Directional Derivatives and the Gradient Vector Part 2
|
|
|
- Madeline Malone
- 5 years ago
- Views:
Transcription
1 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, / 12
2 Recall Fact Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
3 Recall Fact If f is a dierentiable function of x and y, then f has a directional derivative in the direction of any unit vector u = a, b and D u f (x, y) = f x (x, y)a + f y (x, y)b. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
4 Recall Fact If f is a dierentiable function of x and y, then f has a directional derivative in the direction of any unit vector u = a, b and D u f (x, y) = f x (x, y)a + f y (x, y)b. If f is a function of two variables x and y, then the gradient of f is the vector function f dened by f (x, y) = f x (x, y), f y (x, y) = f x i + f y j. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
5 Maximizing the Directional Derivative Theorem Suppose that f is a dierentiable function of two (or three) variables. The maximum value of the directional derivative D u f (x, y) is f and it occurs when u has the same direction as the gradient vector f (x). Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
6 Example Example Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
7 Example Example If f (x, y) = xe y, nd the rate of change of f at the point P(2, 0) in the direction from P to Q( 1 2, 2). Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
8 Example Example If f (x, y) = xe y, nd the rate of change of f at the point P(2, 0) in the direction from P to Q( 1 2, 2). In what direction does f have the maximum rate of change? What is this maximum rate of change? Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
9 Example Example Suppose that the temperature at a point (x, y, z) in space is given by T (x, y, z) = x 2 + 2y 2 + 3z 2, where T is measured in degree Celsius and x, y, z in meters. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
10 Example Example Suppose that the temperature at a point (x, y, z) in space is given by T (x, y, z) = x 2 + 2y 2 + 3z 2, where T is measured in degree Celsius and x, y, z in meters. In which direction does the temperature increase fastest at the point (1, 1, 2)? Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
11 Example Example Suppose that the temperature at a point (x, y, z) in space is given by T (x, y, z) = x 2 + 2y 2 + 3z 2, where T is measured in degree Celsius and x, y, z in meters. In which direction does the temperature increase fastest at the point (1, 1, 2)? What is the maximum rate of increase? Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
12 Tangent Planes to Level Surfaces Denition Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
13 Tangent Planes to Level Surfaces Denition A level surface is a surface with equation F (x, y, z) = k. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
14 Tangent Planes to Level Surfaces Denition A level surface is a surface with equation F (x, y, z) = k. Let P(x 0, y 0, z 0 ) be a point on S and let C be any curve that lies on S and passes trough P. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
15 Tangent Planes to Level Surfaces Denition A level surface is a surface with equation F (x, y, z) = k. Let P(x 0, y 0, z 0 ) be a point on S and let C be any curve that lies on S and passes trough P. Recall that C is described by a continuous vector function r(t) = x(t), y(t), z(t). Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
16 Tangent Planes to Level Surfaces Fact Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
17 Tangent Planes to Level Surfaces Fact If x, y, and z are dierentiable and F is also dierentiable, we can apply the Chain Rule: F x dx + F dy + F dz = 0; dt y dt z dt Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
18 Tangent Planes to Level Surfaces Fact If x, y, and z are dierentiable and F is also dierentiable, we can apply the Chain Rule: F x dx + F dy + F dz = 0; dt y dt z dt Or F r (t) = 0. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
19 Tangent Planes to Level Surfaces Fact If x, y, and z are dierentiable and F is also dierentiable, we can apply the Chain Rule: F x dx + F dy + F dz = 0; dt y dt z dt Or F r (t) = 0. The gradient vector at P, F (x 0, y 0 z 0 ) is perpendicular to the tangent vector r (t 0 ) to any curve C on S that passes through P. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
20 The Tangent Plane Denition Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
21 The Tangent Plane Denition We dene the tangent plane to the level surface F (x, y, z) = k at P(x 0, y 0, z 0 ) as the plane passes through P and has normal vector F (x 0, y 0, z 0 ). Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
22 The Tangent Plane Denition We dene the tangent plane to the level surface F (x, y, z) = k at P(x 0, y 0, z 0 ) as the plane passes through P and has normal vector F (x 0, y 0, z 0 ). It has equation F x (x 0, y 0, z 0 )(x x 0 )+F y (x 0, y 0, z 0 )(y y 0 )+F z (z 0, y 0, z 0 )(z z 0 ) = 0. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
23 The Normal Line Denition Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
24 The Normal Line Denition The normal line to S at P is the line passing through P and perpendicular to the tangent plane. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
25 The Normal Line Denition The normal line to S at P is the line passing through P and perpendicular to the tangent plane. The symmetric equations are x x 0 F x (x 0, y 0, z 0 ) = y y 0 F y (x 0, y 0, z 0 ) = z z 0 F z (x 0, y 0, z 0 ) Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
26 Special Case Denition If the equation of the surface S is of the form z = f (x, y), that is F (x, y, z) = f (x, y) z = 0 then the equation of the tangent plane becomes f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) (z z 0 ) = 0. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
27 Examples Examples Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
28 Examples Examples Find the tangent plane and normal line of the surface F (x, y, z) = x 2 + y 2 + z 9 = 0 at the point P 0 (1, 2, 4). Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
29 Examples Examples Find the tangent plane and normal line of the surface F (x, y, z) = x 2 + y 2 + z 9 = 0 at the point P 0 (1, 2, 4). Find the equation of the tangent plane at the point ( 2, 1, 3) to the ellipsoid x y 2 + z 2 9 = 3. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
30 Signicance of the Gradient Vector Fact Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
31 Signicance of the Gradient Vector Fact The gradient f gives the direction of fastest increase of f. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
32 Signicance of the Gradient Vector Fact The gradient f gives the direction of fastest increase of f. The gradient f is orthogonal to the level surface S of f through a point P. Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, / 12
Directional 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
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 ()
Tangent Planes and Linear Approximations
February 21, 2007 Tangent Planes Tangent Planes Let S be a surface with equation z = f (x, y). 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. Tangent
Directional 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
Kevin James. MTHSC 206 Section 15.6 Directional Derivatives and the Gra
MTHSC 206 Section 15.6 Directional Derivatives and the Gradient Vector Definition We define the directional derivative of the function f (x, y) at the point (x 0, y 0 ) in the direction of the unit vector
14.6 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
What 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
Equation 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
Lagrange 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
Kevin 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
while 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
Gradient 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
Topic 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)
12.7 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
Curves, 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
14.5 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
Directional 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
Math 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
Functions 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
The 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
d f(g(t), h(t)) = x dt + f ( y dt = 0. Notice that we can rewrite the relationship on the left hand side of the equality using the dot product: ( f
Gradients and the Directional Derivative In 14.3, we discussed the partial derivatives f f and, which tell us the rate of change of the x y height of the surface defined by f in the x direction and the
Math 5BI: Problem Set 2 The Chain Rule
Math 5BI: Problem Set 2 The Chain Rule April 5, 2010 A Functions of two variables Suppose that γ(t) = (x(t), y(t), z(t)) is a differentiable parametrized curve in R 3 which lies on the surface S defined
Constrained Optimization and Lagrange Multipliers
Constrained Optimization and Lagrange Multipliers MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Constrained Optimization In the previous section we found the local or absolute
f 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,
Math 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
REVIEW 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.
Lagrange multipliers October 2013
Lagrange multipliers 14.8 14 October 2013 Example: Optimization with constraint. Example: Find the extreme values of f (x, y) = x + 2y on the ellipse 3x 2 + 4y 2 = 3. 3/2 1 1 3/2 Example: Optimization
Lagrange multipliers 14.8
Lagrange multipliers 14.8 14 October 2013 Example: Optimization with constraint. Example: Find the extreme values of f (x, y) = x + 2y on the ellipse 3x 2 + 4y 2 = 3. 3/2 Maximum? 1 1 Minimum? 3/2 Idea:
Surfaces 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
Daily 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
The 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
Multivariate 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,
Practice problems. 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b.
Practice problems 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b. 1, 1 = c 1 3, 2 + c 2 2, 1. Solve c 1, c 2. 2. Suppose a is a vector in the plane. If the component of the a in
Math 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
Parametric Surfaces. Substitution
Calculus Lia Vas Parametric Surfaces. Substitution Recall that a curve in space is given by parametric equations as a function of single parameter t x = x(t) y = y(t) z = z(t). A curve is a one-dimensional
Solution of final examination
of final examination Math 20, pring 201 December 9, 201 Problem 1 Let v(t) (2t e t ) i j + π cos(πt) k be the velocity of a particle with initial position r(0) ( 1, 0, 2). Find the accelaration at the
Background for Surface Integration
Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to
27. 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,
= w. w u. u ; u + w. x x. z z. y y. v + w. . Remark. The formula stated above is very important in the theory of. surface integral.
1 Chain rules 2 Directional derivative 3 Gradient Vector Field 4 Most Rapid Increase 5 Implicit Function Theorem, Implicit Differentiation 6 Lagrange Multiplier 7 Second Derivative Test Theorem Suppose
The 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.
Chapter 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
Preliminary 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
MULTIPLE 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 = (-,
. 1. Chain rules. Directional derivative. Gradient Vector Field. Most Rapid Increase. Implicit Function Theorem, Implicit Differentiation
1 Chain rules 2 Directional derivative 3 Gradient Vector Field 4 Most Rapid Increase 5 Implicit Function Theorem, Implicit Differentiation 6 Lagrange Multiplier 7 Second Derivative Test Theorem Suppose
Multivariate 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
ds dt ds 1 dt 1 dt v v v dt ds and the normal vector is given by N
Normal Vectors and Curvature In the last section, we stated that reparameterization by arc length would help us analyze the twisting and turning of a curve. In this section, we ll see precisely how to
This exam will be cumulative. Consult the review sheets for the midterms for reviews of Chapters
Final exam review Math 265 Fall 2007 This exam will be cumulative. onsult the review sheets for the midterms for reviews of hapters 12 15. 16.1. Vector Fields. A vector field on R 2 is a function F from
Let and be a differentiable function. Let Then be the level surface given by
Module 12 : Total differential, Tangent planes and normals Lecture 35 : Tangent plane and normal [Section 35.1] > Objectives In this section you will learn the following : The notion tangent plane to a
1. Gradient and Chain Rule. First a reminder about the gradient f of a smooth function f. In R n, the gradient operator or grad has the form
REVIEW OF CRITICAL OINTS, CONVEXITY AND INTEGRALS KATHERINE DONOGHUE & ROB KUSNER 1. Gradient and Chain Rule. First a reminder about the gradient f of a smooth function f. In R n, the gradient operator
Lecture 15. Lecturer: Prof. Sergei Fedotov Calculus and Vectors. Length of a Curve and Parametric Equations
Lecture 15 Lecturer: Prof. Sergei Fedotov 10131 - Calculus and Vectors Length of a Curve and Parametric Equations Sergei Fedotov (University of Manchester) MATH10131 2011 1 / 5 Lecture 15 1 Length of a
Practice problems from old exams for math 233 William H. Meeks III December 21, 2009
Practice problems from old exams for math 233 William H. Meeks III December 21, 2009 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These
University 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
MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points.
MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points. 1. Evaluate the area A of the triangle with the vertices
Differentiability 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
Grad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures
Grad operator, triple and line integrals Notice: this material must not be used as a substitute for attending the lectures 1 .1 The grad operator Let f(x 1, x,..., x n ) be a function of the n variables
Chapter 6 Some Applications of the Integral
Chapter 6 Some Applications of the Integral More on Area More on Area Integrating the vertical separation gives Riemann Sums of the form More on Area Example Find the area A of the set shaded in Figure
Graded Assignment 2 Maple plots
Graded Assignment 2 Maple plots The Maple part of the assignment is to plot the graphs corresponding to the following problems. I ll note some syntax here to get you started see tutorials for more. Problem
3.6 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
MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals)
MA 43 Calculus III Fall 8 Dr. E. Jacobs Assignments Reading assignments are found in James Stewart s Calculus (Early Transcendentals) Assignment. Spheres and Other Surfaces Read. -. and.6 Section./Problems
Chapter 15 Vector Calculus
Chapter 15 Vector Calculus 151 Vector Fields 152 Line Integrals 153 Fundamental Theorem and Independence of Path 153 Conservative Fields and Potential Functions 154 Green s Theorem 155 urface Integrals
MATH2111 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
UNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS ) CHAPTER 3: Partial derivatives and differentiation
UNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES SOLUTIONS ) 3-1. Find, for the following functions: a) fx, y) x cos x sin y. b) fx, y) e xy. c) fx, y) x + y ) lnx + y ). CHAPTER 3: Partial derivatives
Math 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
Tangent & normal vectors The arc-length parameterization of a 2D curve The curvature of a 2D curve
Local Analysis of 2D Curve Patches Topic 4.2: Local analysis of 2D curve patches Representing 2D image curves Estimating differential properties of 2D curves Tangent & normal vectors The arc-length parameterization
Functions 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
5. 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
Plane Curve [Parametric Equation]
![Plane Curve [Parametric Equation]](/thumbs/89/98510023.jpg "Plane 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
x 6 + λ 2 x 6 = for the curve y = 1 2 x3 gives f(1, 1 2 ) = λ actually has another solution besides λ = 1 2 = However, the equation λ
Math 0 Prelim I Solutions Spring 010 1. Let f(x, y) = x3 y for (x, y) (0, 0). x 6 + y (4 pts) (a) Show that the cubic curves y = x 3 are level curves of the function f. Solution. Substituting y = x 3 in
Engineering Mathematics (4)
Engineering Mathematics (4) Zhang, Xinyu Department of Computer Science and Engineering, Ewha Womans University, Seoul, Korea zhangxy@ewha.ac.kr Example With respect to parameter: s (arc length) r( t)
Polar (BC Only) They are necessary to find the derivative of a polar curve in x- and y-coordinates. The derivative
Polar (BC Only) Polar coordinates are another way of expressing points in a plane. Instead of being centered at an origin and moving horizontally or vertically, polar coordinates are centered at the pole
13.1. Functions of Several Variables. Introduction to Functions of Several Variables. Functions of Several Variables. Objectives. Example 1 Solution
13 Functions of Several Variables 13.1 Introduction to Functions of Several Variables Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives Understand
Mat 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
Line Integration in the Complex Plane
න f z dz C Engineering Math EECE 3640 1 A Review of Integration of Functions of Real Numbers: a b f x dx is the area under the curve in the figure below: The interval of integration is the path along the
. Tutorial Class V 3-10/10/2012 First Order Partial Derivatives;...
Tutorial Class V 3-10/10/2012 1 First Order Partial Derivatives; Tutorial Class V 3-10/10/2012 1 First Order Partial Derivatives; 2 Application of Gradient; Tutorial Class V 3-10/10/2012 1 First Order
Math 241, Final Exam. 12/11/12.
Math, Final Exam. //. No notes, calculator, or text. There are points total. Partial credit may be given. ircle or otherwise clearly identify your final answer. Name:. (5 points): Equation of a line. Find
MATH 261 FALL 2000 FINAL EXAM INSTRUCTIONS. 1. This test booklet has 14 pages including this one. There are 25 questions, each worth 8 points.
MATH 261 FALL 2 FINAL EXAM STUDENT NAME - STUDENT ID - RECITATION HOUR - RECITATION INSTRUCTOR INSTRUCTOR - INSTRUCTIONS 1. This test booklet has 14 pages including this one. There are 25 questions, each
Suggested problems - solutions
Suggested problems - solutions Writing equations of lines and planes Some of these are similar to ones you have examples for... most of them aren t. P1: Write the general form of the equation of the plane
1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two:
Final Solutions. Suppose that the equation F (x, y, z) implicitly defines each of the three variables x, y, and z as functions of the other two: z f(x, y), y g(x, z), x h(y, z). If F is differentiable
6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.
Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct
Math 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
First of all, we need to know what it means for a parameterize curve to be differentiable. FACT:
CALCULUS WITH PARAMETERIZED CURVES In calculus I we learned how to differentiate and integrate functions. In the chapter covering the applications of the integral, we learned how to find the length of
Math 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
Question Details SCalcET [ ]
![Question Details SCalcET [ ]](/thumbs/86/94735173.jpg "Question Details SCalcET [ ]")
72 Gradient II (10998074) Due: Fri Oct 6 2017 03:00 PM MDT Question 1 2 3 4 5 6 7 8 9 10 11 12 Instructions Notes and Learning Goals 1. Question Details SCalcET8 14.6.001. [3799846] Level curves for barometric
ENGI Parametric & Polar Curves Page 2-01
ENGI 3425 2. Parametric & Polar Curves Page 2-01 2. Parametric and Polar Curves Contents: 2.1 Parametric Vector Functions 2.2 Parametric Curve Sketching 2.3 Polar Coordinates r f 2.4 Polar Curve Sketching
Partial 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
Surfaces. Ron Goldman Department of Computer Science Rice University
Surfaces Ron Goldman Department of Computer Science Rice University Representations 1. Parametric Plane, Sphere, Tensor Product x = f (s,t) y = g(s,t) z = h(s,t) 2. Algebraic Plane, Sphere, Torus F(x,
PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Summary Assignments...2
PURE MATHEMATICS 212 Multivariable Calculus CONTENTS Page 1. Assignment Summary... i 2. Summary...1 3. Assignments...2 i PMTH212, Multivariable Calculus Assignment Summary 2010 Assignment Date to be Posted
Lecture 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
Winter 2012 Math 255 Section 006. Problem Set 7
Problem Set 7 1 a) Carry out the partials with respect to t and x, substitute and check b) Use separation of varibles, i.e. write as dx/x 2 = dt, integrate both sides and observe that the solution also
MATH 19520/51 Class 8
MATH 19520/51 Class 8 Minh-Tam Trinh University of Chicago 2017-10-11 1 Directional derivatives. 2 Gradient vectors. 3 Review level sets. 4 Tangent planes to level surfaces of functions of three variables.
Math 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
Math 126 Final Examination Autumn CHECK that your exam contains 9 problems on 10 pages.
Math 126 Final Examination Autumn 2016 Your Name Your Signature Student ID # Quiz Section Professor s Name TA s Name CHECK that your exam contains 9 problems on 10 pages. This exam is closed book. You
MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS
MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT203 covers essentially the same material as MAT201, but is more in depth and theoretical. Exam problems are often more sophisticated in scope and difficulty
2. Solve for x when x < 22. Write your answer in interval notation. 3. Find the distance between the points ( 1, 5) and (4, 3).
Math 6 Practice Problems for Final. Find all real solutions x such that 7 3 x = 5 x 3.. Solve for x when 0 4 3x
Review 1. Richard Koch. April 23, 2005
Review Richard Koch April 3, 5 Curves From the chapter on curves, you should know. the formula for arc length in section.;. the definition of T (s), κ(s), N(s), B(s) in section.4. 3. the fact that κ =
Calculus III Meets the Final
Calculus III Meets the Final Peter A. Perry University of Kentucky December 7, 2018 Homework Review for Final Exam on Thursday, December 13, 6:00-8:00 PM Be sure you know which room to go to for the final!
MH2800/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
Math 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
MATH 230 FALL 2004 FINAL EXAM DECEMBER 13, :20-2:10 PM
Problem Score 1 2 Name: SID: Section: Instructor: 3 4 5 6 7 8 9 10 11 12 Total MATH 230 FALL 2004 FINAL EXAM DECEMBER 13, 2004 12:20-2:10 PM INSTRUCTIONS There are 12 problems on this exam for a total
MAT175 Overview and Sample Problems
MAT175 Overview and Sample Problems The course begins with a quick review/overview of one-variable integration including the Fundamental Theorem of Calculus, u-substitutions, integration by parts, and