# HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

Save this PDF as:

Size: px
Start display at page:

Download "HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba"

## Transcription

1 HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain rule, w u v = w w y. Thus, i.e., w u v = u w v = w v + w v = w x w y. ( ) w = v u (w x w y ) = u w x u w y ( wy u + w y u = w x u + w x u = w xx + w xy (w yx + w yy ) = w xx w yy w u v = w w y. ) Question :.. (a) : Show that the function g(x, t) = + e t sin x satisfies the heat equation: g t = g xx. [Here g(x, t) represents the temperature in a metal rod at position x and time t.] (b) : Sketch the graph of g for t. (Hint: Look at sections by the planes t =, t =, and t =.) (c) : What happens to g(x, t) as t? Interpret this limit in terms of the behavior of heat in the rod. Solution. (a) : Since g(x, y) = + e t sin x, then g t = e t sin x, g x = e t cos x, and g xx = e t sin x. Therefore, g t = g xx. (b) : The graph of g is shown in Figure. Date: Math c Practical, 8.

2 HOMEWORK SOLUTIONS t = t = t = z x t Figure. The graph of g at t =,, and. (c) : Note that lim g(x, t) = lim ( + t t e t sin x) = This means that the temperature in the rod at position x tends to be a constant (= ) as the time t is large enough. Question :.. Determine the second-order Taylor formula for f(x, y) = x + y + about x =, y =. Solution. We first compute the partial derivatives up through second order: f x = f xy = f xx = f yy = x ( + x + y ), f y y = ( + x + y ) 8xy ( + x + y ), f 8xy yx = ( + x + y ) ( + x + y ) + 8x ( + x + y ) ( + x + y ) + 8y ( + x + y ).

3 HOMEWORK SOLUTIONS Next, we evaluate these derivatives at (, ), obtaining and f x (, ) = f y (, ) =, f xy (, ) = f yx (, ) = f xx (, ) = f yy (, ) =. Therefore, the second order Taylor formula is where h = (h, h ) and where f(h) = h h + R (, h), R (, h) h as h. Question :..6 Determine the second-order Taylor formula for the function expanded about the point x =, y =. f(x, y) = e (x ) cos y Solution. The ingredients needed in the second-order Taylor formula are computed as follows: f x = (x )e (x ) cos y f y = e (x ) sin y f xx = e (x ) cos y + (x ) e (x ) cos y f xy = (x )e (x ) sin y = f yx f yy = e (x ) cos y. Evaluating the function and these derivatives at the point (, ) gives f(, ) = f x (, ) = f y (, ) = f xx (, ) = f xy (, ) = f yx (, ) = and f yy (, ) =. Consequently, the second order Taylor formula is where h = (h, h ) and where f(h) = + h h + R ((, ), h), R ((, ), h) h as h. Question 5:..7 Find the critical points for the function f(x, y) = x + xy + x + y + y +. and then determine whether they are local maxima, local minima, or saddle points.

4 HOMEWORK SOLUTIONS Solution. Here, We have = 6x + y +, =, y = x + y +. y = when x = y = /. Therefore, the only critical point is ( /, /). Now, ( /, /) = 6, f y ( /, /) =, and f y ( /, /) =, which yields D = 6. = >. Therefore ( /, /) is a local minimum. Question 6:..7 Find the local maxima and minima for z = (x +y )e x y. Solution. We first locate the critical points of f(x, y) = (x + y )e x y. f(x, y) = e x y (x( y x )i + y( x y )j) Thus, f(x, y) = if and only if (x, y) = (, ), (, ±), or (±, ). To determine whether they are maxima or minima, we need to calculate the second partial derivatives. = ( + x y + x (6y 5))e x y = ( 5y + 6y + x (y ))e x y, and y y = (y + x )e x y. Therefore, f (, ) = e, f y (, ) = 6e, and f y (, ) =, which yields D = (e)(6e) = e >, and (, ) is a local minimum. (, ±) =, f y (, ±) =, and f y (, ±) =, which yields D = ( )( ) = >, and (, ±) are local maxima. (±, ) =, f y (±, ) =, and f y (, ±) =, which yields D = ( )() = 6 <, and (±, ) are saddle points. Question 7:..5 Write the number as a sum of three numbers so that the sum of the products taken two at a time is a maximum. Solution. Let the three numbers be x, y, z. Thus, We want to find the maximum value for We differentiate to get x + y + z =, z = x y. S(x, y) = xy + yz + xz = xy + (x + y)( x y) = x xy y + x + y. S S = x y +, y = x y +. These vanish when x = y =, then z = (x + y) =. Therefore, when x = y = z = is the only critical point. The condition x, y, z describes a cube in R and on the boundary of the cube (either x =, x =, y =, y =, z =, z = ), S is zero. Therefore the maximum of S occurs on the interior of this cube, i.e., at a local maximum. Since x =, y =, z = is the only critical point, it must be a maximum.

5 HOMEWORK SOLUTIONS 5 Question 8:.. Find the extrema of f(x, y) = x y subject to the constraint x y =. Solution. By the method of Lagrange multipliers, we write the constraint as g =, where g(x, y) = x y and then write the Lagrange multiplier equations as f = λ g. Thus, we get = λ x = λ y x y =. First of all, the first two equations imply that x and y. Hence we can eliminate λ, giving x = y. From the last equation this would imply that =. Hence there are no extrema. Question 9:.. Let P be a point on a surface S in R defined by the equation f(x, y, z) =, where f is of class C. Suppose that P is a point where the distance from the origin to S is maximized. Show that the vector emanating from the origin and ending at P is perpendicular to S. Solution. We want to maximize the function g(x, y, z) = x + y + z subject to the constraint f(x, y, z) =. Suppose this maximum occurs at P = (x, y, z ), then by the method of Lagrange multipliers we have the equations x = λ { f(x, y, z )} y = λ { f(x, y, z )} z = λ { f(x, y, z )} where { f(x, y, z )} i denotes the ith component of f(x, y, z ), i. If v = (x, y, z ) is the vector from the origin ending at P, then these equations say that v = ( ) λ f(x, y, z ). But f(x, y, z ) is perpendicular to S at P, and since v is a scalar multiple of f(x, y, z ) it is also perpendicular to S at P. Question :..8 A company s production function is Q(x, y) = xy. The cost of production is C(x, y) = x + y. If this company can spend C(x, y) =, what is the maximum quantity that can be produced? Solution. We want to maximize Q subject to the constraint C(x, y) =. Since both x, y, this imposes the condition that x 5, y /. Thus, we wish to maximize Q on the line segment x+y =, x, y. If the maximum occurs at an interior point (x, y ) of this segment, then Q(x, y ) = λ C(x, y ); that is, y = λ x = λ x + y =. Thus 6λ + 6λ =, λ = 5/6, y = 5/, x = 5/, Q(x, y ) = 5/6. The value of Q at the endpoints of this segment are Q(, ) = = Q(5, ). Consequently the maximum occurs at (5/, 5/) and the maximum value of Q is 5/6.

### PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

### 1 3 4 = 8i + 20j 13k. x + w. y + w

) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations

### Maximum and Minimum Values

Jim Lambers MAT 280 Spring Semester 2009-10 Lecture 8 Notes These notes correspond to Section 11.7 in Stewart and Section 3.3 in Marsden and Tromba. Maximum and Minimum Values In single-variable calculus,

### Math 53 Worksheet Solutions- Minmax and Lagrange

Math 5 Worksheet Solutions- Minmax and Lagrange. Find the local maximum and minimum values as well as the saddle point(s) of the function f(x, y) = e y (y x ). Solution. First we calculate the partial

### Multivariable Calculus Practice Midterm 2 Solutions Prof. Fedorchuk

Multivariable Calculus Practice Midterm Solutions Prof. Fedorchuk. ( points) Let f(x, y, z) xz + e y x. a. (4 pts) Compute the gradient f. b. ( pts) Find the directional derivative D,, f(,, ). c. ( pts)

### SOLUTIONS TO HOMEWORK ASSIGNMENT #5, Math 253

SOLUTIONS TO HOMEWORK ASSIGNMENT #5, Math 53. For what values of the constant k does the function f(x, y) =kx 3 + x +y 4x 4y have (a) no critical points; (b) exactly one critical point; (c) exactly two

### SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

### (a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,

Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We

### Math 21a Review Session for Exam 2 Solutions to Selected Problems

Math 1a Review Session for Exam Solutions to Selected Problems John Hall April 5, 9 Note: Problems which do not have solutions were done in the review session. 1. Suppose that the temperature distribution

### The variable λ is a dummy variable called a Lagrange multiplier ; we only really care about the values of x, y, and z.

Math a Lagrange Multipliers Spring, 009 The method of Lagrange multipliers allows us to maximize or minimize functions with the constraint that we only consider points on a certain surface To find critical

### Section 3.5. Extreme Values

The Calculus of Functions of Several Variables Section 3.5 Extreme Values After a few preliminary results definitions, we will apply our work from the previous sections to the problem of finding maximum

### Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve. cosh y = x + sin y + cos y

Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve at the point (0, 0) is cosh y = x + sin y + cos y Answer : y = x Justification: The equation of the

### The Method of Lagrange Multipliers

The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### Section 12.6: Directional Derivatives and the Gradient Vector

Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

### ( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those

1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make

### Differential Equations

Differential Equations A differential equation is an equation that contains an unknown function and one or more of its derivatives. Here are some examples: y = 1, y = x, y = xy y + 2y + y = 0 d 3 y dx

### Practice Problems for Midterm 2

Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,

### DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

### Name: ID: Discussion Section:

Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS:

### SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA

SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the

### ECON Mathematical Economics - ANSWERS FINAL EXAM. 1. (a)- The consumer's utility maximization problem is written as:

ECON 331 - Mathematical Economics - ANSWERS FINAL EXAM 1. a- The consumer's utility maximization problem is written as: max x,y [xy + y2 + 2x + 2y] s.t. 6x + 10y = m, x 0, y 0 - The associated Lagrangian

### Linear and quadratic Taylor polynomials for functions of several variables.

ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### f(x) = lim 2) = 2 2 = 0 (c) Provide a rough sketch of f(x). Be sure to include your scale, intercepts and label your axis.

Math 16 - Final Exam Solutions - Fall 211 - Jaimos F Skriletz 1 Answer each of the following questions to the best of your ability. To receive full credit, answers must be supported by a sufficient amount

### MATHEMATICS FOR ENGINEERS & SCIENTISTS 7

MATHEMATICS FOR ENGINEERS & SCIENTISTS 7 We stress that f(x, y, z) is a scalar-valued function and f is a vector-valued function. All of the above works in any number of dimensions. For instance, consider

### Constrained optimization.

ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

### A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

### Limits and Continuity

Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

### MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

### Absolute Maxima and Minima

Absolute Maxima and Minima Definition. A function f is said to have an absolute maximum on an interval I at the point x 0 if it is the largest value of f on that interval; that is if f( x ) f() x for all

### Math 113 HW #10 Solutions

Math HW #0 Solutions. Exercise 4.5.4. Use the guidelines of this section to sketch the curve Answer: Using the quotient rule, y = x x + 9. y = (x + 9)(x) x (x) (x + 9) = 8x (x + 9). Since the denominator

### Solutions to Homework 5

Solutions to Homework 5 1. Let z = f(x, y) be a twice continously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular

### 1 Lecture 19: Implicit differentiation

Lecture 9: Implicit differentiation. Outline The technique of implicit differentiation Tangent lines to a circle Examples.2 Implicit differentiation Suppose we have two quantities or variables x and y

### Math 265 (Butler) Practice Midterm II B (Solutions)

Math 265 (Butler) Practice Midterm II B (Solutions) 1. Find (x 0, y 0 ) so that the plane tangent to the surface z f(x, y) x 2 + 3xy y 2 at ( x 0, y 0, f(x 0, y 0 ) ) is parallel to the plane 16x 2y 2z

### Math 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).

Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field

### MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages

MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem

### Section 4.7. More on Area

Difference Equations to Differential Equations Section 4.7 More on Area In Section 4. we motivated the definition of the definite integral with the idea of finding the area of a region in the plane. However,

### Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

### Chapter 1 Vectors, lines, and planes

Simplify the following vector expressions: 1. a (a + b). (a + b) (a b) 3. (a b) (a + b) Chapter 1 Vectors, lines, planes 1. Recall that cross product distributes over addition, so a (a + b) = a a + a b.

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

Math 210 Quadratic Polynomials Jerry L. Kazdan Polynomials in One Variable. After studying linear functions y = ax + b, the next step is to study quadratic polynomials, y = ax 2 + bx + c, whose graphs

### Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### The Gradient and Level Sets

The Gradient and Level Sets. Let f(x, y) = x + y. (a) Find the gradient f. Solution. f(x, y) = x, y. (b) Pick your favorite positive number k, and let C be the curve f(x, y) = k. Draw the curve on the

### since by using a computer we are limited to the use of elementary arithmetic operations

> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

### Interpolating Polynomials Handout March 7, 2012

Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ),

### 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

### MATH 425, HOMEWORK 7, SOLUTIONS

MATH 425, HOMEWORK 7, SOLUTIONS Each problem is worth 10 points. Exercise 1. (An alternative derivation of the mean value property in 3D) Suppose that u is a harmonic function on a domain Ω R 3 and suppose

### Lecture 4: Equality Constrained Optimization. Tianxi Wang

Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x

Mathematics 0N1 Solutions 1 1. Write the following sets in list form. 1(i) The set of letters in the word banana. {a, b, n}. 1(ii) {x : x 2 + 3x 10 = 0}. 3(iv) C A. True 3(v) B = {e, e, f, c}. True 3(vi)

### Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

### Math 432 HW 2.5 Solutions

Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/

### MVE041 Flervariabelanalys

MVE041 Flervariabelanalys 2015-16 This document contains the learning goals for this course. The goals are organized by subject, with reference to the course textbook Calculus: A Complete Course 8th ed.

### Q ( q(m, t 0 ) n) S t.

THE HEAT EQUATION The main equations that we will be dealing with are the heat equation, the wave equation, and the potential equation. We use simple physical principles to show how these equations are

### Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6

### MA261-A Calculus III 2006 Fall Homework 8 Solutions Due 10/30/2006 8:00AM

MA61-A Calculus III 006 Fall Homework Solutions Due 10/0/006 :00AM 116 # Let f (x; y) = y ln x (a) Find the gradient of f (b) Evaluate the gradient at the oint P (1; ) (c) Find the rate of change of f

### INTERMEDIATE MICROECONOMICS MATH REVIEW

INTERMEDIATE MICROECONOMICS MATH REVIEW August 31, 2008 OUTLINE 1. Functions Definition Inverse functions Convex and Concave functions 2. Derivative of Functions of One variable Definition Rules for finding

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Chapter 7 Div, grad, and curl 7.1 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: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

### This makes sense. t 2 1 + 1/t 2 dt = 1. t t 2 + 1dt = 2 du = 1 3 u3/2 u=5

1. (Line integrals Using parametrization. Two types and the flux integral) Formulas: ds = x (t) dt, d x = x (t)dt and d x = T ds since T = x (t)/ x (t). Another one is Nds = T ds ẑ = (dx, dy) ẑ = (dy,

### Section 3.2 Polynomial Functions and Their Graphs

Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)

### Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

### Differential Equations BERNOULLI EQUATIONS. Graham S McDonald. A Tutorial Module for learning how to solve Bernoulli differential equations

Differential Equations BERNOULLI EQUATIONS Graham S McDonald A Tutorial Module for learning how to solve Bernoulli differential equations Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk

### R is a function which is of class C 1. We have already thought about its level sets, sets of the form

Manifolds 1 Chapter 5 Manifolds We are now going to begin our study of calculus on curved spaces. Everything we have done up to this point has been concerned with what one might call the flat Euclidean

### + 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider

Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake

### HIGHER ORDER DIFFERENTIAL EQUATIONS

HIGHER ORDER DIFFERENTIAL EQUATIONS 1 Higher Order Equations Consider the differential equation (1) y (n) (x) f(x, y(x), y (x),, y (n 1) (x)) 11 The Existence and Uniqueness Theorem 12 The general solution

### GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics

GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### 1 Functions of Several Variables

Chain Rule for Functions of Several Variables June, 0 Functions of Several Variables We write f : R n R m for a rule assigning to each vector in a domain D R n a unique vector in R m Examples: Suppose

### Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

### 2 Topics in 3D Geometry

2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and

### 4.3 Lagrange Approximation

206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

### In this section, we will consider techniques for solving problems of this type.

Constrained optimisation roblems in economics typically involve maximising some quantity, such as utility or profit, subject to a constraint for example income. We shall therefore need techniques for solving

### Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,

### 8 Polynomials Worksheet

8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions - Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph

### Module 3: Second-Order Partial Differential Equations

Module 3: Second-Order Partial Differential Equations In Module 3, we shall discuss some general concepts associated with second-order linear PDEs. These types of PDEs arise in connection with various

### Math 1B, lecture 14: Taylor s Theorem

Math B, lecture 4: Taylor s Theorem Nathan Pflueger 7 October 20 Introduction Taylor polynomials give a convenient way to describe the local behavior of a function, by encapsulating its first several derivatives

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

### In this chapter we turn to surfaces in general. We discuss the following topics. Describing surfaces with equations and parametric descriptions.

Chapter 4 Surfaces In this chapter we turn to surfaces in general. We discuss the following topics. Describing surfaces with equations and parametric descriptions. Some constructions of surfaces: surfaces

Introduction to Calculus for Business and Economics by Stephen J. Silver Department of Business Administration The Citadel I. Functions Introduction to Calculus for Business and Economics y = f(x) is a

### Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10

Dirk Bergemann Department of Economics Yale University s by Olga Timoshenko Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10 This problem set is due on Wednesday, 1/27/10. Preliminary

### SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

L SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS SECOND-ORDER LINEAR HOOGENEOUS DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS A second-order linear differential equation is one of the form d

### 5.3 The Cross Product in R 3

53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or

### FINAL EXAM SOLUTIONS Math 21a, Spring 03

INAL EXAM SOLUIONS Math 21a, Spring 3 Name: Start by printing your name in the above box and check your section in the box to the left. MW1 Ken Chung MW1 Weiyang Qiu MW11 Oliver Knill h1 Mark Lucianovic

### Introduction to Calculus

Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative

### Introduction to polynomials

Worksheet 4.5 Polynomials Section 1 Introduction to polynomials A polynomial is an expression of the form p(x) = p 0 + p 1 x + p 2 x 2 + + p n x n, (n N) where p 0, p 1,..., p n are constants and x os

### Scalar Valued Functions of Several Variables; the Gradient Vector

Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector) valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x) = φ(x 1,

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Solution: 2. Sketch the graph of 2 given the vectors and shown below.

7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit

### 21-114: Calculus for Architecture Homework #1 Solutions

21-114: Calculus for Architecture Homework #1 Solutions November 9, 2004 Mike Picollelli 1.1 #26. Find the domain of g(u) = u + 4 u. Solution: We solve this by considering the terms in the sum separately:

### Math 241: More heat equation/laplace equation

Math 241: More heat equation/aplace equation D. DeTurck University of Pennsylvania September 27, 2012 D. DeTurck Math 241 002 2012C: Heat/aplace equations 1 / 13 Another example Another heat equation problem: