Directional Derivatives and the Gradient Vector
Marius Ionescu
October 24, 2012
Marius Ionescu ()
Directional Derivatives and the Gradient Vector
October 24, 2012
1 / 12
Directional Derivatives
Fact
Recall:
f (x0 + h, y0 ) f (x0 , y0 )
h
h 0
f (x0 , y0 +
Math 113 - Calculus 11] Practice Exam I Spring 2003
Name:
1. Match each function to one of the given graphs. and to one of the given commu-
diagrams. The plots are gin-n in a separate handout. 2. (3) Consider the function
103.! : 12 - y 16'
Find a functi
Math 113 - Calculus II] Exam 2 Practice Problems Fall 2005
I. For each oi the following. nd the indicated partial derivative.
(I) line) = 29 - alum). fulll) =
(b) gum) = 0+". MM) =
9 =
= ob
(c)R mu. an
2. Let
1(1v)= xv + 2n: - 3v-
(a) What In the rate of
Math 3 - Calculus lll Chapter 16 Pracca Problems Fall 2003
1. Suppose the integral of some function I over a region R in the plane is given in polar mor-
dinntm 33 v
.3 .1
l I r2 d0dr.
. u . 0
(:5) Sketch the region of integration R in the my plane.
(b) C
Math 113 - Calcullm II] Final Exam Practice Problems Spring 2003
I. Let
giny. :) = V213 + y? + 49.
(8) Describe the shapes of the level surfaces of 9
(b) In three different graphs. sketch the three tress sections to the level surface
g(r.y.:) ' l for Wlll
Math 113 - Calculus 11] Exam 1 Practice Problems Fall 2005
Name: _
1. Match each function to one of the given graphs, and to one of the given contour
diagrams. The plots are given in a separate handout.
Contout
Graph Diagram 2. (3) Consider the function
Math 113 - Calculus II] Practice Exam 1 Spring 2003
Name: SOLUTIONS
1. Match each function to one of the given graphs. and no one of the glwn contour
diagrams. The plots ate given In a semi-we handout.
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0:
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ulss
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In the book How to slowly kill yourself and others in America, the author
Laymon uses the concept of an album to connect all tracks which are his
experiences with different people to show how a black person suffers from the
racism in America. Through the
2011 AP STATISTICS FREE-RESPONSE QUESTIONS
Formulas
(I)
Descriptive Statistics
xi
n
x
1
sx
1
n
xi
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xi
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b1
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2011 AP STATISTI
AP Statis
tics
Chen Tingjing
chentingjing024@hotmail.com
9.1 Point Estimation
A point estimate of a population characte
ristic is a single number that is
based on sample data and represents a pl
ausible value of the characteristic.
How to get a point est
Summary for discussion (page 63)
1. From our discussion, we reached a consensus that it would be easy to obtain
a simple random sample of students at our school. From our perspective, the
most challenging work for simple random sample of all students in t
AP Statis
tics
Chen Tingjing
chentingjing024@hotmail.com
Large-Sample Confidence Interval for a
Population Proportion
Because of sampling variability, rarely is t
he point estimate from a sample exactly e
qual to the true value of the population cha
ract
AP
Statistic
s
Chen Tingjing
chentingjing024@hotmail.com
Large-Sample Confidence Interval for a
Population Proportion
Because of sampling variability, rarely is
the point estimate from a sample exactly
equal to the true value of the population
characteris
AP
Statistic
s
Chen Tingjing
chentingjing024@hotmail.com
Confidence Interval for a
Population Mean
2
Example 9.7 Cosmic Radiation
t Distributions
One-Sample t Confidence Interval
15/9/27
homework
9.34
15/9/27
9.41
Questionnaire
1. Have you registered for any AP Exam?
A. Yes (Please continue answering this questionnaire) 42
B. No (This is the end of the questionnaire. Thanks for your participation. ) 8
2. How many AP exams did you register?
A. 1-2 21
B. 3-4 17
C. 5-
Group 5
!
SURVEY OVERVIEW
!
2012530513 2012530122
2012530412
2013530424
Our group choose The Survey and Analysis on the AP Exams Registration of
Shenzhen Middle School IC Freshmen as our topic. We analyze and research the
senior one students attitudes, u
Math 113 - CalcuIns lll Exam 3 Practice Problems Spring 2004
l. Lot.
9(I,y.z) = e" + 2%: + y}.
Suppose that a piece of fruit 18 sitting on a table in a room. and at each point (any, 2)
in the space within the room. 9(1. y, :) gives the atrength of the od
The Chain Rule
Marius Ionescu
October 19, 2012
Marius Ionescu ()
The Chain Rule
October 19, 2012
1/5
The Chain Rule (case 1)
Denition
Suppose that z = f (x , y ) is a dierentiable function of x and y , where
x = g (t ) and y = h(t ) are both dierentiable
Directional Derivatives and the Gradient Vector
Part 2
Marius Ionescu
October 26, 2012
Marius Ionescu ()
Directional Derivatives and the Gradient Vector Part 2
October 26, 2012
1 / 12
Recall
Fact
Marius Ionescu ()
Directional Derivatives and the Gradient
14.7: Maxima and Minima
Marius Ionescu
October 29, 2012
Marius Ionescu ()
14.7: Maxima and Minima
October 29, 2012
1 / 13
Local Maximum and Local Minimum
Denition
Marius Ionescu ()
14.7: Maxima and Minima
October 29, 2012
2 / 13
Local Maximum and Local Mi
15.1: Double Integrals over Rectangles
Marius Ionescu
November 14, 2012
Marius Ionescu ()
15.1: Double Integrals over Rectangles
November 14, 2012
1 / 12
Volumes and Double Integrals
Let
f
be a function of two variables dened on a closed rectangle:
R
= cf
Maxima and Minima
Marius Ionescu
November 5, 2012
Marius Ionescu ()
Maxima and Minima
November 5, 2012
1/7
Second Derivative Test
Fact
Suppose the second partial derivatives of f are continuous on a disk with
center (a, b), and suppose that fx (a, b) = 0
14.4: Tangent Planes and Linear Approximations
Marius Ionescu
October 15, 2012
Marius Ionescu ()
14.4: Tangent Planes and Linear Approximations
October 15, 2012
1 / 13
Tangent Planes
Denition
Marius Ionescu ()
14.4: Tangent Planes and Linear Approximation
14.3: Partial Derivatives
Marius Ionescu
October 12, 2012
Marius Ionescu ()
14.3: Partial Derivatives
October 12, 2012
1 / 13
Partial derivative of f
with respect to x
Denition
Marius Ionescu ()
14.3: Partial Derivatives
October 12, 2012
2 / 13
Partial de
The Cross Product
Lecture 3
September 7, 2012
()
The Cross Product
September 7, 2012
1/7
Properties
Given two vectors a and b we want to build a vector a b called
the cross product of a and b with the following properties:
The vector a b is orthogonal to
The Dot Product
Lecture 2
Marius Ionescu
August 31, 2012
Marius Ionescu ()
The Dot Product
August 31, 2012
1/9
The Dot Product
If a = a1 , a2 , a3 and b = b1 , b2 , b3 , then the dot product
of a and b is the number
a b = a1 b1 + a2 b2 + a3 b3 .
Marius Io
12.5: Lines and Planes in R3
Lecture 4
Marius Ionescu
September 10, 2012
Marius Ionescu ()
12.5: Lines and Planes in R3
September 10, 2012
1 / 13
The vector equation of a line
A line L is determined when we know a point P0 (x0 , y0 , z0 ) on
L and the dir