Tangent Planes and Linear Approximations
|
|
- Trevor Singleton
- 5 years ago
- Views:
Transcription
1 February 21, 2007
2 Tangent Planes
3 Tangent Planes Let S be a surface with equation z = f (x, y).
4 Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S.
5 Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S. Let C 1 and C 2 be the curves obtained by intersecting the vertical planes y = y 0 and x = x 0 with the surface S.
6 Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S. Let C 1 and C 2 be the curves obtained by intersecting the vertical planes y = y 0 and x = x 0 with the surface S. Let T 1 and T 2 be the tangent lines to the curves C 1 and C 2.
7 Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S. Let C 1 and C 2 be the curves obtained by intersecting the vertical planes y = y 0 and x = x 0 with the surface S. Let T 1 and T 2 be the tangent lines to the curves C 1 and C 2. The tangent plane to the surface S at the point P is dened to be the plane that contains both tangent lines T 1 and T 2.
8 Equations of the tangent plane
9 Equations of the tangent plane Suppose f has a continuous partial derivatives.
10 Equations of the tangent plane Suppose f has a continuous partial derivatives. An equation of the tangent plane to the surface z = f (x, y) at the point P(x 0, y 0, z 0 ) is z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 )
11 Examples Example Find the tangent plane to the elliptic paraboloid z = 2x 2 + y 2 at the point (1, 1, 3) y
12 Linear Approximations y x
13 Linear Approximations The linear function whose graph is this tangent plane L(x, y) = f (a, b) + f x (a, b)(x a) + f y (a, b)(y b) is called the linearization of f at (a, b) and the approximation f (x, y) f (a, b) + f x (a, b)(x a) + f y (a, b)(y b) is called the linear approximation or the tangent plane approximation of f at (a, b).
14 Examples Examples
15 Examples Examples Find the linearization of the function f (x, y) = xy at the point (4, 16).
16 Examples Examples Find the linearization of the function f (x, y) = xy at the point (4, 16). Find the linearization of the function f (x, y) = 1 + y + x cos y at P 0 (0, 0).
17 The increment of z
18 The increment of z Recall that for a function of one variable, y = f (x), if x changes from a to a + x, we dened the increment of y as y = f (a + x) f (a).
19 The increment of z Recall that for a function of one variable, y = f (x), if x changes from a to a + x, we dened the increment of y as y = f (a + x) f (a). If f is dierentiable at a, then where ɛ 0 as x 0. y = f (a) x + ɛ x,
20 The increment of z
21 The increment of z If z = f (x, y) and x changes from (a, b) to (a + x, b + y), then the increment of z is z = f (a + x, b + y) f (a, b)
22 The increment of z If z = f (x, y) and x changes from (a, b) to (a + x, b + y), then the increment of z is z = f (a + x, b + y) f (a, b) If z = f (x, y), then f expressed in the form is dierentiable at (a, b) if z can be z = f x (a, b) x + f y (a, b) y + ɛ 1 x + ɛ 2 y, where ɛ 1 and ɛ 2 0 as ( x, y) (0, 0).
23 Fact If the partial derivatives f x and f y exist near (a, b) and are continuous at (a, b), then f is dierentiable at (a, b).
24 Example Example Show that f (x, y) = xe xy linearization there. is dierentiable at (1, 0) and nd its
25 Dierentials
26 Dierentials For a dierentiable function z = f (x, y) we dene the dierential dz, also called the total dierential, is dened by dz = f x (x, y)dx + f y (x, y)dy = z x dz + z y dy, where the dierentials dx and dy are independent variables.
27 Dierentials For a dierentiable function z = f (x, y) we dene the dierential dz, also called the total dierential, is dened by dz = f x (x, y)dx + f y (x, y)dy = z x dz + z y dy, where the dierentials dx and dy are independent variables. If dx = x = x a and dy = y = y b the the dierential of z is dz = f x (a, b)(x a) + f y (a, b)(y b)
28 Examples Examples
29 Examples Examples If f (x, y) = x 2 + 3xy y 2, nd the dierential dz.
30 Examples Examples If f (x, y) = x 2 + 3xy y 2, nd the dierential dz. If x changes from 2 to 2.05 and y changes 3 to 2.96, compare the values of z and dz.
14.4: Tangent Planes and Linear Approximations
14.4: Tangent Planes and Linear Approximations Marius Ionescu October 15, 2012 Marius Ionescu () 14.4: Tangent Planes and Linear Approximations October 15, 2012 1 / 13 Tangent Planes Marius Ionescu ()
More informationDirectional Derivatives and the Gradient Vector Part 2
Directional Derivatives and the Gradient Vector Part 2 Lecture 25 February 28, 2007 Recall Fact Recall Fact If f is a dierentiable function of x and y, then f has a directional derivative in the direction
More informationDirectional Derivatives and the Gradient Vector Part 2
Directional Derivatives and the Gradient Vector Part 2 Marius Ionescu October 26, 2012 Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, 2012 1 / 12 Recall Fact Marius
More informationCurves, Tangent Planes, and Differentials ( ) Feb. 26, 2012 (Sun) Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent
Lecture 9. Partial Derivatives: Signs on Level Curves, Tangent Planes, and Differentials ( 11.3-11.4) Feb. 26, 2012 (Sun) Signs of Partial Derivatives on Level Curves Level curves are shown for a function
More informationWhat you will learn today
What you will learn today Tangent Planes and Linear Approximation and the Gradient Vector Vector Functions 1/21 Recall in one-variable calculus, as we zoom in toward a point on a curve, the graph becomes
More informationDifferentiability and Tangent Planes October 2013
Differentiability and Tangent Planes 14.4 04 October 2013 Differentiability in one variable. Recall for a function of one variable, f is differentiable at a f (a + h) f (a) lim exists and = f (a) h 0 h
More informationPartial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives. Partial Derivatives
In general, if f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. By the definition of a derivative, we have Then we are really
More informationMATH 19520/51 Class 6
MATH 19520/51 Class 6 Minh-Tam Trinh University of Chicago 2017-10-06 1 Review partial derivatives. 2 Review equations of planes. 3 Review tangent lines in single-variable calculus. 4 Tangent planes to
More informationREVIEW I MATH 254 Calculus IV. Exam I (Friday, April 29) will cover sections
REVIEW I MATH 254 Calculus IV Exam I (Friday, April 29 will cover sections 14.1-8. 1. Functions of multivariables The definition of multivariable functions is similar to that of functions of one variable.
More informationEquation of tangent plane: for implicitly defined surfaces section 12.9
Equation of tangent plane: for implicitly defined surfaces section 12.9 Some surfaces are defined implicitly, such as the sphere x 2 + y 2 + z 2 = 1. In general an implicitly defined surface has the equation
More informationf xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is
1. Let f(x, y) = 5 + 3x 2 + 3y 2 + 2y 3 + x 3. (a) Final all critical points of f. (b) Use the second derivatives test to classify the critical points you found in (a) as a local maximum, local minimum,
More informationPlane Curve [Parametric Equation]
Plane Curve [Parametric Equation] Bander Almutairi King Saud University December 1, 2015 Bander Almutairi (King Saud University) Plane Curve [Parametric Equation] December 1, 2015 1 / 8 1 Parametric Equation
More informationDirectional Derivatives as Vectors
Directional Derivatives as Vectors John Ganci 1 Al Lehnen 2 1 Richland College Dallas, TX jganci@dcccd.edu 2 Madison Area Technical College Madison, WI alehnen@matcmadison.edu Statement of problem We are
More informationTangent Planes/Critical Points
Tangent Planes/Critical Points Christopher Croke University of Pennsylvania Math 115 UPenn, Fall 2011 Problem: Find the tangent line to the curve of intersection of the surfaces xyz = 1 and x 2 + 2y 2
More informationThe directional derivative of f x, y in the direction of at x, y u. f x sa y sb f x y (, ) (, ) 0 0 y 0 0
Review: 0, lim D f u 0 0 0 0 u The directional derivative of f, in the direction of at, is denoted b D f, : u a, b must a unit vector u f sa sb f s 0 (, ) (, ) s f (, ) a f (, ) b 0 0 0 0 0 0 D f, f u
More informationTopic 2.3: Tangent Planes, Differentiability, and Linear Approximations. Textbook: Section 14.4
Topic 2.3: Tangent Planes, Differentiability, and Linear Approximations Textbook: Section 14.4 Warm-Up: Graph the Cone & the Paraboloid paraboloid f (x, y) = x 2 + y 2 cone g(x, y) = x 2 + y 2 Do you notice
More informationKevin James. MTHSC 206 Section 14.5 The Chain Rule
MTHSC 206 Section 14.5 The Chain Rule Theorem (Chain Rule - Case 1) Suppose that z = f (x, y) is a differentiable function and that x(t) and y(t) are both differentiable functions as well. Then, dz dt
More informationName: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.
. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f (x, y) = x y, x + y = 8. Set up the triple integral of an arbitrary continuous function
More informationGradient and Directional Derivatives
Gradient and Directional Derivatives MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Background Given z = f (x, y) we understand that f : gives the rate of change of z in
More informationMath 126 Winter CHECK that your exam contains 8 problems.
Math 126 Winter 2016 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name CHECK that your exam contains 8 problems. This exam is closed book. You may use one 8 1 11 sheet of hand-written
More informationMath 21a Tangent Lines and Planes Fall, What do we know about the gradient f? Tangent Lines to Curves in the Plane.
Math 21a Tangent Lines and Planes Fall, 2016 What do we know about the gradient f? Tangent Lines to Curves in the Plane. 1. For each of the following curves, find the tangent line to the curve at the point
More informationFunctions of Several Variables
Functions of Several Variables Directional Derivatives and the Gradient Vector Philippe B Laval KSU April 7, 2012 Philippe B Laval (KSU) Functions of Several Variables April 7, 2012 1 / 19 Introduction
More informationMath 113 Calculus III Final Exam Practice Problems Spring 2003
Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross
More informationf (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim
Chapter 14 Multiple Integrals..1 Double Integrals, Iterated Integrals, Cross-sections.2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals.3
More informationDetermine whether or not F is a conservative vector field. If it is, find a function f such that F = enter NONE.
Ch17 Practice Test Sketch the vector field F. F(x, y) = (x - y)i + xj Evaluate the line integral, where C is the given curve. C xy 4 ds. C is the right half of the circle x 2 + y 2 = 4 oriented counterclockwise.
More informationMultivariate Calculus: Review Problems for Examination Two
Multivariate Calculus: Review Problems for Examination Two Note: Exam Two is on Tuesday, August 16. The coverage is multivariate differential calculus and double integration. You should review the double
More informationMultivariate Calculus Review Problems for Examination Two
Multivariate Calculus Review Problems for Examination Two Note: Exam Two is on Thursday, February 28, class time. The coverage is multivariate differential calculus and double integration: sections 13.3,
More informationwhile its direction is given by the right hand rule: point fingers of the right hand in a 1 a 2 a 3 b 1 b 2 b 3 A B = det i j k
I.f Tangent Planes and Normal Lines Again we begin by: Recall: (1) Given two vectors A = a 1 i + a 2 j + a 3 k, B = b 1 i + b 2 j + b 3 k then A B is a vector perpendicular to both A and B. Then length
More information----- o Implicit Differentiation ID: A. dy r.---; d 2 Y 2. If- = '" 1-y- then - = dx 'dx 2. a c. -1 d. -2 e.
Name: Class: Date: ----- ID: A Implicit Differentiation Multiple Choice Identify the choice that best completes the statement or answers the question.. The slope of the line tangent to the curve y + (xy
More informationMath 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate.
Math 10 Practice Problems Sec 1.-1. Name Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1) 5 5 - x dy dx -5 0 A) 5 B) C) 15 D) 5 ) 0 0-8 - 6 - x (8 + ln 9) A) 1 1 + x
More informationAP Calculus AB Unit 2 Assessment
Class: Date: 203-204 AP Calculus AB Unit 2 Assessment Multiple Choice Identify the choice that best completes the statement or answers the question. A calculator may NOT be used on this part of the exam.
More information12.7 Tangent Planes and Normal Lines
.7 Tangent Planes and Normal Lines Tangent Plane and Normal Line to a Surface Suppose we have a surface S generated by z f(x,y). We can represent it as f(x,y)-z 0 or F(x,y,z) 0 if we wish. Hence we can
More informationDaily WeBWorK, #1. This means the two planes normal vectors must be multiples of each other.
Daily WeBWorK, #1 Consider the ellipsoid x 2 + 3y 2 + z 2 = 11. Find all the points where the tangent plane to this ellipsoid is parallel to the plane 2x + 3y + 2z = 0. In order for the plane tangent to
More informationHw 4 Due Feb 22. D(fg) x y z (
Hw 4 Due Feb 22 2.2 Exercise 7,8,10,12,15,18,28,35,36,46 2.3 Exercise 3,11,39,40,47(b) 2.4 Exercise 6,7 Use both the direct method and product rule to calculate where f(x, y, z) = 3x, g(x, y, z) = ( 1
More informationMath 209, Fall 2009 Homework 3
Math 209, Fall 2009 Homework 3 () Find equations of the tangent plane and the normal line to the given surface at the specified point: x 2 + 2y 2 3z 2 = 3, P (2,, ). Solution Using implicit differentiation
More informationUpdated: March 31, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University
Updated: March 3, 26 Calculus III Section 5.6 Math 232 Calculus III Brian Veitch Fall 25 Northern Illinois University 5.6 Triple Integrals In order to build up to a triple integral let s start back at
More informationLecture 6: Chain rule, Mean Value Theorem, Tangent Plane
Lecture 6: Chain rule, Mean Value Theorem, Tangent Plane Rafikul Alam Department of Mathematics IIT Guwahati Chain rule Theorem-A: Let x : R R n be differentiable at t 0 and f : R n R be differentiable
More information3.6 Directional Derivatives and the Gradient Vector
288 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES 3.6 Directional Derivatives and te Gradient Vector 3.6.1 Functions of two Variables Directional Derivatives Let us first quickly review, one more time, te
More informationMATH 116 REVIEW PROBLEMS for the FINAL EXAM
MATH 116 REVIEW PROBLEMS for the FINAL EXAM The following questions are taken from old final exams of various calculus courses taught in Bilkent University 1. onsider the line integral (2xy 2 z + y)dx
More informationChapter 15 Notes, Stewart 7e
Contents 15.2 Iterated Integrals..................................... 2 15.3 Double Integrals over General Regions......................... 5 15.4 Double Integrals in Polar Coordinates..........................
More informationMath 213 Calculus III Practice Exam 2 Solutions Fall 2002
Math 13 Calculus III Practice Exam Solutions Fall 00 1. Let g(x, y, z) = e (x+y) + z (x + y). (a) What is the instantaneous rate of change of g at the point (,, 1) in the direction of the origin? We want
More informationMath 253, Section 102, Fall 2006 Practice Final Solutions
Math 253, Section 102, Fall 2006 Practice Final Solutions 1 2 1. Determine whether the two lines L 1 and L 2 described below intersect. If yes, find the point of intersection. If not, say whether they
More informationDirectional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives
Recall that if z = f(x, y), then the partial derivatives f x and f y are defined as and represent the rates of change of z in the x- and y-directions, that is, in the directions of the unit vectors i and
More information4 = 1 which is an ellipse of major axis 2 and minor axis 2. Try the plane z = y2
12.6 Quadrics and Cylinder Surfaces: Example: What is y = x? More correctly what is {(x,y,z) R 3 : y = x}? It s a plane. What about y =? Its a cylinder surface. What about y z = Again a cylinder surface
More informationThe Differential df, the Gradient f, & the Directional Derivative Dû f sec 14.4 (cont), Goals. Warm-up: Differentiability. Notes. Notes.
The Differential df, the Gradient f, & the Directional Derivative Dû f sec 14.4 (cont), 14.5 10 March 2014 Goals. We will: Define the differential df and use it to approximate changes in a function s value.
More informationMath 210, Exam 2, Spring 2010 Problem 1 Solution
Math, Exam, Spring Problem Solution. Find and classify the critical points of the function f(x,y) x 3 +3xy y 3. Solution: By definition, an interior point (a,b) in the domain of f is a critical point of
More information(1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives.
Section 11.3 Partial Derivatives (1) Tangent Lines on Surfaces, (2) Partial Derivatives, (3) Notation and Higher Order Derivatives. MATH 127 (Section 11.3) Partial Derivatives The University of Kansas
More informationMATH2111 Higher Several Variable Calculus Lagrange Multipliers
MATH2111 Higher Several Variable Calculus Lagrange Multipliers Dr. Jonathan Kress School of Mathematics and Statistics University of New South Wales Semester 1, 2016 [updated: February 29, 2016] JM Kress
More informationDirection Fields; Euler s Method
Direction Fields; Euler s Method It frequently happens that we cannot solve first order systems dy (, ) dx = f xy or corresponding initial value problems in terms of formulas. Remarkably, however, this
More information5. y 2 + z 2 + 4z = 0 correct. 6. z 2 + x 2 + 2x = a b = 4 π
M408D (54690/95/00), Midterm #2 Solutions Multiple choice questions (20 points) See last two pages. Question #1 (25 points) Dene the vector-valued function r(t) = he t ; 2; 3e t i: a) At what point P (x
More informationMath Exam III Review
Math 213 - Exam III Review Peter A. Perry University of Kentucky April 10, 2019 Homework Exam III is tonight at 5 PM Exam III will cover 15.1 15.3, 15.6 15.9, 16.1 16.2, and identifying conservative vector
More informationPreliminary Mathematics of Geometric Modeling (3)
Preliminary Mathematics of Geometric Modeling (3) Hongxin Zhang and Jieqing Feng 2006-11-27 State Key Lab of CAD&CG, Zhejiang University Differential Geometry of Surfaces Tangent plane and surface normal
More informationWe can conclude that if f is differentiable in an interval containing a, then. f(x) L(x) = f(a) + f (a)(x a).
= sin( x) = 8 Lecture :Linear Approximations and Differentials Consider a point on a smooth curve y = f(x), say P = (a, f(a)), If we draw a tangent line to the curve at the point P, we can see from the
More informationUniversity of California, Berkeley
University of California, Berkeley FINAL EXAMINATION, Fall 2012 DURATION: 3 hours Department of Mathematics MATH 53 Multivariable Calculus Examiner: Sean Fitzpatrick Total: 100 points Family Name: Given
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Calculus III-Final review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the corresponding position vector. 1) Define the points P = (-,
More informationMat 241 Homework Set 7 Due Professor David Schultz
Mat 41 Homework Set 7 Due Professor David Schultz Directions: Show all algebraic steps neatly and concisely using proper mathematical symbolism When graphs and technology are to be implemented, do so appropriately
More information2.9 Linear Approximations and Differentials
2.9 Linear Approximations and Differentials 2.9.1 Linear Approximation Consider the following graph, Recall that this is the tangent line at x = a. We had the following definition, f (a) = lim x a f(x)
More informationName: Date: 1. Match the equation with its graph. Page 1
Name: Date: 1. Match the equation with its graph. y 6x A) C) Page 1 D) E) Page . Match the equation with its graph. ( x3) ( y3) A) C) Page 3 D) E) Page 4 3. Match the equation with its graph. ( x ) y 1
More informationMH2800/MAS183 - Linear Algebra and Multivariable Calculus
MH28/MAS83 - Linear Algebra and Multivariable Calculus SEMESTER II EXAMINATION 2-22 Solved by Tao Biaoshuai Email: taob@e.ntu.edu.sg QESTION Let A 2 2 2. Solve the homogeneous linear system Ax and write
More informationChapter 10 Homework: Parametric Equations and Polar Coordinates
Chapter 1 Homework: Parametric Equations and Polar Coordinates Name Homework 1.2 1. Consider the parametric equations x = t and y = 3 t. a. Construct a table of values for t =, 1, 2, 3, and 4 b. Plot the
More informationDerivatives. Day 8 - Tangents and Linearizations
Derivatives Day 8 - Tangents and Linearizations Learning Objectives Write an equation for the tangent line to a graph Write an equation for the normal line to a graph Find the locations of horizontal and
More informationSurfaces and Integral Curves
MODULE 1: MATHEMATICAL PRELIMINARIES 16 Lecture 3 Surfaces and Integral Curves In Lecture 3, we recall some geometrical concepts that are essential for understanding the nature of solutions of partial
More informationTopic 5.1: Line Elements and Scalar Line Integrals. Textbook: Section 16.2
Topic 5.1: Line Elements and Scalar Line Integrals Textbook: Section 16.2 Warm-Up: Derivatives of Vector Functions Suppose r(t) = x(t) î + y(t) ĵ + z(t) ˆk parameterizes a curve C. The vector: is: r (t)
More informationc x y f() f (x) Determine the Determine the Approximate c : Replacin on the AP exam: under-approximation
Tangent Lines and Linear Approximations Students should be able to: Determine the slope of tangent line to a curve at a point Determine the equations of tangent lines and normal lines Approximate a value
More informationWorksheet 2.7: Critical Points, Local Extrema, and the Second Derivative Test
Boise State Math 275 (Ultman) Worksheet 2.7: Critical Points, Local Extrema, and the Second Derivative Test From the Toolbox (what you need from previous classes) Algebra: Solving systems of two equations
More informationMATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM
MATH 4: CALCULUS 3 MAY 9, 7 FINAL EXAM I have neither given nor received aid on this exam. Name: 1 E. Kim................ (9am) E. Angel.............(1am) 3 I. Mishev............ (11am) 4 M. Daniel...........
More informationChapter 5 Partial Differentiation
Chapter 5 Partial Differentiation For functions of one variable, y = f (x), the rate of change of the dependent variable can dy be found unambiguously by differentiation: f x. In this chapter we explore
More information(c) 0 (d) (a) 27 (b) (e) x 2 3x2
1. Sarah the architect is designing a modern building. The base of the building is the region in the xy-plane bounded by x =, y =, and y = 3 x. The building itself has a height bounded between z = and
More informationEquation of tangent plane: for explicitly defined surfaces
Equation of tangent plane: for explicitly defined surfaces Suppose that the surface z = f(x,y) has a non-vertical tangent plane at a point (a, b, f(a,b)). The plane y = b intersects the surface at a curve
More informationFunctions of Two variables.
Functions of Two variables. Ferdinánd Filip filip.ferdinand@bgk.uni-obuda.hu siva.banki.hu/jegyzetek 27 February 217 Ferdinánd Filip 27 February 217 Functions of Two variables. 1 / 36 Table of contents
More informationLagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers. Lagrange Multipliers
In this section we present Lagrange s method for maximizing or minimizing a general function f(x, y, z) subject to a constraint (or side condition) of the form g(x, y, z) = k. Figure 1 shows this curve
More information14.6 Directional Derivatives and the Gradient Vector
14 Partial Derivatives 14.6 and the Gradient Vector Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. and the Gradient Vector In this section we introduce
More information14.5 Directional Derivatives and the Gradient Vector
14.5 Directional Derivatives and the Gradient Vector 1. Directional Derivatives. Recall z = f (x, y) and the partial derivatives f x and f y are defined as f (x 0 + h, y 0 ) f (x 0, y 0 ) f x (x 0, y 0
More informationThe Divergence Theorem
The Divergence Theorem MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Summer 2011 Green s Theorem Revisited Green s Theorem: M(x, y) dx + N(x, y) dy = C R ( N x M ) da y y x Green
More informationUNIVERSITI TEKNOLOGI MALAYSIA SSCE 1993 ENGINEERING MATHEMATICS II TUTORIAL 2. 1 x cos dy dx x y dy dx. y cosxdy dx
UNIVESITI TEKNOLOI MALAYSIA SSCE 99 ENINEEIN MATHEMATICS II TUTOIAL. Evaluate the following iterated integrals. (e) (g) (i) x x x sinx x e x y dy dx x dy dx y y cosxdy dx xy x + dxdy (f) (h) (y + x)dy
More information= f (a, b) + (hf x + kf y ) (a,b) +
Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals
More informationThere are 10 problems, with a total of 150 points possible. (a) Find the tangent plane to the surface S at the point ( 2, 1, 2).
Instructions Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. You may use a scientific
More informationMATH 2023 Multivariable Calculus
MATH 2023 Multivariable Calculus Problem Sets Note: Problems with asterisks represent supplementary informations. You may want to read their solutions if you like, but you don t need to work on them. Set
More informationToday s outline: pp
Chapter 3 sections We will SKIP a number of sections Random variables and discrete distributions Continuous distributions The cumulative distribution function Bivariate distributions Marginal distributions
More informationExact equations are first order DEs of the form M(x, y) + N(x, y) y' = 0 for which we can find a function f(x, φ(x)) so that
Section 2.6 Exact Equations (ONLY) Key Terms: Exact equations are first order DEs of the form M(x, y) + N(x, y) y' = 0 for which we can find a function f(x, φ(x)) so that The construction of f(x, φ(x))
More informationLINEAR PROGRAMMING: A GEOMETRIC APPROACH. Copyright Cengage Learning. All rights reserved.
3 LINEAR PROGRAMMING: A GEOMETRIC APPROACH Copyright Cengage Learning. All rights reserved. 3.1 Graphing Systems of Linear Inequalities in Two Variables Copyright Cengage Learning. All rights reserved.
More informationQuiz 6 Practice Problems
Quiz 6 Practice Problems Practice problems are similar, both in difficulty and in scope, to the type of problems you will see on the quiz. Problems marked with a are for your entertainment and are not
More informationName: Final Exam Review. (b) Reparameterize r(t) with respect to arc length measured for the point (1, 0, 1) in the direction of increasing t.
MATH 127 ALULU III Name: 1. Let r(t) = e t i + e t sin t j + e t cos t k (a) Find r (t) Final Exam Review (b) Reparameterize r(t) with respect to arc length measured for the point (1,, 1) in the direction
More informationMath 126C: Week 3 Review
Math 126C: Week 3 Review Note: These are in no way meant to be comprehensive reviews; they re meant to highlight the main topics and formulas for the week. Doing homework and extra problems is always the
More informationSection 12.2: Quadric Surfaces
Section 12.2: Quadric Surfaces Goals: 1. To recognize and write equations of quadric surfaces 2. To graph quadric surfaces by hand Definitions: 1. A quadric surface is the three-dimensional graph of an
More informationChapter 3. Quadric hypersurfaces. 3.1 Quadric hypersurfaces Denition.
Chapter 3 Quadric hypersurfaces 3.1 Quadric hypersurfaces. 3.1.1 Denition. Denition 1. In an n-dimensional ane space A; given an ane frame fo;! e i g: A quadric hypersurface in A is a set S consisting
More information38. Triple Integration over Rectangular Regions
8. Triple Integration over Rectangular Regions A rectangular solid region S in R can be defined by three compound inequalities, a 1 x a, b 1 y b, c 1 z c, where a 1, a, b 1, b, c 1 and c are constants.
More informationQuadric Surfaces. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24
Quadric Surfaces Philippe B. Laval KSU Today Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24 Introduction A quadric surface is the graph of a second degree equation in three variables. The general
More informationTriple Integrals. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Triple Integrals
Triple Integrals MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 211 Riemann Sum Approach Suppose we wish to integrate w f (x, y, z), a continuous function, on the box-shaped region
More information27. Tangent Planes & Approximations
27. Tangent Planes & Approximations If z = f(x, y) is a differentiable surface in R 3 and (x 0, y 0, z 0 ) is a point on this surface, then it is possible to construct a plane passing through this point,
More informationQuadric Surfaces. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Quadric Surfaces Spring /
.... Quadric Surfaces Philippe B. Laval KSU Spring 2012 Philippe B. Laval (KSU) Quadric Surfaces Spring 2012 1 / 15 Introduction A quadric surface is the graph of a second degree equation in three variables.
More informationPART I. Answer each of the following. è1è Let! u = h2;,1;,2i and! v = h3; 1;,1i. Calculate: èaè! u, 2! v èbè the dot product of! u and! v ècè! u æ! v
MATH 241: FINAL EXAM Name Instructions and Point Values: Put your name in the space provided above. Check that your test contains 14 diæerent pages including one blank page. Work each problem below and
More informationPartial Derivatives (Online)
7in x 10in Felder c04_online.tex V3 - January 21, 2015 9:44 A.M. Page 1 CHAPTER 4 Partial Derivatives (Online) 4.7 Tangent Plane Approximations and Power Series It is often helpful to use a linear approximation
More informationApplications of Triple Integrals
Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals
More informationTHE TANGENT PLANE 6.1 INTRODUCTION TO THE TANGENT PLANE CHAPTER 6:
CHAPTER 6: THE TANGENT PLANE 6.1 INTRODUCTION TO THE TANGENT PLANE The following diagram contains the graph of the function y = x 2 and the graph of its tangent line at the point (1, 1) With these graphs,
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationSYSTEMS OF NONLINEAR EQUATIONS
SYSTEMS OF NONLINEAR EQUATIONS Widely used in the mathematical modeling of real world phenomena. We introduce some numerical methods for their solution. For better intuition, we examine systems of two
More informationTHE TANGENT PLANE 6.1 INTRODUCTION TO THE TANGENT PLANE CHAPTER 6:
CHAPTER 6: THE TANGENT PLANE 6.1 INTRODUCTION TO THE TANGENT PLANE The following diagram contains the graph of the function y = x 2 and the graph of its tangent line at the point (1, 1) With these graphs,
More informationUniversity of Saskatchewan Department of Mathematics & Statistics MATH Final Instructors: (01) P. J. Browne (03) B. Friberg (05) H.
University of Saskatchewan Department of Mathematics & Statistics MATH. Final Instructors: (0) P. J. Browne (0) B. Friberg (0) H. Teismann December 9, 000 Time: :00-:00 pm This is an open book exam. Students
More informationAnnouncements. Topics: To Do:
Announcements Topics: In the Functions of Several Variables module: - Section 3: Limits and Continuity - Section 4: Partial Derivatives - Section 5: Tangent Plane, Linearization, and Differentiability
More information