and
I Cur
Area and arc length with parametric curves 9.2
14
Area under a parametric curve
Given y = f (x), the area under the curve from x = a to x = b is
Z x=b
Z t=
right endpoint
right endpoint
Area =
y dx
=
g (t) f 0 (t) dt
x=a
left endpoint
t=
left endpoint
Exa
0
Course Notes
Multivariable Calculus, Fall 2015
Queens College, Math 201
Prof. Christopher Hanusa
http:/qc.edu/~chanusa/courses/201/15/
Introduction
1
Class Introductions
Arrange yourselves into groups of four or five people,
With people you dont know.
I
Equations of Lines 10.5
32
Lines, Planes, and Automobiles!
Lines in 2D Coordinates
Two common formats:
Lines in 3D Coordinates
y = mx + b (slope-intercept) or
(y y0 ) = m(x x0 ) (pt-slope)
Given a point and a direction, you
hx, y , zi = hx0 , y0 , z0 i +
Limits and Continuity 11.2
58
Limits
Function of one variable
lim f (x) = L
x!a
Function of several variables
lim
f (x, y ) = L
(x,y )!(a,b)
Visually:
Visually:
Interpretation:
Interpretation:
However you approach x = a, the
value f (x) always approaches
Chain Rule 11.5
71
Chain Rule
Function of one variable
Suppose y = f (x) and x = g (t).
That is, y = f g (t) .
The chain rule gives:
dy
dy dx
=
dt
dx dt
dy
= f 0 (g (t) g 0 (t)
dt
Key idea:
You must add contributions
from all dependencies.
Function of sev
Dot products 10.3
25
What else can we do with vectors?
How to multiply two vectors:
~u ~v In any dimension: dot product. Answer is a number. Easy.
~u ~v In 3 dimensions: cross product. Answer is a vector. Memorize.
Dot product
Let ~a and ~b be vectors of
Functions of Several Variables 11.1
54
Functions of Several Variables
Function of one variable
f :R!R
f : x 7! f (x)
f takes in a real number x
outputs real number y = f (x)
Domain: x-vals where f defined.
Range: y -vals that f can output.
Function of sev
Vector functions 10.7
42
Functions
Single-variable functions
f :R!R
f : x 7! f (x)
f takes in a real number x
outputs a real number f (x)
Vector functions
~r : R ! R3
(or R2 or Rn )
~r : t 7! hf (t), g (t), h(t)i
~r takes in a real number t
outputs a vect
Three-dimensional coordinate system 10.1
19
Coordinates, Now with More Dimensions.
2D Coordinates
3D Coordinates
Variables: x (indep), y (dep)
Axes: x-axis ? y -axis
Variables: x, y (indep), z (dep)
Axes: x-axis ? y -axis ? z-axis
(drop a line ? from poin
Directional Derivatives 11.6
74
Definition of the directional derivative
Partial derivatives allow us to see how fast a function changes.
Dx f =fx (x, y ) is the rate of change of f in the x-direction. Toward ~i = (1, 0)
Dy f =fy (x, y ) is the rate of ch
Arc length 10.8
49
Arc length
The arc length of a vector function is calculated by:
Z t=b q
Z t=b
f 0 (t)2 + g 0 (t)2 + h0 (t)2 dt =
|~r 0 (t)| dt
t=a
The arc length function is s(t) =
Z
t=a
u=t
u=a
|~r 0 (u)| du.
I We are using u as the parametrization v
Quadric Surfaces 10.6
Drawing simple 3-D surfaces
Definition: Cylinders are surfaces where all slices are the same.
Example. z = x 2 .
y is not in this equation; y can be anything.
For any choice of y = k (parallel to
-plane),
the surface looks like a par
END-OF-TERM EXCEL PROJECT FALL 2016
Your end-of-term Excel project consists of the following problem and exercise from the text:
Problem 4-4 (part a only)
Exercise 5-12
The presentation of each exercise and problem should have appropriate arithmetic formu