The Gradient and Directional Derivative - (12.6)
|
|
|
- Joel Fowler
- 5 years ago
- Views:
Transcription
1 Te Gradient and Directional Deriatie - (.6). Directional Deriaties Definition: Te directional deriatie of f x,y at te point a,b andintedirectionofteunit ector u u,u, denoted as D u f a,b, is defined by proided te limit exists. Note tat If u, 0, and if u 0,, f a u,b u f a,b f a,b f a,b f a,b f a,b f x a,b f y a,b If D u f a,b 0, ten f x,y is increasing at a,b in te direction u. If D u f a,b 0, ten f x,y is decreasing at a,b in te direction u. Property: Suppose tat f is differentiable at a,b and u u, u is any unit ector. Ten D u f a,b f x a,b u f y a,b u f x a,b, f y a,b u, u. Let g f a u,b u. Ten g 0 f a,b. So, f a u,b u f a,b g g 0 lim g 0 Let x a u, y b u. g f x dx d f dy y d f x a u,b u u f y a u,b u u g 0 f x a,b u f y a,b u. In a similar way, we can derie te directional deriatie of f x,y,z at a,b,c and in te unit direction u u, u, u D u f a,b,c f x a,b,c u f y a,b,c u f z a,b,c u f x a,b,c, f y a,b,c, f z a,b,c u, u, u. In wic direction u, isd u f a,b,c te largest (te smallest)? Recall: u u cos were 0. Hence, u is te largest if u (te smallest if u ). Tat is, D u f a,b,c is te largest (te smallest) along te direction u (u ) were f x a,b,c, f y a,b,c, f z a,b,c. Example Let f x,y x y y. Find D u f, were a. u,
2 b. d, c. u is in te direction from te point, to,0. f x x,y x f y x,y x y, f x, f y, 8 a. D u f, 8 b. u,, D u f, 8 0 c. d,, d, u, D u f, Example Let te temperature at point x,y,z is gien by T x,y,z 8 z 00 e x y in degrees y x y x0 - - Grap of T x,y,0 8 e x y Grap of T x,y, e a. Find te temperature at 0,0,0 and at,,0. T 0,0,0 8 0 e 0 86, T,, e 8 e b. Find all points at wic te temperature is 87 degrees. rcos t,rsin t,00 e r
3 T x,y,z 90 8 z 00 e x y 87 sole for z in terms of x,y z 00 e x y z 00 e x y e x y z 00 z 00 e x y Te temperatures at all ordered pairs: x,y,z on te surface are z t 0r c. Find te rate of cange of te temperature at,,0 in te direction to te point.,., 60. d.,., 60,,0 0.,0.,0, u d d ,0., ,0.,0 T x,y,z 8 00 z y e x T x x,y,z x 00 z y e x, T x,,0 e e T y x,y,z y 00 z y e x, T y,,0 e e T z x,y,z 00 y e x, T z,,0 00 D u f,,0 e, e, ,0., e d. Find te direction from te point,0,99 in wic te temperature increases most rapidly. T x,0, e e, T y,0,99 0, T z,0,99 00 e,0, 00 e, u e 00 e u 00 7 e,0, 00 7 e e,0, 00 e. Gradients Definition: Te gradient of f x,y is te ector-alued function, denoted by f x,y, is defined by f x,y f x x,y, f y x,y proided partial deriaties exist.
4 D u f x,y f x x,y u f y x,y u f x x,y, f y x,y u, u f x,u u f x,y u cos f x,y cos Note tat Since cos, f x,y D u f x,y f x,y f x,y increases te most at a,b along te direction u f x,y and decreases tat f x,y most along te direction u f x,y. f x,y Let r t x t, y t be a parametric representation of te leel cure: f x,y f a,b. Obsere tat d dt f x t, y t dt d f a,b. f x a,b dx f dy dt y a,b 0 f x a,b, f y a,b dx dy, f a,b r t 0. dt dt dt So, te gradient ector at a,b is perpendicular to te tangent ector of te leel cure at a,b. Example Find te maximum and minimum rates of cange of f x,y x y at,. Know tat f x,y x,8y, f,, and f, Te maximum rate of cange of f x,y at, is and te minimum rate of cange is. Example Let f x,y x y. Sketc te leel cures: f x,y, and f x,y and te gradients of f at,,,,, and,. Sketc in te pat of steepest decent from,. f x,y x y f x,y x y Te pat will be graped on te board. y t - -. Tangent Planes and Normal Lines to a Surface Let f x,y,z be a differentiable function and a,b,c be a point in te domain of f. Consider te leel surface: f x,y,z f a,b,c
5 Let r u, x u,, y u,, z u, be a parametric representation of te leel surface. Obsere tat f u f x a,b,c x u f y a,b,c y u f z a,b,c z u f a,b,c x u, y u, z u f a,b,c r u 0 f f x a,b,c x f y a,b,c y f z a,b,c z f a,b,c x, y, z f a,b,c r 0 So, f a,b,c is perpendicular to te plane containing ectors r u and r wic is te tangent plane of te surface at a,b,c.. Hence, f a,b,c is a normal ector of te tangent plane at a,b,c and te tangent plane can be defined as: f x a,b,c x a f y a,b,c y b f z a,b,c z c 0. Te line in te direction of f a,b,c is called te normal line of te surface at a,b,c and is defined as: r t a f x a,b,c t, b f y a,b,c t, c f z a,b,c t, t. or x a f x a,b,c t, y b f y a,b,c t, z c f z a,b,c t. Note: For leel cures: f x,y f a,b, we can consider te equation: z f x,y or f x,y z 0. Ten f x,y z z and te normal ector is: f x a,b, f y a,b,. Example Find te equation of te tangent plane and normal line of te surface f x,y,z x ye z at,,0. f x 8xye z f x,,0 6 f y x e z f z x y e z, f y,,0 f z,,0 6 Te tangent plane at,,0 : 6 x y 6z 0 Te normal line at,,0 : r t 6t, t, 6t, t Example Find all points at wic te tangent plane to te surface z cosxsiny is parallel to te xy plane. Let F x,y,z cosxsiny z. Te tangent plane at a point a,b,c on te surface is parallel to te xy plane if its normal ector F a,b,c is in te direction k,tatis, F a,b,c 0,0,. F x,y,z sinxsiny, cosxcosy, 0,0, sinxsiny 0,, sinx 0, or siny 0 and cosx 0, or cosy 0 cosxcosy 0 x n or y n, weren 0,,,... and x m,ory m were m,,... x n were n 0,,,... y m were m,,... z, x m were n,,... y n were m 0,,,... z 0 x,y,z ; or
The Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )
Test 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 =
Test Review Find te determinant of te matrix below using (a cofactor expansion and (b row reduction Answer: (a det + = (b Observe R R R R R R R R R Ten det B = (((det Hence det Use Cramer s rule to solve:
Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.
Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope
Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is
Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit
Section 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,
Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.
Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative
1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.
Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of
MATH 15a: Linear Algebra Exam 1, Solutions
MATH 5a: Linear Algebra Exam, Solutions. Let T : R 3 R 4 be the linear transformation with T( e ) = 2 e + e 2 + 3 e 3 4 e 4, T( e 2 ) = e e 2 +2 e 3 +6 e 4, and T( e 3 ) = 4 e e 2 +7 e 3 +8 e 4. (a) (6
Higher Derivatives. Differentiable Functions
Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.
Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.
December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need
CHAPTER 3: DERIVATIVES
(Answers to Exercises for Capter 3: Derivatives) A.3.1 CHAPTER 3: DERIVATIVES SECTION 3.1: DERIVATIVES, TANGENT LINES, and RATES OF CHANGE 1) a) f ( 3) f 3.1 3.1 3 f 3.01 f ( 3) 3.01 3 f 3.001 f ( 3) 3.001
LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT
LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as
MATH CALCULUS I 2.1: Derivatives and Rates of Change
MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Cange Professor Donald L. Wite Department of Matematical Sciences Kent State University D.L. Wite (Kent State University) 1 / 1 Introduction Our main
2.2 Derivative. 1. Definition of Derivative at a Point: The derivative of the function f x at x a is defined as
. Derivative. Definition of Derivative at a Point: Te derivative of te function f at a is defined as f fa fa a lim provided te limit eists. If te limit eists, we sa tat f is differentiable at a, oterwise,
Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).
Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate
qwertyuiopasdfghjklzxcvbnmqwerty uiopasdfghjklzxcvbnmqwertyuiopasd fghjklzxcvbnmqwertyuiopasdfghjklzx cvbnmqwertyuiopasdfghjklzxcvbnmq
qwertyuiopasdfgjklzxcbnmqwerty uiopasdfgjklzxcbnmqwertyuiopasd fgjklzxcbnmqwertyuiopasdfgjklzx cbnmqwertyuiopasdfgjklzxcbnmq Projectile Motion Quick concepts regarding Projectile Motion wertyuiopasdfgjklzxcbnmqwertyui
First we will go over the following derivative rule. Theorem
Tuesday, Feb 1 Tese slides will cover te following 1 d [cos(x)] = sin(x) iger-order derivatives 3 tangent line problems 4 basic differential equations First we will go over te following derivative rule
. Compute the following limits.
Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim
Notes: DERIVATIVES. Velocity and Other Rates of Change
Notes: DERIVATIVES Velocity and Oter Rates of Cange I. Average Rate of Cange A.) Def.- Te average rate of cange of f(x) on te interval [a, b] is f( b) f( a) b a secant ( ) ( ) m troug a, f ( a ) and b,
13 Implicit Differentiation
- 13 Implicit Differentiation This sections highlights the difference between explicit and implicit expressions, and focuses on the differentiation of the latter, which can be a very useful tool in mathematics.
Worksheet 9. Math 1B, GSI: Andrew Hanlon. 1 Ce 3t 1/3 1 = Ce 3t. 4 Ce 3t 1/ =
Worksheet 9 Math B, GSI: Andrew Hanlon. Show that for each of the following pairs of differential equations and functions that the function is a solution of a differential equation. (a) y 2 y + y 2 ; Ce
Section 3: The Derivative Definition of the Derivative
Capter 2 Te Derivative Business Calculus 85 Section 3: Te Derivative Definition of te Derivative Returning to te tangent slope problem from te first section, let's look at te problem of finding te slope
Math 1310 Final Exam
Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space
f a h f a h h lim lim
Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point
Function Composition and Chain Rules
Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function
Lecture 4: Partial and Directional derivatives, Differentiability
Lecture 4: Partial and Directional derivatives, Differentiability Rafikul Alam Department of Mathematics IIT Guwahati Differential Calculus Task: Extend differential calculus to the functions: Case I:
MATH 200 WEEK 5 - WEDNESDAY DIRECTIONAL DERIVATIVE
WEEK 5 - WEDNESDAY DIRECTIONAL DERIVATIVE GOALS Be able to compute a gradient vector, and use it to compute a directional derivative of a given function in a given direction. Be able to use the fact that
PRACTICE 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
MATH 3208 MIDTERM REVIEW. (B) {x 4 x 5 ; x ʀ} (D) {x x ʀ} Use the given functions to answer questions # 3 5. determine the value of h(7).
MATH 08 MIDTERM REVIEW. If () = (f + g)() wat is te domain of () { 5 4 ; ʀ} { 4 4 ; ʀ} { 4 5 ; ʀ} { ʀ}. Given p() = and g() = wic function represents k() k() = p() g() + + Use te given functions to answer
DEFINITION OF A DERIVATIVE
DEFINITION OF A DERIVATIVE Section 2.1 Calculus AP/Dual, Revised 2017 viet.dang@umbleisd.net 2.1: Definition of a Derivative 1 DEFINITION A. Te derivative of a function allows you to find te SLOPE OF THE
The Derivative The rate of change
Calculus Lia Vas Te Derivative Te rate of cange Knowing and understanding te concept of derivative will enable you to answer te following questions. Let us consider a quantity wose size is described by
The derivative of a function f is a new function defined by. f f (x + h) f (x)
Derivatives Definition Te erivative of a function f is a new function efine by f f (x + ) f (x) (x). 0 We will say tat a function f is ifferentiable over an interval (a, b) if if te erivative function
Lesson 4 - Limits & Instantaneous Rates of Change
Lesson Objectives Lesson 4 - Limits & Instantaneous Rates of Cange SL Topic 6 Calculus - Santowski 1. Calculate an instantaneous rate of cange using difference quotients and limits. Calculate instantaneous
= h. Geometrically this quantity represents the slope of the secant line connecting the points
Section 3.7: Rates of Cange in te Natural and Social Sciences Recall: Average rate of cange: y y y ) ) ), ere Geometrically tis quantity represents te slope of te secant line connecting te points, f (
Gradient Descent etc.
1 Gradient Descent etc EE 13: Networked estimation and control Prof Kan) I DERIVATIVE Consider f : R R x fx) Te derivative is defined as d fx) = lim dx fx + ) fx) Te cain rule states tat if d d f gx) )
LECTURE 3 3.1Rules of Vector Differentiation
LETURE 3 3.1Rules of Vector Differentiation We hae defined three kinds of deriaties inoling the operator grad( ) i j k, x y z 1 3 di(., x y z curl( i x 1 j y k z 3 d The good news is that you can apply
1 Lecture 13: The derivative as a function.
1 Lecture 13: Te erivative as a function. 1.1 Outline Definition of te erivative as a function. efinitions of ifferentiability. Power rule, erivative te exponential function Derivative of a sum an a multiple
Multiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question
MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
2.1 THE DEFINITION OF DERIVATIVE
2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative
Solutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
Lesson 6: The Derivative
Lesson 6: Te Derivative Def. A difference quotient for a function as te form f(x + ) f(x) (x + ) x f(x + x) f(x) (x + x) x f(a + ) f(a) (a + ) a Notice tat a difference quotient always as te form of cange
Solution to Review Problems for Midterm II
Solution to Review Problems for Midterm II Midterm II: Monday, October 18 in class Topics: 31-3 (except 34) 1 Use te definition of derivative f f(x+) f(x) (x) lim 0 to find te derivative of te functions
Exam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0
3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,
Differential Equations: Homework 2
Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y
Final Review Worksheet
Score: Name: Final Review Worksheet Math 2110Q Fall 2014 Professor Hohn Answers (in no particular order): f(x, y) = e y + xe xy + C; 2; 3; e y cos z, e z cos x, e x cos y, e x sin y e y sin z e z sin x;
Continuity and Differentiability Worksheet
Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;
AP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions
AP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions Question No. 1 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as
2.11 That s So Derivative
2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point
11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is
Page Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 2014 MTH 234 FINAL EXAM December 8, 2014 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 2 5 1 3 5 4 5 5 5 6 5 7 5 2 8 5 9 5 10
Tuesday, Feb 12. These slides will cover the following. [cos(x)] = sin(x) 1 d. 2 higher-order derivatives. 3 tangent line problems
![Tuesday, Feb 12. These slides will cover the following. [cos(x)] = sin(x) 1 d. 2 higher-order derivatives. 3 tangent line problems](/thumbs/96/127321199.jpg "Tuesday, Feb 12. These slides will cover the following. [cos(x)] = sin(x) 1 d. 2 higher-order derivatives. 3 tangent line problems")
Tuesday, Feb 12 These slides will cover the following. 1 d dx [cos(x)] = sin(x) 2 higher-order derivatives 3 tangent line problems 4 basic differential equations Proof First we will go over the following
(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?
Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is
x + ye z2 + ze y2, y + xe z2 + ze x2, z and where T is the
1.(8pts) Find F ds where F = x + ye z + ze y, y + xe z + ze x, z and where T is the T surface in the pictures. (The two pictures are two views of the same surface.) The boundary of T is the unit circle
Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk
Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 1 Te Tangent Line Problem Te idea of a tangent line first arises in geometry in te context of a circle. But before we jump into a discussion of
Math 263 Assignment #4 Solutions. 0 = f 1 (x,y,z) = 2x 1 0 = f 2 (x,y,z) = z 2 0 = f 3 (x,y,z) = y 1
Math 263 Assignment #4 Solutions 1. Find and classify the critical points of each of the following functions: (a) f(x,y,z) = x 2 + yz x 2y z + 7 (c) f(x,y) = e x2 y 2 (1 e x2 ) (b) f(x,y) = (x + y) 3 (x
Geostrophy & Thermal wind
Lecture 10 Geostrophy & Thermal wind 10.1 f and β planes These are planes that are tangent to the earth (taken to be spherical) at a point of interest. The z ais is perpendicular to the plane (anti-parallel
Precalculus Test 2 Practice Questions Page 1. Note: You can expect other types of questions on the test than the ones presented here!
Precalculus Test 2 Practice Questions Page Note: You can expect oter types of questions on te test tan te ones presented ere! Questions Example. Find te vertex of te quadratic f(x) = 4x 2 x. Example 2.
Directional 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
Page Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a
Math 4A03: Practice problems on Multivariable Calculus
Mat 4A0: Practice problems on Mltiariable Calcls Problem Consider te mapping f, ) : R R defined by fx, y) e y + x, e x y) x, y) R a) Is it possible to express x, y) as a differentiable fnction of, ) near
Vector Calculus handout
Vector Calculus handout The Fundamental Theorem of Line Integrals Theorem 1 (The Fundamental Theorem of Line Integrals). Let C be a smooth curve given by a vector function r(t), where a t b, and let f
MAT 1339-S14 Class 2
MAT 1339-S14 Class 2 July 07, 2014 Contents 1 Rate of Cange 1 1.5 Introduction to Derivatives....................... 1 2 Derivatives 5 2.1 Derivative of Polynomial function.................... 5 2.2 Te
Math 1241 Calculus Test 1
February 4, 2004 Name Te first nine problems count 6 points eac and te final seven count as marked. Tere are 120 points available on tis test. Multiple coice section. Circle te correct coice(s). You do
MATH1901 Differential Calculus (Advanced)
MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction
Tangent 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,
Math 115 Test 1 Sample Problems for Dr. Hukle s Class
Mat 5 Test Sample Problems for Dr. Hukle s Class. Demand for a Jayawk pen at te Union is known to be D(p) = 26 pens per mont wen te selling p price is p dollars and eac p 3. A supplier for te bookstore
2.3 More Differentiation Patterns
144 te derivative 2.3 More Differentiation Patterns Polynomials are very useful, but tey are not te only functions we need. Tis section uses te ideas of te two previous sections to develop tecniques for
Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014
Solutions to te Multivariable Calculus and Linear Algebra problems on te Compreensive Examination of January 3, 24 Tere are 9 problems ( points eac, totaling 9 points) on tis portion of te examination.
Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector
Calculus 3 Lia Vas Review for Exam 1 1. Surfaces. Describe the following surfaces. (a) x + y = 9 (b) x + y + z = 4 (c) z = 1 (d) x + 3y + z = 6 (e) z = x + y (f) z = x + y. Review of Vectors. (a) Let a
MVT and Rolle s Theorem
AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state
. h I B. Average velocity can be interpreted as the slope of a tangent line. I C. The difference quotient program finds the exact value of f ( a)
Capter Review Packet (questions - ) KEY. In eac case determine if te information or statement is correct (C) or incorrect (I). If it is incorrect, include te correction. f ( a ) f ( a) I A. represents
Differential Calculus (The basics) Prepared by Mr. C. Hull
Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit
SCORE. Exam 3. MA 114 Exam 3 Fall 2016
Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing
Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
Practice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29
Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)
Math 10C - Fall Final Exam
Math 1C - Fall 217 - Final Exam Problem 1. Consider the function f(x, y) = 1 x 2 (y 1) 2. (i) Draw the level curve through the point P (1, 2). Find the gradient of f at the point P and draw the gradient
Chapter 2: 1D Kinematics Tuesday January 13th
Chapter : D Kinematics Tuesday January 3th Motion in a straight line (D Kinematics) Aerage elocity and aerage speed Instantaneous elocity and speed Acceleration Short summary Constant acceleration a special
Differentiation. Area of study Unit 2 Calculus
Differentiation 8VCE VCEco Area of stud Unit Calculus coverage In tis ca 8A 8B 8C 8D 8E 8F capter Introduction to limits Limits of discontinuous, rational and brid functions Differentiation using first
Math 102: A Log-jam. f(x+h) f(x) h. = 10 x ( 10 h 1. = 10x+h 10 x h. = 10x 10 h 10 x h. 2. The hyperbolic cosine function is defined by
Mat 102: A Log-jam 1. If f(x) = 10 x, sow tat f(x+) f(x) ( 10 = 10 x ) 1 f(x+) f(x) = 10x+ 10 x = 10x 10 10 x = 10 x ( 10 1 ) 2. Te yperbolic cosine function is defined by cos(x) = ex +e x 2 Te yperbolic
Handout 5, Summer 2014 Math May Consider the following table of values: x f(x) g(x) f (x) g (x)
Handout 5, Summer 204 Math 823-7 29 May 204. Consider the following table of values: x f(x) g(x) f (x) g (x) 3 4 8 4 3 4 2 9 8 8 3 9 4 Let h(x) = (f g)(x) and l(x) = g(f(x)). Compute h (3), h (4), l (8),
Polynomial Functions. Linear Functions. Precalculus: Linear and Quadratic Functions
Concepts: definition of polynomial functions, linear functions tree representations), transformation of y = x to get y = mx + b, quadratic functions axis of symmetry, vertex, x-intercepts), transformations
Calculus I Homework: The Derivative as a Function Page 1
Calculus I Homework: Te Derivative as a Function Page 1 Example (2.9.16) Make a careful sketc of te grap of f(x) = sin x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap.
Answer sheet: Final exam for Math 2339, Dec 10, 2010
Answer sheet: Final exam for Math 9, ec, Problem. Let the surface be z f(x,y) ln(y + cos(πxy) + e ). (a) Find the gradient vector of f f(x,y) y + cos(πxy) + e πy sin(πxy), y πx sin(πxy) (b) Evaluate f(,
University of Babylon College of Engineering Mechanical Engineering Dept. Subject : Mathematics III Class : 2 nd First Semester Year :
Uniersity of Babylon College of Engineering Mechanical Engineering Dept. Subject : Mathematics III Class : nd First Semester Year : 16-17 VECTOR FUNCTIONS SECTION 13. Ideal Projectile Motion Ideal Projectile
MA119-A Applied Calculus for Business Fall Homework 4 Solutions Due 9/29/ :30AM
MA9-A Applied Calculus for Business 006 Fall Homework Solutions Due 9/9/006 0:0AM. #0 Find te it 5 0 + +.. #8 Find te it. #6 Find te it 5 0 + + = (0) 5 0 (0) + (0) + =.!! r + +. r s r + + = () + 0 () +
Motion in Two and Three Dimensions
PH 1-A Fall 014 Motion in Two and Three Dimensions Lectures 4,5 Chapter 4 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 4 Motion in Two and Three Dimensions In this chapter
Continuity. Example 1
Continuity MATH 1003 Calculus and Linear Algebra (Lecture 13.5) Maoseng Xiong Department of Matematics, HKUST A function f : (a, b) R is continuous at a point c (a, b) if 1. x c f (x) exists, 2. f (c)
Practice Problem Solutions: Exam 1
Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical
4 Partial Differentiation
4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which
Motion in Two and Three Dimensions
PH 1-1D Spring 013 Motion in Two and Three Dimensions Lectures 5,6,7 Chapter 4 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 4 Motion in Two and Three Dimensions In this chapter
1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits.
Questions 1. State weter te function is an exponential growt or exponential decay, and describe its end beaviour using its. (a) f(x) = 3 2x (b) f(x) = 0.5 x (c) f(x) = e (d) f(x) = ( ) x 1 4 2. Matc te
Introduction to Derivatives
Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))
MTH-112 Quiz 1 Name: # :
MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation
f(x, y) = 1 sin(2x) sin(2y) and determine whether each is a maximum, minimum, or saddle. Solution: Critical points occur when f(x, y) = 0.
Some Answers to Exam Name: (4) Find all critical points in [, π) [, π) of the function: f(x, y) = sin(x) sin(y) 4 and determine whether each is a maximum, minimum, or saddle. Solution: Critical points
Main Points: 1. Limit of Difference Quotients. Prep 2.7: Derivatives and Rates of Change. Names of collaborators:
Name: Section: Names of collaborators: Main Points:. Definition of derivative as limit of difference quotients. Interpretation of derivative as slope of grap. Interpretation of derivative as instantaneous
Department of Mathematics, K.T.H.M. College, Nashik F.Y.B.Sc. Calculus Practical (Academic Year )
F.Y.B.Sc. Calculus Practical (Academic Year 06-7) Practical : Graps of Elementary Functions. a) Grap of y = f(x) mirror image of Grap of y = f(x) about X axis b) Grap of y = f( x) mirror image of Grap
Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.
For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin
Derivatives. By: OpenStaxCollege
By: OpenStaxCollege Te average teen in te United States opens a refrigerator door an estimated 25 times per day. Supposedly, tis average is up from 10 years ago wen te average teenager opened a refrigerator
158 Calculus and Structures
58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you