Directional Derivatives and the Gradient
|
|
- Raymond Golden
- 5 years ago
- Views:
Transcription
1 Unit #20 : Directional Derivatives and the Gradient Goals: To learn about dot and scalar products of vectors. To introduce the directional derivative and the gradient vector. To learn how to compute the gradient vector and how it relates to the directional derivative. To explore how the gradient vector relates to contours.
2 Vector Multiplication - 1 Vector Multiplication Unlike for addition and subtraction, vector quantities differ from scalars in that vector multiplication can be defined in several ways. There are two such operations that we will need to use: scalar multiplication dot product Scalar multiplication: λ v combines a scalar, e.g. λ, with a vector, e.g. v to produce a new vector, λ v. the magnitude of the new vector is λ times the original vector length e.g. 2 v = v + v twice as long as the original. If λ > 0, λ v is a vector in the same direction as v If λ < 0, λ v is a vector in the opposite direction as v
3 Vector Multiplication - 2 Example: 2 v, Choose a vector v and then draw 0 v, and ( 1.5) v.
4 Vector Multiplication - 3 Example: form: 2 v, For the vector v = 5, 2, express the following in component 0 v, and ( 1.5) v.
5 Vector Multiplication - 4 Linearity of Vector Operations Addition, subtraction, and scalar multiplication all obey consistent rules of operation familiar from your experience with scalar operations. These properties are summarized on page 617 of Hughes-Hallett. For convenience we repeat them here. Commutativity v + w = w + v Associativity u + ( v + w) = ( u + v) + w Distributivity Identity (λ + µ) v = λ v + µ v 1 v = v, 0 v = 0 λ( v + w) = λ v + λ w v + 0 = v Note that for any vector v, ( 1) v is a vector with the same magnitude/length as v and opposite direction. Because of this property we write ( 1) v = v.
6 Dot Product - Angle Definition - 1 Dot Product of Vectors: v w Remember that the scalar product multiplies a scalar times a vector. Another possible multiplication between two vectors is called the dot product. The dot product combines two vectors, e.g. v, w to produce a scalar, v w If θ [0, π] is the angle between two vectors v and w, then v w = v w cos(θ) Question: (a) -1 Use this definition to find i i. (b) 0 (c) 1 (d) 2
7 Dot Product - Angle Definition - 2 Question: (a) -1 v w = v w cos(θ) Use this definition to find i j. (b) 0 (c) 1 (d) 2
8 Dot Product - Angle Definition - 3 v w = v w cos(θ) Suppose that v and w are perpendicular to one another. What can you say about v w? What can you conclude if v w = 0?
9 Dot Product - Component Definition - 1 The previous definition of dot product involved the angle between the two vectors. It is also helpful to compute the dot product purely in terms of the components of the vectors. Component Definition of Dot Product If v = λ 1 i + λ 2 j + λ 3 k (or = λ 1, λ 2, λ 3 ) and w = µ 1 i + µ 2 j + µ 3 k, (or = µ 1, µ 2, µ 3 ) then v w = λ 1 µ 1 + λ 2 µ 2 + λ 3 µ 3. It is not at all obvious that this is the same as the other definition!
10 Dot Product - Component Definition - 2 The fact that the two definitions always give the same result is proven in your textbook. We will study an example demonstrating this general property to see a specific instance of this general rule. Example: Use both definitions of the dot product to calculate in two different ways. 1, 1 0, 3
11 Dot Product - Component Definition - 3 Example: Find a vector u = a, b of magnitude/length 1 which is perpendicular to the vector 3 i + 7 j.
12 Dot Product - Component Definition - 4 u = a, b of magnitude/length 1, perpendicular to 3 i + 7 j.
13 Dot Product - Component Definition - 5 Are there other possibilities than the perpendicular vector you found?
14 Using the Dot Product - 1 Product Confusion Is ( v 1 v 2 ) v 3 = v 1 ( v 2 v 3 )? (a) Yes, the results are equal. (b) No, the results will be different because of the grouping. (c) No,the results will be different because the product types are different.
15 Using the Dot Product - 2 Example: Which pairs (if any) of vectors from the following list (a) Are perpendicular? (b) Have an angle less than π/2 between them? (c) Have an angle of more than π/2 between them? a = 1, 0, 2 b = 1, 3, 0 c = 2, 1, 1
16 Directional Derivative - Concept - 1 Directional Derivative - Concept Now we can return to the study of rates of change of a function f(x, y) whose domain is all or part of IR 2 (in other words, functions of two real variables, x and y). In our new terms, The partial derivative f x is the rate of change of f in the direction of the unit vector i (towards larger x values) The partial derivative f y is the rate of change of f in the direction of the unit vector j (towards larger y values)
17 Directional Derivative - Concept - 2 On the surface below, find a point that has f x < 0 and f y > 0.
18 Directional Derivative - Concept - 3 Suppose we now want to find the rate of change in an arbitrary direction. Any direction can be specified by a vector u of length 1. Vectors of length 1 are called unit vectors. Given a unit vector u, we want to find the rate of change of f(x, y) if we move away from (x, y) in the direction of u. From the same point on the graph, indicate a direction where the slope would be steeper than f y. Indicate another direction where the slope would be close to zero.
19 Directional Derivative - Contour Diagrams - 1 Example: below. Consider the contour diagram for a linear function f(x, y) shown P On the diagram, mark three directions u, v and w at the point P, chosen so that D u f(a, b) > 0 D v f(a, b) < 0 D w f(a, b) = 0
20 Directional Derivative - Contour Diagrams - 2 Example - The following is a contour diagram for a more complex function f(x, y). A = (a, b) is a point in the domain of f. On the diagram, mark three directions u, v and w at the point P, chosen so that D u f(a, b) > 0 D v f(a, b) < 0 D w f(a, b) = 0
21 Directional Derivative - Definition - 1 We now define the slope of f(x, y) in an arbitrary direction, with the direction specified by a unit vector u. Directional Derivative Let u = (u 1 i + u 2 j) = u 1, u 2 with u u 2 2 = 1, so that u = 1. Then, at the point (a, b) in the domain of f, the rate of change of f in the direction of u is f(a + hu 1, b + hu 2 ) f(a, b) lim. h 0 h This is called the directional derivative of f at the point (a, b) in the direction of u and it is denoted by D u f(a, b) or f u (a, b) Note: This formula only applies if u is a unit vector. Unfortunately, this derivative definition is cumbersome as it involves limits. We would prefer to compute these directional derivatives using our simpler derivative rules if we could.
22 Directional Derivative - Definition - 2 Computing D u f(a, b) How can we go about computing values for D u f(a, b) in a systematic way? Keep in mind the ingredients of our calculation: f(x, y) is a function of two variables, (a, b) is a point in the domain of f, u = u 1, u 2 with u u2 2 = 1 is a unit vector. Then D u f(a, b) = lim h 0 f(a + hu 1, b + hu 2 ) f(a, b) h = lim h 0 f(x, y) f(a, b) h (where x = a + hu 1 and y = b + hu 2 ) Use local linearity to find an alternate expression for f(x, y) f(a, b).
23 Directional Derivative - Definition - 3 Use that alternate expression to express the directional derivative in terms of partial derivatives.
24 Directional Derivative - Calculation - 1 Computing the Directional Derivative If u = u 1, u 2 is a unit vector ( u = 1), then D u f(a, b) = f x (a, b)u 1 + f y (a, b)u 2 NOTE: we don t define directional derivatives for non-unit vectors. To find the slope in the direction of a non-unit length vector, v, you must normalize it before computing the directional derivative. If v = v 1, v 2 is not a unit vector, first find u = 1 v v = 1 v v2 v, 2 then compute D u f(a, b) This formula allows us to compute the slope in any direction simply by knowing the partial derivatives.
25 Directional Derivative - Calculation - 2 Example: Let f(x, y) = x 2 xy 2 and let u = 3 5, 4 5. We are going to show the steps required to calculate D u f(2, 2). Question: (a) Yes. First: is u a unit vector? (b) No.
26 Directional Derivative - Calculation - 3 f(x, y) = x 2 xy 2 and u = 3 5, 4 5. f x (x, y) = f y (x, y) = f x (2, 2) = f y (2, 2) = u 1 = u 2 = D u f(2, 2) =
27 Directional Derivative - Calculation - 4 Now compute the slope in the direction opposite of u. What do you notice about the slope?
28 Directional Derivative - Example - 1 Example: Find the slope of the surface f(x, y) = x 2 y 2 at (x, y) = (2, 3) if we were to move directly towards the origin.
29 Directional Derivative - Example - 2 Slope of f(x, y) = x 2 y 2, at (2, 3), moving directly towards the origin.
30 The Gradient Vector - 1 The Gradient Vector Note that the formula for directional derivatives could be written as a dot product if we so desired: D u f(a, b) = f x (a, b)u 1 + f y (a, b)u 2 = f x (a, b), f y (a, b) }{{} new vector u 1 u 2 }{{} u This is the first appearance of an important vector function called the gradient of f. While f assigns a number to each point in its domain, the gradient of f assigns a vector to each point in the domain of f, provided both partial derivatives of f exist at that point. The gradient is denoted by either grad f or f.
31 The Gradient Vector - 2 Gradient Vector Definition gradf = f = f x (x, y) i + f y (x, y) j = f x (x, y), f y (x, y) Alternate Directional Derivative Definition D u f(x, y) = (gradf) u
32 The Gradient Vector - 3 Example - Let f(x, y) = xe y grad f(x, y) = grad f(1, 0) = grad f(0, 1) = grad f(2, 3) = For each point in the domain of f where the partial derivatives are both defined, the gradient vector is also defined.
33 Gradient Vector - Importance - 1 Example: Use the gradient-based definition of the directional derivative to determine the direction in which a surface has the largest positive slope.
34 Gradient Vector - Importance - 2 Relationship between the surface and the gradient at a point (a, b) The direction of grad f(a, b) is the direction of maximum increase of the function f at the point (a, b). or The gradient at (a, b) points in the direction of the steepest uphill slope.
35 Gradient Vector - Importance - 3 Example: Consider the plane z = x + 2y + 3. At the point (x, y) = (1, 1), in which (x, y) direction should we move to move uphill the most quickly?
36 Gradient Vector - Importance - 4 Support your answer, using the contour diagram for z = x + 2y + 3 shown below
37 Gradient Vector - Properties - 1 Properties of the Gradient Vector Use the properties of the directional derivative and the dot product to justify the following conclusions : grad f(a, b) is perpendicular to the contour of f that passes through the point (a, b) grad f(a, b) gives the direction of maximum decrease of the function f at the point (a, b).
38 Gradient Vector - Properties - 2 grad f(a, b) (i.e. the length or magnitude of the gradient vector) is the maximum rate of change of f at (a, b).
39 Gradient Vector - Properties - 3 Reminding ourselves of these properties of the gradient vector, consider the contour diagram for a function f(x, y) For each of the points. A, B, and C, draw a vector that points in the direction of the gradient vector at that point. At which of the points is the gradient vector longest? At which of the points is the gradient vector shortest? Justify your answers.
40 Gradient and Contours - Example - 1 Putting Gradients and Contours Together We said earlier that the gradient is perpendicular to the contour at the same point. However, that isn t very precise, given that the contours are curves themselves. It is better to say that contours, as curves, have tangent lines, and gradient is perpendicular to those tangent lines. Consider the 2-variable function f(x, y) = xy 2. Write an equation for the contour (level curve), C, of f that passes through the point (2, 1).
41 Gradient and Contours - Example - 2 Find grad f(2, 1). Find an equation for the tangent line at (2, 1) to the contour (level curve) C, through (2, 1).
42 Gradient and Contours - Example - 3 f(x, y) = xy 2 On the axes below, sketch the level curve C indicate the vector grad f(2, 1), and draw the tangent line to the contour at (2, 1). y x
43 2 4 8 Gradient and Contours - Example - 4 For reference, here is a more detailed contour diagram of the function f(x, y) = xy 2, used in the previous question
44 Gradient and Contours - Example - 5 Note The idea of directional derivative and gradient are new, and are easily confused at first. The following reminders can be useful to help you check that you are on the right track. D u f(a, b) is a derivative or slope, so is a scalar number. It is a rate of change associated with a specific direction, chosen regardless of the surface. grad f(a, b) is a vector. Its direction is the direction of maximum increase of f at (a, b). Its length is a number which represent the rate of change in the gradient s direction.
Vector Multiplication. Directional Derivatives and the Gradient
Vector Multiplication - 1 Unit # : Goals: Directional Derivatives and the Gradient To learn about dot and scalar products of vectors. To introduce the directional derivative and the gradient vector. To
More informationDirectional Derivatives and Gradient Vectors. Suppose we want to find the rate of change of a function z = f x, y at the point in the
14.6 Directional Derivatives and Gradient Vectors 1. Partial Derivates are nice, but they only tell us the rate of change of a function z = f x, y in the i and j direction. What if we are interested in
More informationOmm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics
LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector
More informationElements of Vector Calculus : Scalar Field & its Gradient
Elements of Vector Calculus : Scalar Field & its Gradient Lecture 1 : Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Introduction : In this set of approximately 40 lectures
More informationVectors for Physics. AP Physics C
Vectors for Physics AP Physics C A Vector is a quantity that has a magnitude (size) AND a direction. can be in one-dimension, two-dimensions, or even three-dimensions can be represented using a magnitude
More informationGradient and Directional Derivatives October 2013
Gradient and Directional Derivatives 14.5 07 October 2013 function of one variable: makes sense to talk about the rate of change function of several variables: rate of change depends on direction slope
More informationV. Graph Sketching and Max-Min Problems
V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.
More informationSection 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
More informationMath 20C. Lecture Examples. Section 14.5, Part 1. Directional derivatives and gradient vectors in the plane
Math 0C. Lecture Examples. (8/7/08) Section 4.5, Part. Directional derivatives and gradient vectors in the plane The x-derivative f x (a,b) is the derivative of f at (a,b) in the direction of the unit
More informationLinear Algebra. 1.1 Introduction to vectors 1.2 Lengths and dot products. January 28th, 2013 Math 301. Monday, January 28, 13
Linear Algebra 1.1 Introduction to vectors 1.2 Lengths and dot products January 28th, 2013 Math 301 Notation for linear systems 12w +4x + 23y +9z =0 2u + v +5w 2x +2y +8z =1 5u + v 6w +2x +4y z =6 8u 4v
More informationElectromagnetic Theory Prof. D. K. Ghosh Department of Physics Indian Institute of Technology, Bombay
Electromagnetic Theory Prof. D. K. Ghosh Department of Physics Indian Institute of Technology, Bombay Lecture -1 Element of vector calculus: Scalar Field and its Gradient This is going to be about one
More informationPRACTICE PROBLEMS FOR MIDTERM I
Problem. Find the limits or explain why they do not exist (i) lim x,y 0 x +y 6 x 6 +y ; (ii) lim x,y,z 0 x 6 +y 6 +z 6 x +y +z. (iii) lim x,y 0 sin(x +y ) x +y Problem. PRACTICE PROBLEMS FOR MIDTERM I
More informationIntroduction to systems of equations
Introduction to systems of equations A system of equations is a collection of two or more equations that contains the same variables. This is a system of two equations with two variables: In solving a
More information26. Directional Derivatives & The Gradient
26. Directional Derivatives & The Gradient Given a multivariable function z = f(x, y) and a point on the xy-plane P 0 = (x 0, y 0 ) at which f is differentiable (i.e. it is smooth with no discontinuities,
More information2.1 Definition. Let n be a positive integer. An n-dimensional vector is an ordered list of n real numbers.
2 VECTORS, POINTS, and LINEAR ALGEBRA. At first glance, vectors seem to be very simple. It is easy enough to draw vector arrows, and the operations (vector addition, dot product, etc.) are also easy to
More informationQuiz No. 1: Tuesday Jan. 31. Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3.
Quiz No. 1: Tuesday Jan. 31 Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3.20 Chapter 3 Vectors and Two-Dimensional Kinematics Properties of
More informationOn the other hand, if we measured the potential difference between A and C we would get 0 V.
DAY 3 Summary of Topics Covered in Today s Lecture The Gradient U g = -g. r and U E = -E. r. Since these equations will give us change in potential if we know field strength and distance, couldn t we calculate
More informationA-Level Notes CORE 1
A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is
More informationVectors Year 12 Term 1
Vectors Year 12 Term 1 1 Vectors - A Vector has Two properties Magnitude and Direction - A vector is usually denoted in bold, like vector a, or a, or many others. In 2D - a = xı + yȷ - a = x, y - where,
More informationVector Operations. Vector Operations. Graphical Operations. Component Operations. ( ) ˆk
Vector Operations Vector Operations ME 202 Multiplication by a scalar Addition/subtraction Scalar multiplication (dot product) Vector multiplication (cross product) 1 2 Graphical Operations Component Operations
More information2. Signal Space Concepts
2. Signal Space Concepts R.G. Gallager The signal-space viewpoint is one of the foundations of modern digital communications. Credit for popularizing this viewpoint is often given to the classic text of
More informationVector Spaces. Chapter 1
Chapter 1 Vector Spaces Linear algebra is the study of linear maps on finite-dimensional vector spaces. Eventually we will learn what all these terms mean. In this chapter we will define vector spaces
More informationExam 1 Review SOLUTIONS
1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make
More information15. LECTURE 15. I can calculate the dot product of two vectors and interpret its meaning. I can find the projection of one vector onto another one.
5. LECTURE 5 Objectives I can calculate the dot product of two vectors and interpret its meaning. I can find the projection of one vector onto another one. In the last few lectures, we ve learned that
More informationVCE. VCE Maths Methods 1 and 2 Pocket Study Guide
VCE VCE Maths Methods 1 and 2 Pocket Study Guide Contents Introduction iv 1 Linear functions 1 2 Quadratic functions 10 3 Cubic functions 16 4 Advanced functions and relations 24 5 Probability and simulation
More informationVectors for Zero Robotics students
Vectors for Zero Robotics students Zero Robotics Australia August 7, 08 Assumed Knowledge The robots used for the Zero Robotics competition (SPHERES) were designed for NASA researchers, and are able to
More information3 Vectors. 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan
Chapter 3 Vectors 3 Vectors 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 2 3 3-2 Vectors and Scalars Physics deals with many quantities that have both size and direction. It needs a special mathematical
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Functions
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 2017/2018 DR. ANTHONY BROWN 4. Functions 4.1. What is a Function: Domain, Codomain and Rule. In the course so far, we
More informationLINEAR ALGEBRA - CHAPTER 1: VECTORS
LINEAR ALGEBRA - CHAPTER 1: VECTORS A game to introduce Linear Algebra In measurement, there are many quantities whose description entirely rely on magnitude, i.e., length, area, volume, mass and temperature.
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More information11.4 Dot Product Contemporary Calculus 1
11.4 Dot Product Contemporary Calculus 1 11.4 DOT PRODUCT In the previous sections we looked at the meaning of vectors in two and three dimensions, but the only operations we used were addition and subtraction
More informationA quadratic expression is a mathematical expression that can be written in the form 2
118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is
More informationn=0 ( 1)n /(n + 1) converges, but not
Math 07H Topics for the third exam (and beyond) (Technically, everything covered on the first two exams plus...) Absolute convergence and alternating series A series a n converges absolutely if a n converges.
More informationCHAPTER ONE FUNCTIONS AND GRAPHS. In everyday life, many quantities depend on one or more changing variables eg:
CHAPTER ONE FUNCTIONS AND GRAPHS 1.0 Introduction to Functions In everyday life, many quantities depend on one or more changing variables eg: (a) plant growth depends on sunlight and rainfall (b) speed
More informationPage 52. Lecture 3: Inner Product Spaces Dual Spaces, Dirac Notation, and Adjoints Date Revised: 2008/10/03 Date Given: 2008/10/03
Page 5 Lecture : Inner Product Spaces Dual Spaces, Dirac Notation, and Adjoints Date Revised: 008/10/0 Date Given: 008/10/0 Inner Product Spaces: Definitions Section. Mathematical Preliminaries: Inner
More informationContravariant and Covariant as Transforms
Contravariant and Covariant as Transforms There is a lot more behind the concepts of contravariant and covariant tensors (of any rank) than the fact that their basis vectors are mutually orthogonal to
More informationThere are additional problems on WeBWorK, under the name Study Guide Still need to know from before last exam: many things.
Math 236 Suggestions for Studying for Midterm 2 1 Time: 5:30-8:30, Thursday 4/10 Location: SC 1313, SC 1314 What It Covers: Mainly Sections 11.2, 11.3, 10.6, 12.1-12.6, and the beginning of 12.7. (Through
More informationUnits, Physical Quantities, and Vectors. 8/29/2013 Physics 208
Chapter 1 Units, Physical Quantities, and Vectors 1 Goals for Chapter 1 To learn three fundamental quantities of physics and the units to measure them To keep track of significant figures in calculations
More informationLB 220 Homework 4 Solutions
LB 220 Homework 4 Solutions Section 11.4, # 40: This problem was solved in class on Feb. 03. Section 11.4, # 42: This problem was also solved in class on Feb. 03. Section 11.4, # 43: Also solved in class
More informationSIMPLE MULTIVARIATE OPTIMIZATION
SIMPLE MULTIVARIATE OPTIMIZATION 1. DEFINITION OF LOCAL MAXIMA AND LOCAL MINIMA 1.1. Functions of variables. Let f(, x ) be defined on a region D in R containing the point (a, b). Then a: f(a, b) is a
More informationLIMITS AND DERIVATIVES
2 LIMITS AND DERIVATIVES LIMITS AND DERIVATIVES 1. Equation In Section 2.7, we considered the derivative of a function f at a fixed number a: f '( a) lim h 0 f ( a h) f ( a) h In this section, we change
More informationReview of Vector Analysis in Cartesian Coordinates
Review of Vector Analysis in Cartesian Coordinates 1 Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers. Scalars are usually
More informationVECTORS. 3-1 What is Physics? 3-2 Vectors and Scalars CHAPTER
CHAPTER 3 VECTORS 3-1 What is Physics? Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language the language of vectors to describe those
More informationTangent Planes, Linear Approximations and Differentiability
Jim Lambers MAT 80 Spring Semester 009-10 Lecture 5 Notes These notes correspond to Section 114 in Stewart and Section 3 in Marsden and Tromba Tangent Planes, Linear Approximations and Differentiability
More information(x + 3)(x 1) lim(x + 3) = 4. lim. (x 2)( x ) = (x 2)(x + 2) x + 2 x = 4. dt (t2 + 1) = 1 2 (t2 + 1) 1 t. f(x) = lim 3x = 6,
Math 140 MT1 Sample C Solutions Tyrone Crisp 1 (B): First try direct substitution: you get 0. So try to cancel common factors. We have 0 x 2 + 2x 3 = x 1 and so the it as x 1 is equal to (x + 3)(x 1),
More informationFalse. 1 is a number, the other expressions are invalid.
Ma1023 Calculus III A Term, 2013 Pseudo-Final Exam Print Name: Pancho Bosphorus 1. Mark the following T and F for false, and if it cannot be determined from the given information. 1 = 0 0 = 1. False. 1
More informationChapter 2. Motion in One Dimension. AIT AP Physics C
Chapter 2 Motion in One Dimension Kinematics Describes motion while ignoring the agents that caused the motion For now, will consider motion in one dimension Along a straight line Will use the particle
More informationVectors a vector is a quantity that has both a magnitude (size) and a direction
Vectors In physics, a vector is a quantity that has both a magnitude (size) and a direction. Familiar examples of vectors include velocity, force, and electric field. For any applications beyond one dimension,
More information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationx 1. x n i + x 2 j (x 1, x 2, x 3 ) = x 1 j + x 3
Version: 4/1/06. Note: These notes are mostly from my 5B course, with the addition of the part on components and projections. Look them over to make sure that we are on the same page as regards inner-products,
More informationFormulas that must be memorized:
Formulas that must be memorized: Position, Velocity, Acceleration Speed is increasing when v(t) and a(t) have the same signs. Speed is decreasing when v(t) and a(t) have different signs. Section I: Limits
More informationUnit IV: Introduction to Vector Analysis
Unit IV: Introduction to Vector nalysis s you learned in the last unit, there is a difference between speed and velocity. Speed is an example of a scalar: a quantity that has only magnitude. Velocity is
More informationMEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions
MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms
More informationWeek 4: Differentiation for Functions of Several Variables
Week 4: Differentiation for Functions of Several Variables Introduction A functions of several variables f : U R n R is a rule that assigns a real number to each point in U, a subset of R n, For the next
More informationMathematical review trigonometry vectors Motion in one dimension
Mathematical review trigonometry vectors Motion in one dimension Used to describe the position of a point in space Coordinate system (frame) consists of a fixed reference point called the origin specific
More informationMaths Higher Prelim Content
Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
More informationChapter 3. Vectors and. Two-Dimensional Motion Vector vs. Scalar Review
Chapter 3 Vectors and Two-Dimensional Motion Vector vs. Scalar Review All physical quantities encountered in this text will be either a scalar or a vector A vector quantity has both magnitude (size) and
More informationAlgebra I - Study Guide for Final
Name: Date: Period: Algebra I - Study Guide for Final Multiple Choice Identify the choice that best completes the statement or answers the question. To truly study for this final, EXPLAIN why the answer
More informationCalculus for the Life Sciences II Assignment 6 solutions. f(x, y) = 3π 3 cos 2x + 2 sin 3y
Calculus for the Life Sciences II Assignment 6 solutions Find the tangent plane to the graph of the function at the point (0, π f(x, y = 3π 3 cos 2x + 2 sin 3y Solution: The tangent plane of f at a point
More informationCourse Notes Math 275 Boise State University. Shari Ultman
Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction
More informationDistances in R 3. Last time we figured out the (parametric) equation of a line and the (scalar) equation of a plane:
Distances in R 3 Last time we figured out the (parametric) equation of a line and the (scalar) equation of a plane: Definition: The equation of a line through point P(x 0, y 0, z 0 ) with directional vector
More informationLecture Notes for MATH6106. March 25, 2010
Lecture Notes for MATH66 March 25, 2 Contents Vectors 4. Points in Space.......................... 4.2 Distance between Points..................... 4.3 Scalars and Vectors........................ 5.4 Vectors
More informationSection 4.3 Vector Fields
Section 4.3 Vector Fields DEFINITION: A vector field in R n is a map F : A R n R n that assigns to each point x in its domain A a vector F(x). If n = 2, F is called a vector field in the plane, and if
More informationSuccessful completion of the core function transformations unit. Algebra manipulation skills with squares and square roots.
Extension A: Circles and Ellipses Algebra ; Pre-Calculus Time required: 35 50 min. Learning Objectives Math Objectives Students will write the general forms of Cartesian equations for circles and ellipses,
More informationPurpose of the experiment
Kinematics PES 1160 Advanced Physics Lab I Purpose of the experiment To measure a value of g, the acceleration of gravity at the Earth s surface. To understand the relationships between position, velocity
More informationStudy Guide/Practice Exam 2 Solution. This study guide/practice exam is longer and harder than the actual exam. Problem A: Power Series. x 2i /i!
Study Guide/Practice Exam 2 Solution This study guide/practice exam is longer and harder than the actual exam Problem A: Power Series (1) Find a series representation of f(x) = e x2 Explain why the series
More information56 CHAPTER 3. POLYNOMIAL FUNCTIONS
56 CHAPTER 3. POLYNOMIAL FUNCTIONS Chapter 4 Rational functions and inequalities 4.1 Rational functions Textbook section 4.7 4.1.1 Basic rational functions and asymptotes As a first step towards understanding
More informationMath 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
More information10.1 Vectors. c Kun Wang. Math 150, Fall 2017
10.1 Vectors Definition. A vector is a quantity that has both magnitude and direction. A vector is often represented graphically as an arrow where the direction is the direction of the arrow, and the magnitude
More informationVectors Part 1: Two Dimensions
Vectors Part 1: Two Dimensions Last modified: 20/02/2018 Links Scalars Vectors Definition Notation Polar Form Compass Directions Basic Vector Maths Multiply a Vector by a Scalar Unit Vectors Example Vectors
More informationProperties of the Gradient
Properties of the Gradient Gradients and Level Curves In this section, we use the gradient and the chain rule to investigate horizontal and vertical slices of a surface of the form z = g (x; y) : To begin
More informationIn this chapter, we study the calculus of vector fields.
16 VECTOR CALCULUS VECTOR CALCULUS In this chapter, we study the calculus of vector fields. These are functions that assign vectors to points in space. VECTOR CALCULUS We define: Line integrals which can
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
More informationDot product. The dot product is an inner product on a coordinate vector space (Definition 1, Theorem
Dot product The dot product is an inner product on a coordinate vector space (Definition 1, Theorem 1). Definition 1 Given vectors v and u in n-dimensional space, the dot product is defined as, n v u v
More informationMath 233. Directional Derivatives and Gradients Basics
Math 233. Directional Derivatives and Gradients Basics Given a function f(x, y) and a unit vector u = a, b we define the directional derivative of f at (x 0, y 0 ) in the direction u by f(x 0 + ta, y 0
More informationVectors in Physics. Topics to review:
Vectors in Physics Topics to review: Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion
More informationComputing Derivatives With Formulas Some More (pages 14-15), Solutions
Computing Derivatives With Formulas Some More pages 14-15), Solutions This worksheet focuses on computing derivatives using the shortcut formulas, including the power rule, product rule, quotient rule,
More informationSlope Fields: Graphing Solutions Without the Solutions
8 Slope Fields: Graphing Solutions Without the Solutions Up to now, our efforts have been directed mainly towards finding formulas or equations describing solutions to given differential equations. Then,
More informationSection 4.2: The Mean Value Theorem
Section 4.2: The Mean Value Theorem Before we continue with the problem of describing graphs using calculus we shall briefly pause to examine some interesting applications of the derivative. In previous
More informationAnalytic Geometry and Calculus I Exam 1 Practice Problems Solutions 2/19/7
Analytic Geometry and Calculus I Exam 1 Practice Problems Solutions /19/7 Question 1 Write the following as an integer: log 4 (9)+log (5) We have: log 4 (9)+log (5) = ( log 4 (9)) ( log (5)) = 5 ( log
More informationPre-AP Algebra 2 Lesson 1-5 Linear Functions
Lesson 1-5 Linear Functions Objectives: Students will be able to graph linear functions, recognize different forms of linear functions, and translate linear functions. Students will be able to recognize
More informationVECTORS. Given two vectors! and! we can express the law of vector addition geometrically. + = Fig. 1 Geometrical definition of vector addition
VECTORS Vectors in 2- D and 3- D in Euclidean space or flatland are easy compared to vectors in non- Euclidean space. In Cartesian coordinates we write a component of a vector as where the index i stands
More informationPre-Algebra 2. Unit 9. Polynomials Name Period
Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:
More informationM. Matrices and Linear Algebra
M. Matrices and Linear Algebra. Matrix algebra. In section D we calculated the determinants of square arrays of numbers. Such arrays are important in mathematics and its applications; they are called matrices.
More informationSTEP 1: Ask Do I know the SLOPE of the line? (Notice how it s needed for both!) YES! NO! But, I have two NO! But, my line is
EQUATIONS OF LINES 1. Writing Equations of Lines There are many ways to define a line, but for today, let s think of a LINE as a collection of points such that the slope between any two of those points
More informationVectors Summary. can slide along the line of action. not restricted, defined by magnitude & direction but can be anywhere.
Vectors Summary A vector includes magnitude (size) and direction. Academic Skills Advice Types of vectors: Line vector: Free vector: Position vector: Unit vector (n ): can slide along the line of action.
More informationProperties of Derivatives
6 CHAPTER Properties of Derivatives To investigate derivatives using first principles, we will look at the slope of f ( ) = at the point P (,9 ). Let Q1, Q, Q, Q4, be a sequence of points on the curve
More informationReplacing the a in the definition of the derivative of the function f at a with a variable x, gives the derivative function f (x).
Definition of The Derivative Function Definition (The Derivative Function) Replacing the a in the definition of the derivative of the function f at a with a variable x, gives the derivative function f
More informationMAT 1339-S14 Class 8
MAT 1339-S14 Class 8 July 28, 2014 Contents 7.2 Review Dot Product........................... 2 7.3 Applications of the Dot Product..................... 4 7.4 Vectors in Three-Space.........................
More informationMATH 12 CLASS 4 NOTES, SEP
MATH 12 CLASS 4 NOTES, SEP 28 2011 Contents 1. Lines in R 3 1 2. Intersections of lines in R 3 2 3. The equation of a plane 4 4. Various problems with planes 5 4.1. Intersection of planes with planes or
More informationMathematics Review. Sid Rudolph
Physics 2010 Sid Rudolph General Physics Mathematics Review These documents in mathematics are intended as a brief review of operations and methods. Early in this course, you should be totally familiar
More informationStudent Exploration: Vectors
Name: Date: Student Exploration: Vectors Vocabulary: component, dot product, magnitude, resultant, scalar, unit vector notation, vector Prior Knowledge Question (Do this BEFORE using the Gizmo.) An airplane
More informationTangent Plane. Linear Approximation. The Gradient
Calculus 3 Lia Vas Tangent Plane. Linear Approximation. The Gradient The tangent plane. Let z = f(x, y) be a function of two variables with continuous partial derivatives. Recall that the vectors 1, 0,
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS The fundamental objects that we deal with in calculus are functions. FUNCTIONS AND MODELS This chapter prepares the way for calculus by discussing: The basic
More informationCalculus (Math 1A) Lecture 4
Calculus (Math 1A) Lecture 4 Vivek Shende August 31, 2017 Hello and welcome to class! Last time We discussed shifting, stretching, and composition. Today We finish discussing composition, then discuss
More informationChapter 3 Vectors. 3.1 Vector Analysis
Chapter 3 Vectors 3.1 Vector nalysis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Coordinate Systems... 6 3.2.1 Cartesian Coordinate System... 6 3.2.2 Cylindrical Coordinate
More informationCalculus (Math 1A) Lecture 4
Calculus (Math 1A) Lecture 4 Vivek Shende August 30, 2017 Hello and welcome to class! Hello and welcome to class! Last time Hello and welcome to class! Last time We discussed shifting, stretching, and
More informationInstructor Quick Check: Question Block 12
Instructor Quick Check: Question Block 2 How to Administer the Quick Check: The Quick Check consists of two parts: an Instructor portion which includes solutions and a Student portion with problems for
More informationNumbers and Operations Review
C H A P T E R 5 Numbers and Operations Review This chapter reviews key concepts of numbers and operations that you need to know for the SAT. Throughout the chapter are sample questions in the style of
More informationNote: Every graph is a level set (why?). But not every level set is a graph. Graphs must pass the vertical line test. (Level sets may or may not.
Curves in R : Graphs vs Level Sets Graphs (y = f(x)): The graph of f : R R is {(x, y) R y = f(x)} Example: When we say the curve y = x, we really mean: The graph of the function f(x) = x That is, we mean
More information