MATH 19520/51 Class 5

Size: px
Start display at page:

Download "MATH 19520/51 Class 5"

Transcription

1 MATH 19520/51 Class 5 Minh-Tam Trinh University of Chicago

2 1 Definition of partial derivatives. 2 Geometry of partial derivatives. 3 Higher derivatives. 4 Definition of a partial differential equation (PDE). 5 Cobb Douglas production function revisited.

3 Partial Derivatives What is... The partial derivative of x y with respect to x? The partial derivative of log(ab) with respect to a? The partial derivative of 1 xy+xz+yz with respect to x?

4 Partial Derivatives What is... The partial derivative of x y with respect to x? yx y 1 The partial derivative of log(ab) with respect to a? The partial derivative of 1 xy+xz+yz with respect to x?

5 Partial Derivatives What is... The partial derivative of x y with respect to x? yx y 1 The partial derivative of log(ab) with respect to a? 1 a The partial derivative of 1 xy+xz+yz with respect to x?

6 Partial Derivatives What is... The partial derivative of x y with respect to x? yx y 1 The partial derivative of log(ab) with respect to a? 1 a The partial derivative of 1 xy+xz+yz with respect to x? y+z (xy+xz+yz) 2

7 In general, let f be a function of n variables x 1,..., x n.

8 In general, let f be a function of n variables x 1,..., x n. Formally, the partial derivative of f with respect to x i is (1) lim ɛ 0 f (x 1,..., x i + ɛ,..., x n ) f (x 1,..., x i,..., x n ). ɛ Informally, it s the derivative we get by treating the other variables x 1,..., x i 1, x i+1,..., x n as constants.

9 There are several different notations for partial derivatives: (2) f x, xf, D x f, f x all mean the partial derivative of f with respect to x. Notice that we use instead of d.

10 Example If f (x, y, z) = x yz, then what is f z (e, e, e)?

11 Example If f (x, y, z) = x yz, then what is f z (e, e, e)? Use the chain rule: (3) f z = (xyz log x) z (yz ) = (x yz log x)(y z log y). Then f z (e, e, e) = (e ee log e)(e e log e) = e ee e e = e ee +e.

12 Example ( Let f (x, y) = sin x 1 + y ). Evaluate x f and y f at (x, y) = (π, 2).

13 Example ( Let f (x, y) = sin x 1 + y We compute ( ) f x = cos x (4) 1 + y ). Evaluate x f and y f at (x, y) = (π, 2). ( x ) x 1 + y = ( ) y cos x 1 + y. (5) ( ) f y = cos x 1 + y ( x ) y 1 + y ( x = (1 + y) 2 cos x 1 + y ).

14 Example ( Let f (x, y) = sin x 1 + y We compute ( ) f x = cos x (4) 1 + y ). Evaluate x f and y f at (x, y) = (π, 2). ( x ) x 1 + y = ( ) y cos x 1 + y. (5) ( ) f y = cos x 1 + y ( x ) y 1 + y ( x = (1 + y) 2 cos x 1 + y ). Thus, f x (π, 2) = 1 3 cos π 3 = 1 6 and f y(π, 2) = π 9 cos π 3 = π 18.

15 Geometry of Partial Derivatives In 1-variable calculus, df dx (a) measures the rate of change of f in the positive x-direction at a. This is the slope of the tangent line to the graph of f at a.

16 Geometry of Partial Derivatives In 1-variable calculus, df dx (a) measures the rate of change of f in the positive x-direction at a. This is the slope of the tangent line to the graph of f at a. In 2-variable calculus, f f x (a, b) and y (a, b) respectively measure the rate of change of f in the positive x- and y-directions at (a, b). These are the slopes of tangent lines to the graph of f at (a, b). Which tangent lines?

17

18 Higher Derivatives A higher derivative is when we apply several partial derivatives in succession. Stewart s notation: 2 f x 2 = ( ) f 2 f x x y x = ( ) f y x (6) and 2 f x y = ( ) f x y 2 f y 2 = ( ) f y y (7) f xx = (f x ) x f yx = (f y ) x f xy = (f x ) y f yy = (f y ) y

19 f xx, f xy, etc. are called 2 nd -order partial derivatives. In 2 variables, there are 4 kinds of 2 nd -order partial derivatives we can take: xx, xy, yx, yy.

20 f xx, f xy, etc. are called 2 nd -order partial derivatives. In 2 variables, there are 4 kinds of 2 nd -order partial derivatives we can take: xx, xy, yx, yy. In 3 variables, how many kinds of 2 nd -order partial derivatives can we take?

21 f xx, f xy, etc. are called 2 nd -order partial derivatives. In 2 variables, there are 4 kinds of 2 nd -order partial derivatives we can take: xx, xy, yx, yy. In 3 variables, how many kinds of 2 nd -order partial derivatives can we take? xx, xy, xz, yx, yy, yz, zx, zy, zz.

22 Example Find all 2 nd -order partial derivatives of f (x, y) = x 2 + 2y 2 + 3xy.

23 Example Find all 2 nd -order partial derivatives of f (x, y) = x 2 + 2y 2 + 3xy. Compute: (8) f xx = 2 f xy = 3 f yx = 3 f yy = 4 Notice that f xy = f yx.

24 Theorem (Clairaut) Let f be a differentiable function of x and y. If f x and f y exist and are continuous on a disk around (a, b), then (9) f xy (a, b) = f yx (a, b).

25 Theorem (Clairaut) Let f be a differentiable function of x and y. If f x and f y exist and are continuous on a disk around (a, b), then (9) f xy (a, b) = f yx (a, b). Warning! If f x and f y are not continuous at (0, 0), then Clairaut doesn t apply. If f (x, y) = xy(x2 y 2 ), then we can check that f x 2 +y 2 x and f y are discontinuous at (0, 0). We compute f xy (0, 0) = 1 1 = f yx (0, 0). s_theorem_where_ both_mixed_partials_are_defined_but_not_equal

26 Similarly, we can talk about 3 rd -order partial derivatives.

27 Similarly, we can talk about 3 rd -order partial derivatives. Example Find g xyz where g(x, y, z) = e xyz.

28 Similarly, we can talk about 3 rd -order partial derivatives. Example Find g xyz where g(x, y, z) = e xyz. Compute z ( y ( x (e xyz ))) = z ( y (e xyz yz)) (10)

29 Similarly, we can talk about 3 rd -order partial derivatives. Example Find g xyz where g(x, y, z) = e xyz. Compute (10) z ( y ( x (e xyz ))) = z ( y (e xyz yz)) = z (e xyz xz yz + e xyz z) = z (e xyz (xyz 2 + z))

30 Similarly, we can talk about 3 rd -order partial derivatives. Example Find g xyz where g(x, y, z) = e xyz. Compute (10) z ( y ( x (e xyz ))) = z ( y (e xyz yz)) = z (e xyz xz yz + e xyz z) = z (e xyz (xyz 2 + z)) = e xyz xy (xyz 2 + z) + e xyz (2xyz + 1) = e xyz (x 2 y 2 z 2 + 3xyz + 1).

31 Partial Differential Equations A partial differential equation (PDE) is any equation involving the partial derivatives of a function. The order of the equation is the highest order among the partial derivatives involved.

32 Partial Differential Equations A partial differential equation (PDE) is any equation involving the partial derivatives of a function. The order of the equation is the highest order among the partial derivatives involved. Example Laplace s equation for a function u(x, y) is a 2 nd -order PDE: (11) u xx + u yy = 0. Think of the function u as a variable in its own right.

33 Solution to PDEs are functions, not numbers. Just as an ordinary equation can have many numbers as solutions, a PDE can have many functions as solutions.

34 Solution to PDEs are functions, not numbers. Just as an ordinary equation can have many numbers as solutions, a PDE can have many functions as solutions. Example The function u(x, y) = x solves Laplace s equation: (12) (x) xx + (x) yy = = 0. But so does u(x, y) = x 2 y 2 : (13) (x 2 y 2 ) xx + (x 2 y 2 ) yy = 2 2 = 0.

35 Cobb Douglas Revisited Recall the Cobb Douglas production function: (14) P(L, K) = bl α K 1 α.

36 Cobb Douglas Revisited Recall the Cobb Douglas production function: (14) P(L, K) = bl α K 1 α. It is a solution to the system of equations P(L, 0) = P(0, K) = 0 (15) P L = α P L P K = β P K α + β = 1 P P L is marginal productivity of labor and K capital. is marginal productivity of

37 What do these equations mean? P(L, 0) = P(0, K) = 0 P L = α P L P K = β P K α + β = 1

38 What do these equations mean? P(L, 0) = P(0, K) = 0 To be productive, you need both labor and capital. P L = α P L P K = β P K α + β = 1

39 What do these equations mean? P(L, 0) = P(0, K) = 0 P L = α P L To be productive, you need both labor and capital. The rate at which labor increases productivity is proportional to the ratio of productivity to labor. P K = β P K α + β = 1

40 What do these equations mean? P(L, 0) = P(0, K) = 0 P L = α P L P K = β P K To be productive, you need both labor and capital. The rate at which labor increases productivity is proportional to the ratio of productivity to labor. The rate at which capital increases productivity is proportional to the ratio of productivity to capital. α + β = 1

41 What do these equations mean? P(L, 0) = P(0, K) = 0 P L = α P L P K = β P K α + β = 1 To be productive, you need both labor and capital. The rate at which labor increases productivity is proportional to the ratio of productivity to labor. The rate at which capital increases productivity is proportional to the ratio of productivity to capital. We want scaling factors to work: P(aL, ak) = ap(l, K).

MATH 19520/51 Class 4

MATH 19520/51 Class 4 MATH 19520/51 Class 4 Minh-Tam Trinh University of Chicago 2017-10-02 1 Functions and independent ( nonbasic ) vs. dependent ( basic ) variables. 2 Cobb Douglas production function and its interpretation.

More information

14.3 Partial Derivatives

14.3 Partial Derivatives 14 14.3 Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. On a hot day, extreme humidity makes us think the temperature is higher than it really is, whereas

More information

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics 3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)

More information

Functions of Several Variables

Functions of Several Variables Functions of Several Variables Partial Derivatives Philippe B Laval KSU March 21, 2012 Philippe B Laval (KSU) Functions of Several Variables March 21, 2012 1 / 19 Introduction In this section we extend

More information

g(t) = f(x 1 (t),..., x n (t)).

g(t) = f(x 1 (t),..., x n (t)). Reading: [Simon] p. 313-333, 833-836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018

Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018 Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent

More information

Sec. 14.3: Partial Derivatives. All of the following are ways of representing the derivative. y dx

Sec. 14.3: Partial Derivatives. All of the following are ways of representing the derivative. y dx Math 2204 Multivariable Calc Chapter 14: Partial Derivatives I. Review from math 1225 A. First Derivative Sec. 14.3: Partial Derivatives 1. Def n : The derivative of the function f with respect to the

More information

Complex Variables. Chapter 2. Analytic Functions Section Harmonic Functions Proofs of Theorems. March 19, 2017

Complex Variables. Chapter 2. Analytic Functions Section Harmonic Functions Proofs of Theorems. March 19, 2017 Complex Variables Chapter 2. Analytic Functions Section 2.26. Harmonic Functions Proofs of Theorems March 19, 2017 () Complex Variables March 19, 2017 1 / 5 Table of contents 1 Theorem 2.26.1. 2 Theorem

More information

Comparative Statics. Autumn 2018

Comparative Statics. Autumn 2018 Comparative Statics Autumn 2018 What is comparative statics? Contents 1 What is comparative statics? 2 One variable functions Multiple variable functions Vector valued functions Differential and total

More information

100 CHAPTER 4. SYSTEMS AND ADAPTIVE STEP SIZE METHODS APPENDIX

100 CHAPTER 4. SYSTEMS AND ADAPTIVE STEP SIZE METHODS APPENDIX 100 CHAPTER 4. SYSTEMS AND ADAPTIVE STEP SIZE METHODS APPENDIX.1 Norms If we have an approximate solution at a given point and we want to calculate the absolute error, then we simply take the magnitude

More information

Multivariable Calculus and Matrix Algebra-Summer 2017

Multivariable Calculus and Matrix Algebra-Summer 2017 Multivariable Calculus and Matrix Algebra-Summer 017 Homework 4 Solutions Note that the solutions below are for the latest version of the problems posted. For those of you who worked on an earlier version

More information

Limits and Continuity/Partial Derivatives

Limits and Continuity/Partial Derivatives Christopher Croke University of Pennsylvania Limits and Continuity/Partial Derivatives Math 115 Limits For (x 0, y 0 ) an interior or a boundary point of the domain of a function f (x, y). Definition:

More information

Partial Derivatives October 2013

Partial Derivatives October 2013 Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a

More information

ENGI Partial Differentiation Page y f x

ENGI Partial Differentiation Page y f x ENGI 344 4 Partial Differentiation Page 4-0 4. Partial Differentiation For functions of one variable, be found unambiguously by differentiation: y f x, the rate of change of the dependent variable can

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known

More information

PARTIAL DERIVATIVES AND THE MULTIVARIABLE CHAIN RULE

PARTIAL DERIVATIVES AND THE MULTIVARIABLE CHAIN RULE PARTIAL DERIVATIVES AND THE MULTIVARIABLE CHAIN RULE ADRIAN PĂCURAR LAST TIME We saw that for a function z = f(x, y) of two variables, we can take the partial derivatives with respect to x or y. For the

More information

z 2 = 1 4 (x 2) + 1 (y 6)

z 2 = 1 4 (x 2) + 1 (y 6) MA 5 Fall 007 Exam # Review Solutions. Consider the function fx, y y x. a Sketch the domain of f. For the domain, need y x 0, i.e., y x. - - - 0 0 - - - b Sketch the level curves fx, y k for k 0,,,. The

More information

Example: Limit definition. Geometric meaning. Geometric meaning, y. Notes. Notes. Notes. f (x, y) = x 2 y 3 :

Example: Limit definition. Geometric meaning. Geometric meaning, y. Notes. Notes. Notes. f (x, y) = x 2 y 3 : Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a

More information

MA102: Multivariable Calculus

MA102: Multivariable Calculus MA102: Multivariable Calculus Rupam Barman and Shreemayee Bora Department of Mathematics IIT Guwahati Differentiability of f : U R n R m Definition: Let U R n be open. Then f : U R n R m is differentiable

More information

MATH20411 PDEs and Vector Calculus B

MATH20411 PDEs and Vector Calculus B MATH2411 PDEs and Vector Calculus B Dr Stefan Güttel Acknowledgement The lecture notes and other course materials are based on notes provided by Dr Catherine Powell. SECTION 1: Introctory Material MATH2411

More information

Module Two: Differential Calculus(continued) synopsis of results and problems (student copy)

Module Two: Differential Calculus(continued) synopsis of results and problems (student copy) Module Two: Differential Calculus(continued) synopsis of results and problems (student copy) Srikanth K S 1 Syllabus Taylor s and Maclaurin s theorems for function of one variable(statement only)- problems.

More information

2.12: Derivatives of Exp/Log (cont d) and 2.15: Antiderivatives and Initial Value Problems

2.12: Derivatives of Exp/Log (cont d) and 2.15: Antiderivatives and Initial Value Problems 2.12: Derivatives of Exp/Log (cont d) and 2.15: Antiderivatives and Initial Value Problems Mathematics 3 Lecture 14 Dartmouth College February 03, 2010 Derivatives of the Exponential and Logarithmic Functions

More information

MATH 31BH Homework 5 Solutions

MATH 31BH Homework 5 Solutions MATH 3BH Homework 5 Solutions February 4, 204 Problem.8.2 (a) Let x t f y = x 2 + y 2 + 2z 2 and g(t) = t 2. z t 3 Then by the chain rule a a a D(g f) b = Dg f b Df b c c c = [Dg(a 2 + b 2 + 2c 2 )] [

More information

Math 212-Lecture 8. The chain rule with one independent variable

Math 212-Lecture 8. The chain rule with one independent variable Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle

More information

Partial derivatives BUSINESS MATHEMATICS

Partial derivatives BUSINESS MATHEMATICS Partial derivatives BUSINESS MATHEMATICS 1 CONTENTS Derivatives for functions of two variables Higher-order partial derivatives Derivatives for functions of many variables Old exam question Further study

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

More information

ENGI Partial Differentiation Page y f x

ENGI Partial Differentiation Page y f x ENGI 3424 4 Partial Differentiation Page 4-01 4. Partial Differentiation For functions of one variable, be found unambiguously by differentiation: y f x, the rate of change of the dependent variable can

More information

Partial Derivatives for Math 229. Our puropose here is to explain how one computes partial derivatives. We will not attempt

Partial Derivatives for Math 229. Our puropose here is to explain how one computes partial derivatives. We will not attempt Partial Derivatives for Math 229 Our puropose here is to explain how one computes partial derivatives. We will not attempt to explain how they arise or why one would use them; that is left to other courses

More information

Math Review ECON 300: Spring 2014 Benjamin A. Jones MATH/CALCULUS REVIEW

Math Review ECON 300: Spring 2014 Benjamin A. Jones MATH/CALCULUS REVIEW MATH/CALCULUS REVIEW SLOPE, INTERCEPT, and GRAPHS REVIEW (adapted from Paul s Online Math Notes) Let s start with some basic review material to make sure everybody is on the same page. The slope of a line

More information

ECON 186 Class Notes: Optimization Part 2

ECON 186 Class Notes: Optimization Part 2 ECON 186 Class Notes: Optimization Part 2 Jijian Fan Jijian Fan ECON 186 1 / 26 Hessians The Hessian matrix is a matrix of all partial derivatives of a function. Given the function f (x 1,x 2,...,x n ),

More information

MATH The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input,

MATH The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input, MATH 20550 The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input, F(x 1,, x n ) F 1 (x 1,, x n ),, F m (x 1,, x n ) where each

More information

Lecture 5 - Logarithms, Slope of a Function, Derivatives

Lecture 5 - Logarithms, Slope of a Function, Derivatives Lecture 5 - Logarithms, Slope of a Function, Derivatives 5. Logarithms Note the graph of e x This graph passes the horizontal line test, so f(x) = e x is one-to-one and therefore has an inverse function.

More information

MATH 255 Applied Honors Calculus III Winter Homework 5 Solutions

MATH 255 Applied Honors Calculus III Winter Homework 5 Solutions MATH 255 Applied Honors Calculus III Winter 2011 Homework 5 Solutions Note: In what follows, numbers in parentheses indicate the problem numbers for users of the sixth edition. A * indicates that this

More information

Math 1314 Test 4 Review Lesson 16 Lesson Use Riemann sums with midpoints and 6 subdivisions to approximate the area between

Math 1314 Test 4 Review Lesson 16 Lesson Use Riemann sums with midpoints and 6 subdivisions to approximate the area between Math 1314 Test 4 Review Lesson 16 Lesson 24 1. Use Riemann sums with midpoints and 6 subdivisions to approximate the area between and the x-axis on the interval [1, 9]. Recall: RectangleSum[,

More information

Lecture notes: Introduction to Partial Differential Equations

Lecture notes: Introduction to Partial Differential Equations Lecture notes: Introduction to Partial Differential Equations Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 Classification of Partial Differential

More information

3 Applications of partial differentiation

3 Applications of partial differentiation Advanced Calculus Chapter 3 Applications of partial differentiation 37 3 Applications of partial differentiation 3.1 Stationary points Higher derivatives Let U R 2 and f : U R. The partial derivatives

More information

Lagrange Multipliers

Lagrange Multipliers Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each

More information

Differentiation - Quick Review From Calculus

Differentiation - Quick Review From Calculus Differentiation - Quick Review From Calculus Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 1 / 13 Introduction In this section,

More information

Module 2: First-Order Partial Differential Equations

Module 2: First-Order Partial Differential Equations Module 2: First-Order Partial Differential Equations The mathematical formulations of many problems in science and engineering reduce to study of first-order PDEs. For instance, the study of first-order

More information

Engg. Math. I. Unit-I. Differential Calculus

Engg. Math. I. Unit-I. Differential Calculus Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle

More information

e x3 dx dy. 0 y x 2, 0 x 1.

e x3 dx dy. 0 y x 2, 0 x 1. Problem 1. Evaluate by changing the order of integration y e x3 dx dy. Solution:We change the order of integration over the region y x 1. We find and x e x3 dy dx = y x, x 1. x e x3 dx = 1 x=1 3 ex3 x=

More information

Bessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics

Bessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy

More information

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort - 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

More information

7.1 Functions of Two or More Variables

7.1 Functions of Two or More Variables 7.1 Functions of Two or More Variables Hartfield MATH 2040 Unit 5 Page 1 Definition: A function f of two variables is a rule such that each ordered pair (x, y) in the domain of f corresponds to exactly

More information

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles.

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles. » ~ $ ) 7 x X ) / ( 8 2 X 39 ««x» ««! «! / x? \» «({? «» q «(? (?? x! «? 8? ( z x x q? ) «q q q ) x z x 69 7( X X ( 3»«! ( ~«x ««x ) (» «8 4 X «4 «4 «8 X «x «(» X) ()»» «X «97 X X X 4 ( 86) x) ( ) z z

More information

Section 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 23 Motivation Goal: We want to derive rules to find the derivative

More information

CHAPTER 4: HIGHER ORDER DERIVATIVES. Likewise, we may define the higher order derivatives. f(x, y, z) = xy 2 + e zx. y = 2xy.

CHAPTER 4: HIGHER ORDER DERIVATIVES. Likewise, we may define the higher order derivatives. f(x, y, z) = xy 2 + e zx. y = 2xy. April 15, 2009 CHAPTER 4: HIGHER ORDER DERIVATIVES In this chapter D denotes an open subset of R n. 1. Introduction Definition 1.1. Given a function f : D R we define the second partial derivatives as

More information

DIPOLES III. q const. The voltage produced by such a charge distribution is given by. r r'

DIPOLES III. q const. The voltage produced by such a charge distribution is given by. r r' DIPOLES III We now consider a particularly important charge configuration a dipole. This consists of two equal but opposite charges separated by a small distance. We define the dipole moment as p lim q

More information

Two Posts to Fill On School Board

Two Posts to Fill On School Board Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

More information

Practice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29

Practice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29 Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)

More information

MA261-A Calculus III 2006 Fall Midterm 2 Solutions 11/8/2006 8:00AM ~9:15AM

MA261-A Calculus III 2006 Fall Midterm 2 Solutions 11/8/2006 8:00AM ~9:15AM MA6-A Calculus III 6 Fall Midterm Solutions /8/6 8:AM ~9:5AM. Find the it xy cos y (x;y)(;) 3x + y, if it exists, or show that the it does not exist. Assume that x. The it becomes (;y)(;) y cos y 3 + y

More information

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math 16.

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math 16. Leplace s Analyzing the Analyticity of Analytic Analysis Engineering Math 16.364 1 The Laplace equations are built on the Cauchy- Riemann equations. They are used in many branches of physics such as heat

More information

DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018

DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 2018 DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH 233 SOME SOLUTIONS TO EXAM 2 Fall 208 Version A refers to the regular exam and Version B to the make-up. Version A. A particle

More information

Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0

Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0 3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,

More information

Section 3.5 The Implicit Function Theorem

Section 3.5 The Implicit Function Theorem Section 3.5 The Implicit Function Theorem THEOREM 11 (Special Implicit Function Theorem): Suppose that F : R n+1 R has continuous partial derivatives. Denoting points in R n+1 by (x, z), where x R n and

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

More information

Summer 2017 MATH Solution to Exercise 5

Summer 2017 MATH Solution to Exercise 5 Summer 07 MATH00 Solution to Exercise 5. Find the partial derivatives of the following functions: (a (xy 5z/( + x, (b x/ x + y, (c arctan y/x, (d log((t + 3 + ts, (e sin(xy z 3, (f x α, x = (x,, x n. (a

More information

Differentiation of Multivariable Functions

Differentiation of Multivariable Functions Differentiation of Multivariable Functions 1 Introduction Beginning calculus students identify the derivative of a function either in terms of slope or instantaneous rate of change. When thinking of the

More information

Constrained optimization.

Constrained optimization. ams/econ 11b supplementary notes ucsc Constrained optimization. c 2016, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

More information

Math 222 Spring 2013 Exam 3 Review Problem Answers

Math 222 Spring 2013 Exam 3 Review Problem Answers . (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w

More information

1 Arithmetic calculations (calculator is not allowed)

1 Arithmetic calculations (calculator is not allowed) 1 ARITHMETIC CALCULATIONS (CALCULATOR IS NOT ALLOWED) 1 Arithmetic calculations (calculator is not allowed) 1.1 Check the result Problem 1.1. Problem 1.2. Problem 1.3. Problem 1.4. 78 5 6 + 24 3 4 99 1

More information

Math 1314 Lesson 23 Partial Derivatives

Math 1314 Lesson 23 Partial Derivatives Math 1314 Lesson 3 Partial Derivatives When we are asked to ind the derivative o a unction o a single variable, (x), we know exactly what to do However, when we have a unction o two variables, there is

More information

MATH 425, FINAL EXAM SOLUTIONS

MATH 425, FINAL EXAM SOLUTIONS MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u

More information

CHAPTER 7 DIV, GRAD, AND CURL

CHAPTER 7 DIV, GRAD, AND CURL CHAPTER 7 DIV, GRAD, AND CURL 1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: (1 ϕ = ( ϕ, ϕ,, ϕ x 1 x 2 x n

More information

2.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. differential equations with the initial values y(x 0. ; l.

2.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. differential equations with the initial values y(x 0. ; l. Numerical Methods II UNIT.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS.1.1 Runge-Kutta Method of Fourth Order 1. Let = f x,y,z, = gx,y,z be the simultaneous first order

More information

Tangent Plane. Nobuyuki TOSE. October 02, Nobuyuki TOSE. Tangent Plane

Tangent Plane. Nobuyuki TOSE. October 02, Nobuyuki TOSE. Tangent Plane October 02, 2017 The Equation of a plane Given a plane α passing through P 0 perpendicular to n( 0). For any point P on α, we have n PP 0 = 0 When P 0 has the coordinates (x 0, y 0, z 0 ), P 0 (x, y, z)

More information

Lecture 13 - Wednesday April 29th

Lecture 13 - Wednesday April 29th Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131

More information

g(2, 1) = cos(2π) + 1 = = 9

g(2, 1) = cos(2π) + 1 = = 9 1. Let gx, y 2x 2 cos2πy 2 + y 2. You can use the fact that Dg2, 1 [8 2]. a Find an equation for the tangent plane to the graph z gx, y at the point 2, 1. There are two key parts to this problem. The first,

More information

Math 2163, Practice Exam II, Solution

Math 2163, Practice Exam II, Solution Math 63, Practice Exam II, Solution. (a) f =< f s, f t >=< s e t, s e t >, an v v = , so D v f(, ) =< ()e, e > =< 4, 4 > = 4. (b) f =< xy 3, 3x y 4y 3 > an v =< cos π, sin π >=, so

More information

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:

Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example: Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a

More information

Chapter 3 Differentiation Rules

Chapter 3 Differentiation Rules Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number,

More information

MATH 19520/51 Class 11

MATH 19520/51 Class 11 MATH 1952/51 Class 11 Minh-Tam Trinh University of Chicago 217-1-23 1 Discuss the midterm. 2 Review Riemann sums for functions of one variable. 3 Riemann sums for multivariable functions. 4 Double integrals

More information

13. LECTURE 13. Objectives

13. LECTURE 13. Objectives 13. LECTURE 13 Objectives I can use Clairaut s Theorem to make my calculations easier. I can take higher derivatives. I can check if a function is a solution to a partial differential equation. Higher

More information

PDE: The Method of Characteristics Page 1

PDE: The Method of Characteristics Page 1 PDE: The Method of Characteristics Page y u x (x, y) + x u y (x, y) =, () u(, y) = cos y 2. Solution The partial differential equation given can be rewritten as follows: u(x, y) y, x =, (2) where = / x,

More information

PRACTICE PROBLEMS FOR MIDTERM I

PRACTICE PROBLEMS FOR MIDTERM I Problem. Find the limits or explain why they do not exist (i) lim x,y 0 x +y 6 x 6 +y ; (ii) lim x,y,z 0 x 6 +y 6 +z 6 x +y +z. (iii) lim x,y 0 sin(x +y ) x +y Problem. PRACTICE PROBLEMS FOR MIDTERM I

More information

7a3 2. (c) πa 3 (d) πa 3 (e) πa3

7a3 2. (c) πa 3 (d) πa 3 (e) πa3 1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

More information

Homework 1/Solutions. Graded Exercises

Homework 1/Solutions. Graded Exercises MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both

More information

Contents. 2 Partial Derivatives. 2.1 Limits and Continuity. Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v. 2.

Contents. 2 Partial Derivatives. 2.1 Limits and Continuity. Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v. 2. Calculus III (part 2): Partial Derivatives (by Evan Dummit, 2017, v 260) Contents 2 Partial Derivatives 1 21 Limits and Continuity 1 22 Partial Derivatives 5 23 Directional Derivatives and the Gradient

More information

Mathematical Analysis II, 2018/19 First semester

Mathematical Analysis II, 2018/19 First semester Mathematical Analysis II, 208/9 First semester Yoh Tanimoto Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, I-0033 Roma, Italy email: hoyt@mat.uniroma2.it We basically

More information

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math EECE

Leplace s Equations. Analyzing the Analyticity of Analytic Analysis DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING. Engineering Math EECE Leplace s Analyzing the Analyticity of Analytic Analysis Engineering Math EECE 3640 1 The Laplace equations are built on the Cauchy- Riemann equations. They are used in many branches of physics such as

More information

Chapter 4. Several-variable calculus. 4.1 Derivatives of Functions of Several Variables Functions of Several Variables

Chapter 4. Several-variable calculus. 4.1 Derivatives of Functions of Several Variables Functions of Several Variables Chapter 4 Several-variable calculus 4.1 Derivatives of Functions of Several Variables 4.1.1 Functions of Several Variables ² A function f of n variables (x 1,x 2,...,x n ) in R n is an entity that operates

More information

The Mean Value Theorem. Oct

The Mean Value Theorem. Oct The Mean Value Theorem Oct 14 2011 The Mean Value Theorem Theorem Suppose that f is defined and continuous on a closed interval [a, b], and suppose that f exists on the open interval (a, b). Then there

More information

THE WAVE EQUATION. F = T (x, t) j + T (x + x, t) j = T (sin(θ(x, t)) + sin(θ(x + x, t)))

THE WAVE EQUATION. F = T (x, t) j + T (x + x, t) j = T (sin(θ(x, t)) + sin(θ(x + x, t))) THE WAVE EQUATION The aim is to derive a mathematical model that describes small vibrations of a tightly stretched flexible string for the one-dimensional case, or of a tightly stretched membrane for the

More information

Midterm 1 Solutions Thursday, February 26

Midterm 1 Solutions Thursday, February 26 Math 59 Dr. DeTurck Midterm 1 Solutions Thursday, February 26 1. First note that since f() = f( + ) = f()f(), we have either f() = (in which case f(x) = f(x + ) = f(x)f() = for all x, so c = ) or else

More information

OWELL WEEKLY JOURNAL

OWELL WEEKLY JOURNAL Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --

More information

MANY BILLS OF CONCERN TO PUBLIC

MANY BILLS OF CONCERN TO PUBLIC - 6 8 9-6 8 9 6 9 XXX 4 > -? - 8 9 x 4 z ) - -! x - x - - X - - - - - x 00 - - - - - x z - - - x x - x - - - - - ) x - - - - - - 0 > - 000-90 - - 4 0 x 00 - -? z 8 & x - - 8? > 9 - - - - 64 49 9 x - -

More information

CALCULUS III MATH 265 FALL 2014 (COHEN) LECTURE NOTES

CALCULUS III MATH 265 FALL 2014 (COHEN) LECTURE NOTES CALCULUS III MATH 265 FALL 2014 (COHEN) LECTURE NOTES These lecture notes are intended as an outline for both student and instructor; for much more detailed exposition on the topics contained herein, the

More information

1 Functions of Several Variables Some Examples Level Curves / Contours Functions of More Variables... 6

1 Functions of Several Variables Some Examples Level Curves / Contours Functions of More Variables... 6 Contents 1 Functions of Several Variables 1 1.1 Some Examples.................................. 2 1.2 Level Curves / Contours............................. 4 1.3 Functions of More Variables...........................

More information

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.

(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere. MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.

More information

EconS 301. Math Review. Math Concepts

EconS 301. Math Review. Math Concepts EconS 301 Math Review Math Concepts Functions: Functions describe the relationship between input variables and outputs y f x where x is some input and y is some output. Example: x could number of Bananas

More information

' Liberty and Umou Ono and Inseparablo "

' Liberty and Umou Ono and Inseparablo 3 5? #< q 8 2 / / ) 9 ) 2 ) > < _ / ] > ) 2 ) ) 5 > x > [ < > < ) > _ ] ]? <

More information

The Derivative. Appendix B. B.1 The Derivative of f. Mappings from IR to IR

The Derivative. Appendix B. B.1 The Derivative of f. Mappings from IR to IR Appendix B The Derivative B.1 The Derivative of f In this chapter, we give a short summary of the derivative. Specifically, we want to compare/contrast how the derivative appears for functions whose domain

More information

12. Stresses and Strains

12. Stresses and Strains 12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

More information

Supplementary Questions for HP Chapter Sketch graphs of the following functions: (a) z = y x. (b) z = ye x. (c) z = x 2 y 2

Supplementary Questions for HP Chapter Sketch graphs of the following functions: (a) z = y x. (b) z = ye x. (c) z = x 2 y 2 Supplementary Questions for HP Chapter 17 1. Sketch graphs of the following functions: (a) z = y x (b) z = ye x (c) z = x y. Find the domain and range of the given functions: (a) f(x, y) = 36 9x 4y (b)

More information

Implicit Differentiation and Inverse Trigonometric Functions

Implicit Differentiation and Inverse Trigonometric Functions Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5

More information

Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector

Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector Calculus 3 Lia Vas Review for Exam 1 1. Surfaces. Describe the following surfaces. (a) x + y = 9 (b) x + y + z = 4 (c) z = 1 (d) x + 3y + z = 6 (e) z = x + y (f) z = x + y. Review of Vectors. (a) Let a

More information

Differential Equations. Joe Erickson

Differential Equations. Joe Erickson Differential Equations Joe Erickson Contents 1 Basic Principles 1 1.1 Functions of Several Variables.......................... 1 1.2 Linear Differential Operators........................... 7 1.3 Ordinary

More information