Gradients and Directional Derivatives Chp 15.6. Putting it all together… Over the past several classes you have learned how to use and take partial derivatives.

Slides:

Advertisements

Similar presentations

ESSENTIAL CALCULUS CH11 Partial derivatives

Advertisements

CHAPTER 11 Vector-Valued Functions Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11.1VECTOR-VALUED FUNCTIONS.

Work and Energy Partial Derivatives. Work The force can be three dimensional.

Chapter 14 – Partial Derivatives

PARTIAL DERIVATIVES 14. PARTIAL DERIVATIVES 14.6 Directional Derivatives and the Gradient Vector In this section, we will learn how to find: The rate.

VC.03 Gradient Vectors, Level Curves, Maximums/Minimums/Saddle Points.

EEE340 Lecture : Electric Potential Gradient directed uphill, E-fields directed down-voltage. along fields, voltage drops. Work must be done against.

Final Jeopardy 203. Col Differentials 100 MaxMin 100 MaxMin 100 Curves 100 Col Differentials 200 Partial Deriv’s 200 MaxMin 200 Doub. Ints.

Chapter 14 – Partial Derivatives

9.2 The Directional Derivative 1)gradients 2)Gradient at a point 3)Vector differential operator.

Lecture 16 Today Gradient of a scalar field

Partial Derivatives and the Gradient. Definition of Partial Derivative.

Line integrals (10/22/04) :vector function of position in 3 dimensions. :space curve With each point P is associated a differential distance vector Definition.

The Derivative. Objectives Students will be able to Use the “Newton’s Quotient and limits” process to calculate the derivative of a function. Determine.

CS 376b Introduction to Computer Vision 03 / 04 / 2008 Instructor: Michael Eckmann.

Implicit Differentiation. Objectives Students will be able to Calculate derivative of function defined implicitly. Determine the slope of the tangent.

A Quick Review of Math 200  Calculus rests on 3 critical ideas/concepts: Total accumulated change Instantaneous rates of change Limit.

Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 12 Functions of Several Variables.

Applications of Derivatives

MA Day 25- February 11, 2013 Review of last week’s material Section 11.5: The Chain Rule Section 11.6: The Directional Derivative.

Gradient of a Curve: this is given by the gradient of the tangent

MAT 125 – Applied Calculus 3.2 – The Product and Quotient Rules.

3.1 Derivative of a Function Quick Review In Exercises 1 – 4, evaluate the indicated limit algebraically.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.1 Derivative of a Function.

Slope Fields and Euler’s Method

3.3 Rules for Differentiation Quick Review In Exercises 1 – 6, write the expression as a power of x.

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

Sec 15.6 Directional Derivatives and the Gradient Vector

P7 The Cartesian Plane. Quick review of graphing, axes, quadrants, origin, point plotting.

4-2 Quadratic Functions: Standard Form Today’s Objective: I can graph a quadratic function in standard form.

Uses of the 1 st Derivative – Word Problems.

Section 15.6 Directional Derivatives and the Gradient Vector.

Unit 2: Physics Chapter 3: Describing Motion.

UNIT 2, LESSON 4 THEOREMS ABOUT ZEROS. GETTING STARTED Find the quotient and remainder when is divided by x-3.

Vector Valued Functions

September 24 th, 2015 Questions?  In past years we studied systems of linear equations.  We learned three different methods to solve them.  Elimination,

Chapter 8 – Further Applications of Integration

MA Day 34- February 22, 2013 Review for test #2 Chapter 11: Differential Multivariable Calculus.

Directional Derivatives. Example…What ’ s the slope of at (0,1/2)? What’s wrong with the way the question is posed? What ’ s the slope along the direction.

Functions of Several Variables 13 Copyright © Cengage Learning. All rights reserved.

13.6 Day 2 Gradients For an animation of this concept visit:

10.3 Parametric Arc Length & Area of a Surface of Revolution.

Calculus III Chapter 13. Partial Derivatives of f(x,y) z.z.z.z.

Copyright © Cengage Learning. All rights reserved.

Functions of Several Variables

Mean Value Theorem.

2-4 Rates of change & tangent lines

3.6 Chain Rule.

2.3 Basic Differentiation Formulas

Derivative of a Function

13 Functions of Several Variables

F.1 Social Studies Enrichment Program 4.

Section 4: Topographic Maps

Directional derivative

Derivative of a Function

Derivative of a Function

Click to see each answer.

Directional Derivatives and Gradients

Section 14.4 Gradients and the Directional Derivatives in the Plane

2.1 The Derivative and the Slope of a Graph

Derivative of a Function AP Calculus Honors Ms. Olifer

13.1 Vector Functions.

X and Y Intercepts.

3.1 Derivatives.

Find the directional derivative of the function at the given point in the direction of the vector v. {image}

Derivative of a Function

Directional Derivatives and the Gradient Vector

Lecture 16 Gradient in Cartesian Coordinates

Directional Derivatives

Presentation transcript:

Gradients and Directional Derivatives Chp 15.6

Putting it all together… Over the past several classes you have learned how to use and take partial derivatives Today we look at the essential difference between defining slope along a 2D curve and what it means on a 3D surface or curve

Example…What’s the slope of at (0,1/2)? What’s wrong with the way the question is posed? What’s the slope along the direction of the x- axis? What’s the slope along the direction of the y-axis?

Quick estimate from the contour plot: z Look at this in Excel:

The Directional Derivative We need to specify the direction in which the change occurs… Define, via a slightly modified Newton quotient: This specifies the change in the direction of the vector u =

The Gradient We can write the Directional derivative as: Gradient of f(x,y,z)

A Key Theorem Pg 982 – the Directional derivative is maximum when it is in the same direction as the gradient vector! Example: If your ski begins to slide down a ski slope, it will trace out the gradient for that surface!