Chapter 14 – Partial Derivatives

Slides:



Advertisements
Similar presentations
DERIVATIVE OF A FUNCTION 1.5. DEFINITION OF A DERIVATIVE OTHER FORMS: OPERATOR:,,,
Advertisements

Copyright © Cengage Learning. All rights reserved. 14 Partial Derivatives.
4.5 Linearization & Newton’s Method
Chapter 3 Derivatives Section 1 Derivatives and Rates of Change 1.
ESSENTIAL CALCULUS CH11 Partial derivatives
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)
2.1 The derivative and the tangent line problem
MA Day 24- February 8, 2013 Section 11.3: Clairaut’s Theorem Section 11.4: Differentiability of f(x,y,z) Section 11.5: The Chain Rule.
Derivative Review Part 1 3.3,3.5,3.6,3.8,3.9. Find the derivative of the function p. 181 #1.
Section 2.9 Linear Approximations and Differentials Math 1231: Single-Variable Calculus.
THE DERIVATIVE AND THE TANGENT LINE PROBLEM
Chapter 13 – Vector Functions
Chapter 13 – Vector Functions
Chapter 12 – Vectors and the Geometry of Space
Chapter 14 – Partial Derivatives
Chapter 12 – Vectors and the Geometry of Space 12.4 The Cross Product 1.
Chapter 14 – Partial Derivatives
9.2 The Directional Derivative 1)gradients 2)Gradient at a point 3)Vector differential operator.
DERIVATIVES Linear Approximations and Differentials In this section, we will learn about: Linear approximations and differentials and their applications.
Chapter 15 – Multiple Integrals
Chapter 15 – Multiple Integrals
Chapter 14 – Partial Derivatives 14.8 Lagrange Multipliers 1 Objectives:  Use directional derivatives to locate maxima and minima of multivariable functions.
Chapter 16 – Vector Calculus 16.9 The Divergence Theorem 1 Objectives:  Understand The Divergence Theorem for simple solid regions.  Use Stokes’ Theorem.
Chapter 15 – Multiple Integrals 15.7 Triple Integrals 1 Objectives:  Understand how to calculate triple integrals  Understand and apply the use of triple.
Chapter 12 – Vectors and the Geometry of Space
PARTIAL DERIVATIVES PARTIAL DERIVATIVES One of the most important ideas in single-variable calculus is:  As we zoom in toward a point on the graph.
Mathematics for Business (Finance)
Chapter 14 – Partial Derivatives
DERIVATIVES 3. We have seen that a curve lies very close to its tangent line near the point of tangency. DERIVATIVES.
Copyright © Cengage Learning. All rights reserved. 2 Derivatives.
DERIVATIVES 3. Summary f(x) ≈ f(a) + f’(a)(x – a) L(x) = f(a) + f’(a)(x – a) ∆y = f(x + ∆x) – f(x) dx = ∆x dy = f’(x)dx ∆ y≈ dy.
Chapter 13 – Vector Functions 13.1 Vector Functions and Space Curves 1 Objectives:  Use vector -valued functions to describe the motion of objects through.
Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 12 Functions of Several Variables.
AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.5:
Chapter 14 – Partial Derivatives 14.3 Partial Derivatives 1 Objectives:  Understand the various aspects of partial derivatives Dr. Erickson.
Lesson 3-11 Linear Approximations and Differentials.
MA Day 25- February 11, 2013 Review of last week’s material Section 11.5: The Chain Rule Section 11.6: The Directional Derivative.
MAT 125 – Applied Calculus 3.2 – The Product and Quotient Rules.
Chapter 10 – Parametric Equations & Polar Coordinates 10.2 Calculus with Parametric Curves 1Erickson.
Slope Fields and Euler’s Method
Chapter 13 – Vector Functions 13.2 Derivatives and Integrals of Vector Functions 1 Objectives:  Develop Calculus of vector functions.  Find vector, parametric,
3.1 Definition of the Derivative & Graphing the Derivative
Sec 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and the Gradient Vector.
Linear approximation and differentials (Section 2.9)
Chapter 15 – Multiple Integrals 15.1 Double Integrals over Rectangles 1 Objectives:  Use double integrals to find volumes  Use double integrals to find.
For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point. We call the equation of the.
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 8 – Further Applications of Integration
Section 3.9 Linear Approximation and the Derivative.
Clicker Question 1 If x = e 2t + 1 and y = 2t 2 + t, then what is y as a function of x ? – A. y = (1/2)(ln 2 (x – 1) + ln(x – 1)) – B. y = ln 2 (x – 1)
Copyright © Cengage Learning. All rights reserved. 15 Multiple Integrals.
Section 14.3 Local Linearity and the Differential.
Logarithmic Differentiation 对数求导. Example 16 Example 17.
Chapter 14 – Partial Derivatives 14.1 Functions of Several Variables 1 Objectives:  Use differential calculus to evaluate functions of several variables.
Linear Approximations. In this section we’re going to take a look at an application not of derivatives but of the tangent line to a function. Of course,
Functions of Several Variables
2.1A Tangent Lines & Derivatives
2.9 Linear Approximations and Differentials
THE DERIVATIVE AND THE TANGENT LINE PROBLEM
Question Find the derivative of Sol..
Find the linearization L(x, y) of the function at the given point
Clicker Question 1 If x = e2t + 1 and y = 2t 2 + t , then what is y as a function of x ? A. y = (1/2)(ln2(x – 1) + ln(x – 1)) B. y = ln2(x – 1) + (1/2)ln(x.
Copyright © Cengage Learning. All rights reserved.
Differentials and Linear Approximation
Area & Volume Chapter 6.1 & 6.2 February 20, 2007.
Section 15.1 Functions of Several variables
Copyright © Cengage Learning. All rights reserved.
§3.10 Linear Approximations and Differentials
Find the directional derivative of the function at the given point in the direction of the vector v. {image}
Presentation transcript:

Chapter 14 – Partial Derivatives 14.4 Tangent Planes & Linear Approximations Objectives: Determine how to approximate functions using tangent planes Determine how to approximate functions using linear functions Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Definition – Tangent Plane Suppose a surface S has equation z = f(x, y), where f has continuous first partial derivatives. Let P(x0, y0, z0) be a point on S. let C1 and C2 be the curves obtained by intersecting the vertical planes y = y0 and x = x0 with the surface S. Then, the point P lies on both C1 and C2. Let T1 and T2 be the tangent lines to the curves C1 and C2 at the point P. Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Tangent Plane Then, the tangent plane to the surface S at the point P is defined to be the plane that contains both tangent lines T1 and T2. Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Equation of a tangent plane Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Example 1 Find an equation of the tangent plane to the given surface at the specified point. Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Visualization Tangent Plane of a Surface Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Linearization The linear function whose graph is this tangent plane, namely is called the linearization of f at (a, b). Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Linear Approximation The approximation is called the linear approximation or the tangent plane approximation of f at (a, b). Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Differentiable This means that the tangent plane approximates the graph of f well near the point of tangency. Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Theorem Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Example 2 Find the linear approximation of the function and use it to approximate f (6.9,2.06). Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Total differential For a differentiable function of two variables, z = f(x, y), we define the differentials dx and dy to be independent variables. Then the differential dz, also called the total differential, is defined by: Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Example 3 Find the differential of the function below: Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Example 4 – pg. 923 # 34 Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm think and the metal in the sides is 0.05 cm thick. Dr. Erickson 14.4 Tangent Planes & Linear Approximations

More Examples The video examples below are from section 14.4 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. Example 1 Example 2 Example 4 Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Demonstrations Feel free to explore these demonstrations below. Tangent Planes on a 3D Graph Total Differential of the First Order Limits of a Rational Function of Two Variables Dr. Erickson 14.4 Tangent Planes & Linear Approximations

Feedback: Privacy Policy Feedback
Do Not Sell My Personal Information
About project: SlidePlayer Terms of Service

© 2025 SlidePlayer.com. Inc. All rights reserved.