Reconstructing a Function from its Gradient

Slides:

Advertisements

Similar presentations

Lecture 4 Graphic Primitives, Circle. Drawing Circles.

Advertisements

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)

The Idea of Limits x f(x)f(x)

Rules for Differentiating Univariate Functions Given a univariate function that is both continuous and smooth throughout, it is possible to determine its.

The Envelope Theorem The envelope theorem concerns how the optimal value for a particular function changes when a parameter of the function changes This.

4.2 The Mean Value Theorem.

Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series.

Differentiating the Inverse. Objectives Students will be able to Calculate the inverse of a function. Determine if a function has an inverse. Differentiate.

Representing Functions by Power Series. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series.

THE FUNDAMENTAL THEOREM OF CALCULUS. There are TWO different types of CALCULUS. 1. DIFFERENTIATION: finding gradients of curves. 2. INTEGRATION: finding.

EXAMPLE 4 Find the zeros of a quadratic function Find the zeros of f(x) = x 2 + 6x – 7. SOLUTION Graph the function f(x) = x 2 + 6x –7. The x- intercepts.

(MTH 250) Lecture 24 Calculus. Previous Lecture’s Summary Multivariable functions Limits along smooth curves Limits of multivariable functions Continuity.

Limits and Derivatives Concept of a Function y is a function of x, and the relation y = x 2 describes a function. We notice that with such a relation,

Section 5.3 – The Definite Integral

Section 4.3 Zeros of Polynomials. Approximate the Zeros.

Copyright © Cengage Learning. All rights reserved.

5.4 Fundamental Theorem of Calculus. It is difficult to overestimate the power of the equation: It says that every continuous function f is the derivative.

Section 4.4 The Fundamental Theorem of Calculus Part II – The Second Fundamental Theorem.

SECTION 4-4 A Second Fundamental Theorem of Calculus.

GOAL: USE DEFINITION OF DERIVATIVE TO FIND SLOPE, RATE OF CHANGE, INSTANTANEOUS VELOCITY AT A POINT. 3.1 Definition of Derivative.

Sec 15.6 Directional Derivatives and the Gradient Vector

Mathematics for Economics Beatrice Venturi 1 Economics Faculty EXACT: DIFFERENTIAL EQUATIONS Economic Applications LESSON 5 prof. Beatrice Venturi.

Section 3.3 The Product and Quotient Rule. Consider the function –What is its derivative? –What if we rewrite it as a product –Now what is the derivative?

Confidence Interval & Unbiased Estimator Review and Foreword.

Chapter 5 Integration. Indefinite Integral or Antiderivative.

Section 15.6 Directional Derivatives and the Gradient Vector.

1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

Differential Equations Linear Equations with Variable Coefficients.

Warm up Problems More With Integrals It can be helpful to guess and adjust Ex.

Problem of the Day - Calculator Let f be the function given by f(x) = 2e4x. For what value of x is the slope of the line tangent to the graph of f at (x,

Analyzing Isosceles Triangles Chapter 4, Section 6.

Section 9.4 – Solving Differential Equations Symbolically Separation of Variables.

5.4 The Fundamental Theorem of Calculus. I. The Fundamental Theorem of Calculus Part I. A.) If f is a continuous function on [a, b], then the function.

4.2 The Mean Value Theorem.

Mean Value Theorem 5.4.

Then  a # c in (a, b) such that f  (c) = 0.

Mean Value Theorem.

Section 3.3 The Product and Quotient Rule

Use Properties of Parallelograms

Chapter 3 Overview.

Chain Rules for Functions of Several Variables

Derivative of an Exponential

3.2 Differentiability.

Apply the Remainder and Factor Theorems

Question Find the derivative of Sol..

Section Euler’s Method

Differential Equations Separation of Variables

Applying Differentiation

Chapter 4 Integration.

X and Y's of Scientific Method

13 Functions of Several Variables

Self Assessment 1. Find the absolute extrema of the function

Use Properties of Parallelograms

Warmup 1. What is the interval [a, b] where Rolle’s Theorem is applicable? 2. What is/are the c-values? [-3, 3]

§3.10 Linear Approximations and Differentials

Copyright © Cengage Learning. All rights reserved.

difference in the y-values

Gradient/Concavity 2.5 Geometrical Application of Calculus

Differentiation from first principles

Apply the Fundamental Theorem of Algebra

Section 5.3 – The Definite Integral

The Net Change The net change =.

1. Be able to apply The Mean Value Theorem to various functions.

Section 5.3 – The Definite Integral

Section 4.2 Mean Value Theorem.

Differentiation from first principles

To find the average rate of change of a function between any two points on its graph, we calculate the slope of the line containing the two points.

Do Now: Find all extrema of

LAPLACE TRANSFORMATION

Presentation transcript:

Reconstructing a Function from its Gradient

Differentials We begin by reviewing the one-variable case. If f is differentiable at x, then for small h, the increment Δf = f (x + h) − f (x) can be approximated by the differential d f = f´(x) h.

Differentials

Differentials is called the increment of f , and the dot product is called the differential (more formally, the total differential).

Differentials As in the one-variable case, for small h, the differential and the increment are approximately equal:

Differentials

Example

Theorem

Examples