1 / 14

Chapter 14 – Partial Derivatives

Chapter 14 – Partial Derivatives. 14.8 Lagrange Multipliers. Objectives: Use directional derivatives to locate maxima and minima of multivariable functions Maximize the volume of a box without a lid if we have a fixed amount of cardboard to work with. Lagrange Multiplier.

alvis
Download Presentation

Chapter 14 – Partial Derivatives

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 14 – Partial Derivatives 14.8 Lagrange Multipliers • Objectives: • Use directional derivatives to locate maxima and minima of multivariable functions • Maximize the volume of a box without a lid if we have a fixed amount of cardboard to work with 14.8 Lagrange Multipliers

  2. Lagrange Multiplier • Many optimization problems have restrictions or constraints on the values that can be used to produce the optimal solution. These constraints tend to complicate optimization problems because the optimal solution can occur at a boundary point of the domain. We use Lagrange Multipliers to simplify solitions. 14.8 Lagrange Multipliers

  3. Lagrange Multiplier λ is a Lagrange Multiplier 14.8 Lagrange Multipliers

  4. Method of Lagrange Multipliers 14.8 Lagrange Multipliers

  5. Visualization • Lagrange Multipliers 14.8 Lagrange Multipliers

  6. Example 1 – pg. 963 #6 • Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). 14.8 Lagrange Multipliers

  7. Example 2 – pg. 963 #10 • Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). 14.8 Lagrange Multipliers

  8. Example 3 • Find the minimum value of the function subject to the constraint 14.8 Lagrange Multipliers

  9. Two Constraints • The numbers λ and µ are the Lagrange multipliers such that 14.8 Lagrange Multipliers

  10. Example 4 – pg. 963 #16 • Find the extreme values of f subject to both constraints. 14.8 Lagrange Multipliers

  11. Example 5 – pg. 964 #42 • Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm2 and whose total edge length is 200 cm. 14.8 Lagrange Multipliers

  12. Example 6 – pg. 964 #44 • The plane intersects the cone in an ellipse. • Use Lagrange multipliers to find the highest and lowest points on the ellipse. 14.8 Lagrange Multipliers

  13. More Examples The video examples below are from section 14.6 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 5 14.8 Lagrange Multipliers

  14. Demonstrations • Feel free to explore these demonstrations below. • The Geometry of Lagrange Multipliers • Constrained Optimization • Visualizing the Gradient Vector 14.8 Lagrange Multipliers

More Related