# Maxima and Minima. Marius Ionescu. November 5, Marius Ionescu () Maxima and Minima November 5, / 7

Size: px
Start display at page:

Download "Maxima and Minima. Marius Ionescu. November 5, Marius Ionescu () Maxima and Minima November 5, / 7"

Transcription

1 Maxima and Minima Marius Ionescu November 5, 2012 Marius Ionescu () Maxima and Minima November 5, / 7

2 Second Derivative Test Fact Suppose the second partial derivatives of f are continuous on a disk with center (a, b), and suppose that f x (a, b) = 0 and f y (a, b) = 0. Let D = f xx f yx f xy f yy = f xx f yy (f xy ) 2. 1 If D > 0 and f xx (a, b) > 0, then f (a, b) is a local minimum. 2 If D > 0 and f xx (a, b) < 0, then f (a, b) is a local maximum. 3 If D < 0, then f (a, b) is not a local maximum or minimum. In this case the point (a, b) is called a saddle point of f. Marius Ionescu () Maxima and Minima November 5, / 7

3 Examples Examples Marius Ionescu () Maxima and Minima November 5, / 7

4 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Marius Ionescu () Maxima and Minima November 5, / 7

5 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Find the points on the surface x 2 y 2 z = 1 that are closest to the origin. Marius Ionescu () Maxima and Minima November 5, / 7

6 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Find the points on the surface x 2 y 2 z = 1 that are closest to the origin. Find three positive numbers x, y, and z whose sum is 100 and whose product is maximum. Marius Ionescu () Maxima and Minima November 5, / 7

7 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Find the points on the surface x 2 y 2 z = 1 that are closest to the origin. Find three positive numbers x, y, and z whose sum is 100 and whose product is maximum. A rectangular box without a lid is to be made from 12 m 2 of cardboard. Find the maximum volume of such a box. Marius Ionescu () Maxima and Minima November 5, / 7

8 Examples Examples Find the point on the plane x y + z = 4 that is closest to the point (1, 2, 3). Find the points on the surface x 2 y 2 z = 1 that are closest to the origin. Find three positive numbers x, y, and z whose sum is 100 and whose product is maximum. A rectangular box without a lid is to be made from 12 m 2 of cardboard. Find the maximum volume of such a box. Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid 9x y 2 + 4z 2 = 36. Marius Ionescu () Maxima and Minima November 5, / 7

9 Absolute Maximum and Minimum Values Denition Marius Ionescu () Maxima and Minima November 5, / 7

10 Absolute Maximum and Minimum Values Denition A closed set in R 2 is one that contains all its boundary points. Marius Ionescu () Maxima and Minima November 5, / 7

11 Absolute Maximum and Minimum Values Denition A closed set in R 2 is one that contains all its boundary points. A bounded set in R 2 is one that is contained within some disk. Marius Ionescu () Maxima and Minima November 5, / 7

12 Extreme Value Theorem for Functions of Two Variables Theorem If f is continuous on a closed, bounded set D in R 2, then f attains an absolute maximum value f (x 1, y 1 ) and an absolute minimum value f (x 2, y 2 ) at some points (x 1, y 1 ) and (x 2, y 2 ) in D. Marius Ionescu () Maxima and Minima November 5, / 7

13 Extension of the Closed Interval Method Fact To nd the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: Marius Ionescu () Maxima and Minima November 5, / 7

14 Extension of the Closed Interval Method Fact To nd the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: 1 Find the values of f at the critical points of f in D. Marius Ionescu () Maxima and Minima November 5, / 7

15 Extension of the Closed Interval Method Fact To nd the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: 1 Find the values of f at the critical points of f in D. 2 Find the extreme values of f on the boundary of D. Marius Ionescu () Maxima and Minima November 5, / 7

16 Extension of the Closed Interval Method Fact To nd the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: 1 Find the values of f at the critical points of f in D. 2 Find the extreme values of f on the boundary of D. 3 The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. Marius Ionescu () Maxima and Minima November 5, / 7

17 Examples Examples Marius Ionescu () Maxima and Minima November 5, / 7

18 Examples Examples Find the absolute maximum and minimum values of the function f (x.y) = x 2 2xy + 2y on the rectangle D = {(x, y) 0 x 3, 0 y 2}. Marius Ionescu () Maxima and Minima November 5, / 7

19 Examples Examples Find the absolute maximum and minimum values of the function f (x.y) = x 2 2xy + 2y on the rectangle D = {(x, y) 0 x 3, 0 y 2}. Find the absolute maximum and minimum values of the function f (x, y) = 3 + xy x 2y on the closed triangular region with vertices (1, 0), (5, 0), and (1, 4). Marius Ionescu () Maxima and Minima November 5, / 7

### 14.7: Maxima and Minima

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

### Section 14.8 Maxima & minima of functions of two variables. Learning outcomes. After completing this section, you will inshaallah be able to

Section 14.8 Maxima & minima of functions of two variables 14.8 1 Learning outcomes After completing this section, you will inshaallah be able to 1. explain what is meant by relative maxima or relative

### Math 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.7: Maximum and Minimum Values

Math 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.7: Maximum and Minimum Values I. Review from 1225 A. Definitions 1. Local Extreme Values (Relative) a. A function f has a local

### Math Maximum and Minimum Values, I

Math 213 - Maximum and Minimum Values, I Peter A. Perry University of Kentucky October 8, 218 Homework Re-read section 14.7, pp. 959 965; read carefully pp. 965 967 Begin homework on section 14.7, problems

### Solutions to Homework 7

Solutions to Homework 7 Exercise #3 in section 5.2: A rectangular box is inscribed in a hemisphere of radius r. Find the dimensions of the box of maximum volume. Solution: The base of the rectangular box

### 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

### 2x (x 2 + y 2 + 1) 2 2y. (x 2 + y 2 + 1) 4. 4xy. (1, 1)(x 1) + (1, 1)(y + 1) (1, 1)(x 1)(y + 1) 81 x y y + 7.

Homework 8 Solutions, November 007. (1 We calculate some derivatives: f x = f y = x (x + y + 1 y (x + y + 1 x = (x + y + 1 4x (x + y + 1 4 y = (x + y + 1 4y (x + y + 1 4 x y = 4xy (x + y + 1 4 Substituting

### 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

### Lecture 8: Maxima and Minima

Lecture 8: Maxima and Minima Rafikul Alam Department of Mathematics IIT Guwahati Local extremum of f : R n R Let f : U R n R be continuous, where U is open. Then f has a local maximum at p if there exists

### Review for the Final Exam

Calculus 3 Lia Vas Review for the Final Exam. Sequences. Determine whether the following sequences are convergent or divergent. If they are convergent, find their limits. (a) a n = ( 2 ) n (b) a n = n+

### 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»

### Math 10C - Fall Final Exam

Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient

### f x, y x 2 y 2 2x 6y 14. Then

SECTION 11.7 MAXIMUM AND MINIMUM VALUES 645 absolute minimum FIGURE 1 local maimum local minimum absolute maimum Look at the hills and valles in the graph of f shown in Figure 1. There are two points a,

### 39.1 Absolute maxima/minima

Module 13 : Maxima, Minima Saddle Points, Constrained maxima minima Lecture 39 : Absolute maxima / minima [Section 39.1] Objectives In this section you will learn the following : The notion of absolute

### 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 - - / - -»- --

### Extrema of Functions of Several Variables

Extrema of Functions of Several Variables MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Background (1 of 3) In single-variable calculus there are three important results

### Functions. 1. Any set of ordered pairs is a relation. Graph each set of ordered pairs. A. B. C.

Functions 1. Any set of ordered pairs is a relation. Graph each set of ordered pairs. A. x y B. x y C. x y -3-6 -2-1 -3 6-1 -3-3 0-2 4 0-1 6 3-1 1 3 5-2 1 0 0 5 6-1 4 1 1 A. B. C. 2. A function is a special

### ARNOLD PIZER rochester problib from CVS Summer 2003

WeBWorK assignment VmultivariableFunctions due 3/3/08 at 2:00 AM.( pt) setvmultivariablefunctions/ur VC 5.pg Match the surfaces with the verbal description of the level curves by placing the letter of

### Properties of surfaces II: Second moment of area

Properties of surfaces II: Second moment of area Just as we have discussing first moment of an area and its relation with problems in mechanics, we will now describe second moment and product of area of

### LESSON 23: EXTREMA OF FUNCTIONS OF 2 VARIABLES OCTOBER 25, 2017

LESSON : EXTREMA OF FUNCTIONS OF VARIABLES OCTOBER 5, 017 Just like with functions of a single variable, we want to find the minima (plural of minimum) and maxima (plural of maximum) of functions of several

### c) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0

Some Math 35 review problems With answers 2/6/2005 The following problems are based heavily on problems written by Professor Stephen Greenfield for his Math 35 class in spring 2005. His willingness to

### Calculus 2502A - Advanced Calculus I Fall : Local minima and maxima

Calculus 50A - Advanced Calculus I Fall 014 14.7: Local minima and maxima Martin Frankland November 17, 014 In these notes, we discuss the problem of finding the local minima and maxima of a function.

### 0, otherwise. Find each of the following limits, or explain that the limit does not exist.

Midterm Solutions 1, y x 4 1. Let f(x, y) = 1, y 0 0, otherwise. Find each of the following limits, or explain that the limit does not exist. (a) (b) (c) lim f(x, y) (x,y) (0,1) lim f(x, y) (x,y) (2,3)

### LESSON 25: LAGRANGE MULTIPLIERS OCTOBER 30, 2017

LESSON 5: LAGRANGE MULTIPLIERS OCTOBER 30, 017 Lagrange multipliers is another method of finding minima and maxima of functions of more than one variable. In fact, many of the problems from the last homework

### Optimization. Numerical Optimization. Linear Programming

Mathematical Modeling Lia Vas Optimization. Numerical Optimization. Linear Programming Problems asking for a set of optimal conditions (maxima or minima) in relation to a certain situation appear very

### Partial Derivatives. w = f(x, y, z).

Partial Derivatives 1 Functions of Several Variables So far we have focused our attention of functions of one variable. These functions model situations in which a variable depends on another independent

### 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

### Calculus III. Math 233 Spring Final exam May 3rd. Suggested solutions

alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.

### 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

### 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

### LOWELL WEEKLY JOURNAL

Y G y G Y 87 y Y 8 Y - \$ X ; ; y y q 8 y \$8 \$ \$ \$ G 8 q < 8 6 4 y 8 7 4 8 8 < < y 6 \$ q - - y G y G - Y y y 8 y y y Y Y 7-7- G - y y y ) y - y y y y - - y - y 87 7-7- G G < G y G y y 6 X y G y y y 87 G

### 11.1 Absolute Maximum/Minimum: Definition:

Module 4 : Local / Global Maximum / Minimum and Curve Sketching Lecture 11 : Absolute Maximum / Minimum [Section 111] Objectives In this section you will learn the following : How to find points of absolute

### Study Guide/Practice Exam 3

Study Guide/Practice Exam 3 This study guide/practice exam covers only the material since exam. The final exam, however, is cumulative so you should be sure to thoroughly study earlier material. The distribution

### SOLUTIONS to Problems for Review Chapter 15 McCallum, Hughes, Gleason, et al. ISBN by Vladimir A. Dobrushkin

SOLUTIONS to Problems for Review Chapter 1 McCallum, Hughes, Gleason, et al. ISBN 978-0470-118-9 by Vladimir A. Dobrushkin For Exercises 1, find the critical points of the given function and classify them

### Functions of Several Variables

Functions of Several Variables Extreme Values Philippe B Laval KSU April 9, 2012 Philippe B Laval (KSU) Functions of Several Variables April 9, 2012 1 / 13 Introduction In Calculus I (differential calculus

### Optimization. Sherif Khalifa. Sherif Khalifa () Optimization 1 / 50

Sherif Khalifa Sherif Khalifa () Optimization 1 / 50 Y f(x 0 ) Y=f(X) X 0 X Sherif Khalifa () Optimization 2 / 50 Y Y=f(X) f(x 0 ) X 0 X Sherif Khalifa () Optimization 3 / 50 A necessary condition for

### x 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.

1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x 2 + 7 x + 7 have? 3. How many local maxima and minima does the

### Absolute Maxima and Minima Section 12.11

Absolute Maxima and Minima Section 1.11 Definition (a) A set in the plane is called a closed set if it contains all of its boundary points. (b) A set in the plane is called a bounded set if it is contained

### Unit 3 Factors & Products

1 Unit 3 Factors & Products General Outcome: Develop algebraic reasoning and number sense. Specific Outcomes: 3.1 Demonstrate an understanding of factors of whole number by determining the: o prime factors

### 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

### Math 210, Final Exam, Fall 2010 Problem 1 Solution. v cosθ = u. v Since the magnitudes of the vectors are positive, the sign of the dot product will

Math, Final Exam, Fall Problem Solution. Let u,, and v,,3. (a) Is the angle between u and v acute, obtuse, or right? (b) Find an equation for the plane through (,,) containing u and v. Solution: (a) The

### LESSON 16 More Complicated Evaluations

72 Lesson 16 27. 2 2 + 2 3 2. (1) 3 5 4 2 3 3 1 9 1 5 29. The length of PQ is 5 yards. The length of QR is yards. Find PR. (1) 3 P Q R 30. Find the surface area of this right rectangular prism. Dimensions

### 4.1 - Maximum and Minimum Values

4.1 - Maximum and Minimum Values Calculus I, Section 011 Zachary Cline Temple University October 27, 2017 Maximum and Minimum Values absolute max. of 5 occurs at 3 absolute min. of 2 occurs at 6 Maximum

### Answers to Sample Exam Problems

Math Answers to Sample Exam Problems () Find the absolute value, reciprocal, opposite of a if a = 9; a = ; Absolute value: 9 = 9; = ; Reciprocal: 9 ; ; Opposite: 9; () Commutative law; Associative law;

### 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

### MIDTERM EXAMINATION. Spring MTH301- Calculus II (Session - 3)

ASSALAM O ALAIKUM All Dear fellows ALL IN ONE MTH3 Calculus II Midterm solved papers Created BY Ali Shah From Sahiwal BSCS th semester alaoudin.bukhari@gmail.com Remember me in your prayers MIDTERM EXAMINATION

### Functions of Several Variables

Functions of Several Variables Extreme Values Philippe B. Laval KSU Today Philippe B. Laval (KSU) Extreme Values Today 1 / 18 Introduction In Calculus I (differential calculus for functions of one variable),

### Definition: Let f(x) be a function of one variable with continuous derivatives of all orders at a the point x 0, then the series.

2.4 Local properties o unctions o several variables In this section we will learn how to address three kinds o problems which are o great importance in the ield o applied mathematics: how to obtain the

### 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

### Taylor Series and stationary points

Chapter 5 Taylor Series and stationary points 5.1 Taylor Series The surface z = f(x, y) and its derivatives can give a series approximation for f(x, y) about some point (x 0, y 0 ) as illustrated in Figure

### 7-7 Multiplying Polynomials

Example 1: Multiplying Monomials A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) (6 3)(y 3 y 5 ) 18y 8 Group factors with like bases together. B. (3mn 2 ) (9m 2 n) Example 1C: Multiplying Monomials Group factors with

### Please do not start working until instructed to do so. You have 50 minutes. You must show your work to receive full credit. Calculators are OK.

Loyola University Chicago Math 131, Section 009, Fall 2008 Midterm 2 Name (print): Signature: Please do not start working until instructed to do so. You have 50 minutes. You must show your work to receive

### APPM 2350 FINAL EXAM FALL 2017

APPM 25 FINAL EXAM FALL 27. ( points) Determine the absolute maximum and minimum values of the function f(x, y) = 2 6x 4y + 4x 2 + y. Be sure to clearly give both the locations and values of the absolute

### Transpose & Dot Product

Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix A T whose columns are the rows of A. So: The columns of A T are the rows of A. The rows of A T are the columns of A. Example:

### TEST CODE: MIII (Objective type) 2010 SYLLABUS

TEST CODE: MIII (Objective type) 200 SYLLABUS Algebra Permutations and combinations. Binomial theorem. Theory of equations. Inequalities. Complex numbers and De Moivre s theorem. Elementary set theory.

### 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.

### Optimization: Problem Set Solutions

Optimization: Problem Set Solutions Annalisa Molino University of Rome Tor Vergata annalisa.molino@uniroma2.it Fall 20 Compute the maxima minima or saddle points of the following functions a. f(x y) =

### Mission 1 Factoring by Greatest Common Factor and Grouping

Algebra Honors Unit 3 Factoring Quadratics Name Quest Mission 1 Factoring by Greatest Common Factor and Grouping Review Questions 1. Simplify: i(6 4i) 3+3i A. 4i C. 60 + 3 i B. 8 3 + 4i D. 10 3 + 3 i.

### Kevin James. MTHSC 102 Section 4.3 Absolute Extreme Points

MTHSC 102 Section 4.3 Absolute Extreme Points Definition (Relative Extreme Points and Relative Extreme Values) Suppose that f(x) is a function defined on an interval I (possibly I = (, ). 1 We say that

### Maxima and Minima of Functions

Maxima and Minima of Functions Outline of Section 4.2 of Sullivan and Miranda Calculus Sean Ellermeyer Kennesaw State University October 21, 2015 Sean Ellermeyer (Kennesaw State University) Maxima and

### Maxima and Minima. (a, b) of R if

Maxima and Minima Definition Let R be any region on the xy-plane, a function f (x, y) attains its absolute or global, maximum value M on R at the point (a, b) of R if (i) f (x, y) M for all points (x,

### 14 Increasing and decreasing functions

14 Increasing and decreasing functions 14.1 Sketching derivatives READING Read Section 3.2 of Rogawski Reading Recall, f (a) is the gradient of the tangent line of f(x) at x = a. We can use this fact to

### Faculty of Engineering, Mathematics and Science School of Mathematics

Faculty of Engineering, Mathematics and Science School of Mathematics GROUPS Trinity Term 06 MA3: Advanced Calculus SAMPLE EXAM, Solutions DAY PLACE TIME Prof. Larry Rolen Instructions to Candidates: Attempt

### Projection Theorem 1

Projection Theorem 1 Cauchy-Schwarz Inequality Lemma. (Cauchy-Schwarz Inequality) For all x, y in an inner product space, [ xy, ] x y. Equality holds if and only if x y or y θ. Proof. If y θ, the inequality

### Name: Instructor: 1. a b c d e. 15. a b c d e. 2. a b c d e a b c d e. 16. a b c d e a b c d e. 4. a b c d e... 5.

Name: Instructor: Math 155, Practice Final Exam, December The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for 2 hours. Be sure that your name

### Math 1314 Lesson 24 Maxima and Minima of Functions of Several Variables

Math 1314 Lesson 24 Maxima and Minima of Functions of Several Variables We learned to find the maxima and minima of a function of a single variable earlier in the course We had a second derivative test

### 5-6 The Remainder and Factor Theorems

Use synthetic substitution to find f (4) and f ( 2) for each function. 1. f (x) = 2x 3 5x 2 x + 14 58; 20 2. f (x) = x 4 + 8x 3 + x 2 4x 10 758; 46 3. NATURE The approximate number of bald eagle nesting

### CALCULUS: Math 21C, Fall 2010 Final Exam: Solutions. 1. [25 pts] Do the following series converge or diverge? State clearly which test you use.

CALCULUS: Math 2C, Fall 200 Final Exam: Solutions. [25 pts] Do the following series converge or diverge? State clearly which test you use. (a) (d) n(n + ) ( ) cos n n= n= (e) (b) n= n= [ cos ( ) n n (c)

### Need help? Try or 4.1 Practice Problems

Day Date Assignment (Due the next class meeting) Friday 9/29/17 (A) Monday 10/9/17 (B) 4.1 Operations with polynomials Tuesday 10/10/17 (A) Wednesday 10/11/17 (B) 4.2 Factoring and solving completely Thursday

### TEST CODE: MMA (Objective type) 2015 SYLLABUS

TEST CODE: MMA (Objective type) 2015 SYLLABUS Analytical Reasoning Algebra Arithmetic, geometric and harmonic progression. Continued fractions. Elementary combinatorics: Permutations and combinations,

### MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPPALLI Distinguish between centroid and centre of gravity. (AU DEC 09,DEC 12)

MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPPALLI 621213 Sub Code: GE 6253 Semester: II Subject: ENGINEERING MECHANICS Unit III: PROPERTIES OF SURFACES AND SOLIDS PART A 1. Distinguish between centroid and

### Math Fall 08 Final Exam Review

Math 173.7 Fall 08 Final Exam Review 1. Graph the function f(x) = x 2 3x by applying a transformation to the graph of a standard function. 2.a. Express the function F(x) = 3 ln(x + 2) in the form F = f

### Lesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality

Lesson 6 Diana Pell Monday, March 17 Section 4.1: Solve Linear Inequalities Using Properties of Inequality Example 1. Solve each inequality. Graph the solution set and write it using interval notation.

### f(x, y) = 1 sin(2x) sin(2y) and determine whether each is a maximum, minimum, or saddle. Solution: Critical points occur when f(x, y) = 0.

Some Answers to Exam Name: (4) Find all critical points in [, π) [, π) of the function: f(x, y) = sin(x) sin(y) 4 and determine whether each is a maximum, minimum, or saddle. Solution: Critical points

### Modeling and Optimization (Word Problems ALL DAY) Let the first number and = the second number. The boundaries of this problem are [0, 30].

New Calculus 5.4 Modeling and Optimization (Word Problems ALL DAY) The sum of two non-negative numbers is 30. a) Find the numbers if the sum of their squares is as large as possible. b) Find the numbers

### Partial Derivatives Formulas. KristaKingMath.com

Partial Derivatives Formulas KristaKingMath.com Domain and range of a multivariable function A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D

### Math 10 - Unit 5 Final Review - Polynomials

Class: Date: Math 10 - Unit 5 Final Review - Polynomials Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor the binomial 44a + 99a 2. a. a(44 + 99a)

### 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 +

### Polynomial functions right- and left-hand behavior (end behavior):

Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify

### VANDERBILT UNIVERSITY. MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions

VANDERBILT UNIVERSITY MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions Directions. This practice test should be used as a study guide, illustrating the concepts that will be emphasized in the

### Calculus - II Multivariable Calculus. M.Thamban Nair. Department of Mathematics Indian Institute of Technology Madras

Calculus - II Multivariable Calculus M.Thamban Nair epartment of Mathematics Indian Institute of Technology Madras February 27 Revised: January 29 Contents Preface v 1 Functions of everal Variables 1 1.1

### MAXIMA & MINIMA The single-variable definitions and theorems relating to extermals can be extended to apply to multivariable calculus.

MAXIMA & MINIMA The single-variable definitions and theorems relating to etermals can be etended to appl to multivariable calculus. ( ) is a Relative Maimum if there ( ) such that ( ) f(, for all points

### Lecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then

Lecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then 1. x S is a global minimum point of f over S if f (x) f (x ) for any x S. 2. x S

### Int Math 3 Midterm Review Handout (Modules 5-7)

Int Math 3 Midterm Review Handout (Modules 5-7) 1 Graph f(x) = x and g(x) = 1 x 4. Then describe the transformation from the graph of f(x) = x to the graph 2 of g(x) = 1 2 x 4. The transformations are

### CHAPTER 3 APPLICATIONS OF THE DERIVATIVE

CHAPTER 3 APPLICATIONS OF THE DERIVATIVE 3.1 Maxima and Minima Extreme Values 1. Does f(x) have a maximum or minimum value on S? 2. If it does have a maximum or a minimum, where are they attained? 3. If

### Math 10C Practice Final Solutions

Math 1C Practice Final Solutions March 9, 216 1. (6 points) Let f(x, y) x 3 y + 12x 2 8y. (a) Find all critical points of f. SOLUTION: f x 3x 2 y + 24x 3x(xy + 8) x,xy 8 y 8 x f y x 3 8 x 3 8 x 3 8 2 So

FINAL RVIW Answers and hints Math 3 Fall 7. Let R be a Jordan region and let f : R be integrable. Prove that the graph of f, as a subset of R 3, has zero volume. Let R be a rectangle with R. Since f is

### MATH Max-min Theory Fall 2016

MATH 20550 Max-min Theory Fall 2016 1. Definitions and main theorems Max-min theory starts with a function f of a vector variable x and a subset D of the domain of f. So far when we have worked with functions

### Given a polynomial and one of its factors, find the remaining factors of the polynomial. 4. x 3 6x x 6; x 1 SOLUTION: Divide by x 1.

Use synthetic substitution to find f (4) and f ( 2) for each function. 2. f (x) = x 4 + 8x 3 + x 2 4x 10 Divide the function by x 4. The remainder is 758. Therefore, f (4) = 758. Divide the function by

### MAT 145: Test #3 (50 points)

MAT 145: Test #3 (50 points) Part 2: Calculator OK! Name Calculator Used Score 21. For f (x) = 8x 3 +81x 2 42x 8, defined for all real numbers, use calculus techniques to determine all intervals on which

### Calculus I (Math 241) (In Progress)

Calculus I (Math 241) (In Progress) The following is a collection of Calculus I (Math 241) problems. Students may expect that their final exam is comprised, more or less, of one problem from each section,

### Arnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment VMultIntegrals1Double due 04/03/2008 at 02:00am EST.

WeBWorK assignment VMultIntegralsouble due 04/03/2008 at 02:00am ST.. ( pt) rochesterlibrary/setvmultintegralsouble/ur vc 8.pg Consider the solid that lies above the square = [0,2] [0,2] and below the

### Using the binomial square thm. rewrite the following trinomial into an equivalent expression (The square of a binomial) 25c 2 70cd + 49d 2

Using the binomial square thm. rewrite the following trinomial into an equivalent expression (The square of a binomial) 25c 2 70cd + 49d 2 Dec 6 7:24 AM 1 Notes 6 1 Quadratic Expressions, Rectangles &

### THE UNIVERSITY OF BRITISH COLUMBIA Sample Questions for Midterm 1 - January 26, 2012 MATH 105 All Sections

THE UNIVERSITY OF BRITISH COLUMBIA Sample Questions for Midterm 1 - January 26, 2012 MATH 105 All Sections Closed book examination Time: 50 minutes Last Name First Signature Student Number Special Instructions:

### Solutions to Homework 5

Solutions to Homework 5 1.7 Problem. f = x 2 y, y 2 x. If we solve f = 0, we can find two solutions (0, 0) and (1, 1). At (0, 0), f xx = 0, f yy = 0, f xy = and D = f xx f yy fxy 2 < 0, therefore (0, 0)

### critical points: 1,1, 1, 1, 1,1, 1, 1

Extrema of Functions of Several Variables -(.7). Local Extremes and Critical Points Definition: The point a,b is a critical point of a function f x,y if a,b is in the domain of f and either f a,b or f

### Section 4.7: Optimization Solutions

Section 4.7: Optimization Solutions Methods for finding extreme values have many practical applications Often the hardest part is turning English into math Problem Solving Tips (similar to 3.7): Step 1: