1 Supplement Partial Derivatives. 2 1-D Derivatives (Review) Given f(x). f'(x) = df/dx = Rate of change of f at point x w.r.t. x h f(x + h) – f(x) xx+h.

Slides:

Advertisements

Similar presentations

F(x,y) = x 2 y + y y  f — = f x =  x 2xyx 2 + 3y y ln2 z = f(x,y) = cos(xy) + x cos 2 y – 3  f — = f y =  y  z —(x,y) =  x – y sin(xy)

Advertisements

Recall Taylor’s Theorem from single variable calculus:

Example: Obtain the Maclaurin’s expansion for

Table of Contents Factoring A Difference Of Squares Difference of Squares: A difference of squares is an algebraic expression of the form A 2 – B 2, where.

The Derivative. Definition Example (1) Find the derivative of f(x) = 4 at any point x.

Unit 4 Review Solutions a 5 – 32a 2 4a 2 (3a 3 – 8) 2. a 2 b 2 + ab ab(ab + 1) 3. x(x – 2) + y(2 – x) x(x – 2) – y(x – 2) (x – 2)(x – y)

Functions of Several Variables Copyright © Cengage Learning. All rights reserved.

3.5 Solving Systems of Equations in Three Variables

Sheng-Fang Huang. Definition of Derivative The Basic Concept.

A map f : Rn  R defined by f(x1,x2,…,xn) is called a scalar field.

Section 11.3 Partial Derivatives

Partial Derivatives Determine if a limit exists: 1.First test by substituting, : 2.Use two path test to test for non-existence of a limit at a point Two-Path.

Chapter 14 – Partial Derivatives 14.3 Partial Derivatives 1 Objectives:  Understand the various aspects of partial derivatives Dr. Erickson.

ME 2304: 3D Geometry & Vector Calculus Dr. Faraz Junejo Gradient of a Scalar field & Directional Derivative.

Second Order Partial Derivatives

For any function f(x,y), the first partial derivatives are represented by f f — = fx and — = fy x y For example, if f(x,y) = log(x sin.

Functions of Several Variables Copyright © Cengage Learning. All rights reserved.

9.2 Partial Derivatives Find the partial derivatives of a given function. Evaluate partial derivatives. Find the four second-order partial derivatives.

Copyright © Cengage Learning. All rights reserved Partial Derivatives.

Chapter 15 Section 15.6 Limits and Continuity; Equality of Mixed Partials.

Slide 5- 1 Copyright © 2012 Pearson Education, Inc.

LAGRANGE mULTIPLIERS By Rohit Venkat.

Ms. Battaglia AB/BC Calculus. Up to this point, most functions have been expressed in explicit form. Ex: y=3x 2 – 5 The variable y is explicitly written.

Objectives: 1.Be able to determine if an equation is in explicit form or implicit form. 2.Be able to find the slope of graph using implicit differentiation.

The general linear 2ed order PDE in two variables x, y. Chapter 2:Linear Second-Order Equations Sec 2.2,2.3,2.4:Canonical Form Canonical Form.

MULTIPLE INTEGRALS 2.2 Iterated Integrals In this section, we will learn how to: Express double integrals as iterated integrals.

PGM 2002/03 Langrage Multipliers. The Lagrange Multipliers The popular Lagrange multipliers method is used to find extremum points of a function on a.

Section 15.3 Partial Derivatives. PARTIAL DERIVATIVES If f is a function of two variables, its partial derivatives are the functions f x and f y defined.

CHAPTER 4 DIFFERENTIATION NHAA/IMK/UNIMAP. INTRODUCTION Differentiation – Process of finding the derivative of a function. Notation NHAA/IMK/UNIMAP.

2.4: THE CHAIN RULE. Review: Think About it!!  What is a derivative???

1 Basic Differentiation Rules Lesson 3.2A. 2 Basic Derivatives Constant function – Given f(x) = k Then f’(x) = 0 Power Function –Given f(x) = x n Then.

9.1 Solving Differential Equations Mon Jan 04 Do Now Find the original function if F’(x) = 3x + 1 and f(0) = 2.

Multivariable Calculus f (x,y) = x ln(y 2 – x) is a function of multiple variables. It’s domain is a region in the xy-plane:

Partial Derivatives Written by Dr. Julia Arnold Professor of Mathematics Tidewater Community College, Norfolk Campus, Norfolk, VA With Assistance from.

Calculus Section 5.3 Differentiate exponential functions If f(x) = e x then f’(x) = e x f(x) = x 3 e x y= √(e x – x) Examples. Find the derivative. y =

Chapter 8 Multivariable Calculus Section 2 Partial Derivatives (Part 1)

Lesson 3-7 Higher Order Deriviatives. Objectives Find second and higher order derivatives using all previously learned rules for differentiation.

Warm Up 1.Find the particular solution to the initial value problem 2.Find the general solution to the differential equation.

Introduction to Differential Equations

Section 14.2 Computing Partial Derivatives Algebraically

Introduction to Differential Equations

Differential Equations

Antiderivatives 5.1.

4.2 – Implicit Differentiation

Chain Rules for Functions of Several Variables

Derivative of an Exponential

Algebra with Whole Numbers

4.2 – Implicit Differentiation

Systems of Linear Equations

Chain Rule AP Calculus.

Find the first partial derivatives of the function. {image}

DIFFERENTIATION OF COMPOSITE FUNCTION Let z = f ( x, y) Possesses continuous partial derivatives and let x = g (t) Y = h(t) Possess continuous.

3.6 Solving Systems of Linear Equations in 3 Variables

Engineering Analysis I

Find f x and f y. f ( x, y ) = x 5 + y 5 + x 5y

5.6 More on Polynomials To write polynomials in descending order..

13 Functions of Several Variables

Chapter 8: Partial Derivatives

Objectives The student will be able to:

Bell Ringer 10/27/10 What is the GCF? 18x³y² and 24x² ab and a³b².

Functions of Several Variables

Factoring Polynomials.

Differentiate the function: {image} .

Space groups Start w/ 2s and 21s 222.

Factoring using the greatest common factor (GCF).

Objectives The student will be able to:

3. Differentiation Rules

3. Differentiation Rules

Differential Equations

Differentiation Rules for Products, Quotients,

Presentation transcript:

1 Supplement Partial Derivatives

2 1-D Derivatives (Review) Given f(x). f'(x) = df/dx = Rate of change of f at point x w.r.t. x h f(x + h) – f(x) xx+h

3 1 st -order Partial Derivatives For a function with 2 variables, f(x, y)

4 Evaluating Partial Derivatives To find f x, we differentiate f w.r.t. x and treat all other variables as constants. Example 1: f(x, y) = x 3 y 2 = (y 2 )x 3 To evaluate f x, y 2 is to be treated as a constant term. Thus f x = (y 2 )(3x 2 ) = 3x 2 y 2 Example 2: f(x, y, z) = 10xyz + sin(y)e x z 3 = (10yz)x + (sin(y)z 3 )e x To evaluate f x, (10yz) and (sin(y)z 3 ) are to be treated as constant terms. Thus f x = 10yz + (sin(y)z 3 )e x

5 Exercise 1) f(x, y) = xe xy f x = f y = 2) f(x, y, z) = sin (xy) f x = f y = f z =

6 2 nd -order Partial Derivatives

7 Exercise f(x, y) = x 3 + x 2 y – 3x y 2 + y 3 f x = f y = f xy = f yx =