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
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
.> m w dam 7a m W 0mm ~Hm
PDSHWUQ )(cQJPEC'Q/bn (3:?) buxom Pug: W <7Xjf>"<lla>=c
42 PMLD CEUIKA":Q IQ] cfw_JHe drecl'rgnJ CIQJJYW+ M Hu.
mlw :jrcsLiredTm (3%? LLCML DJ; :_ <1ch7> 6,40; D: 7
3) m F"(
ENDOFTERM EXCEL PROJECT FALL 2016
Your endofterm Excel project consists of the following problem and exercise from the text:
Problem 44 (part a only)
Exercise 512
The presentation of each exerci
Data Analysis
Planning and performing data analysis is an important part of the scientific process. It is a tool
that allows our reasoning to be more objective and less biased. Having a foundation in
Project data collection
You will spend these two class periods on campus collecting data for your group project. Make
sure to checkout and return all instrumentation and equipment. Your TA will be av
South Florida Ecosystems
Visit the link (https:/www.flmnh.ufl.edu/southflorida/home/) to learn about South Floridas
aquatic ecosystems. You can search by habitats and regions.
Your TA will pick an eco
Classroom Resource
Scientific Inquiry Using WildCam Gorongosa
Student Worksheet
INTRODUCTION
Gorongosa National Park is a 1,570squaremile protected area in Mozambique. Lion
researcher Paola Boul
Field methods in biology
Journaling
It is important to keep a detailed record of each time you work in the field or lab. Recording
things like recent weather, habitat descriptions, length of time in f
Systematics
Because life on earth is so diverse and so vast, biologists need a general organized system for
describing and classifying organisms. Systematics is a broad field concerned with classifica
Diversity of Life II
Domain: Eukaryotes
Kingdom: Plants
Division: Green algae
Green Algae (Chlorophyta and Charaphyta) live mostly in
freshwater ecosystems. Green algae exhibit considerable
variabilit
The Scientific Paper
Material from Grinnell College (2015) 11 pgs
III. Communicating the Results of Scientific Investigations
While many people investigate the world for the pure love of discovery, sc
Workshop: building a research project
Guidelines
1. Question, hypothesis, and predictions. Develop a question and hypothesis that interests
you. Think about the testability of your hypothesis.
2. Data
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 C
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
=
Quadric Surfaces 10.6
Drawing simple 3D 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
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

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 xdirection
Threedimensional coordinate system 10.1
19
Coordinates, Now with More Dimensions.
2D Coordinates
3D Coordinates
Variables: x (indep), y (dep)
Axes: xaxis ? y axis
Variables: x, y (indep), z (dep)
A
Vector functions 10.7
42
Functions
Singlevariable 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),
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: xvals where f def
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 vec
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 a
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
Equations of Lines 10.5
32
Lines, Planes, and Automobiles!
Lines in 2D Coordinates
Two common formats:
Lines in 3D Coordinates
y = mx + b (slopeintercept) or
(y y0 ) = m(x x0 ) (ptslope)
Given a poi
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 grou
Reading and Writing Science
Using Scientific Literature
As you explore the scientific literature, it is important to appreciate the distinction between its
different forms. The primary literature cons