{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture9 - Lecture 9 Directional Derivatives and the...

Info icon This preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 9: Directional Derivatives and the Gradient May 19, 2009 Lecture 9: Directional Derivatives and the Gradient
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Objectives 1 Compute the directional derivative of a function. 2 Interpret geometrically the directional derivative. 3 Construct the gradient vector of a multivariable function 4 Use the gradient to maximize the directional derivative. 5 Use the gradient in three dimensions to find tangent planes and normal lines to level surfaces. Lecture 9: Directional Derivatives and the Gradient
Image of page 2
Partial derivatives f x ( x ) = lim h 0 f ( x + h , y ) - f ( x , y ) h f y ( x ) = lim h 0 f ( x , y + h ) - f ( x , y ) h Derivatives in the directions of ˆ i and ˆ j Lecture 9: Directional Derivatives and the Gradient
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Directional derivative Suppose that u = h a , b i is a unit vector. We define the directional derivative D u f ( x 0 , y o ) = lim h 0 f ( x 0 + ah , y 0 + bh ) - f ( x 0 , y 0 ) h Lecture 9: Directional Derivatives and the Gradient
Image of page 4
Directional derivative Suppose that u = h a , b i is a unit vector. We define the directional derivative D u f ( x 0 , y o ) = lim h 0 f ( x 0 + ah , y 0 + bh ) - f ( x 0 , y 0 ) h We compute the directional derivative as follows: D u f ( x 0 , y o ) = f x ( x 0 , y 0 ) a + f y ( x 0 , y 0 ) b Lecture 9: Directional Derivatives and the Gradient
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Directional derivative Suppose that u = h a , b i is a unit vector. We define the directional derivative D u f ( x 0 , y o ) = lim h 0 f ( x 0 + ah , y 0 + bh ) - f ( x 0 , y 0 ) h We compute the directional derivative as follows: D u f ( x 0 , y o ) = f x ( x 0 , y 0 ) a + f y ( x 0 , y 0 ) b IMPORTANT: u must be a unit vector Lecture 9: Directional Derivatives and the Gradient
Image of page 6
Gradient vector If f is a function of two variables, the gradient of f is defined as f ( x , y ) = h f x ( x , y ) , f y ( x , y ) i = f x ˆ i + f y ˆ j Lecture 9: Directional Derivatives and the Gradient
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Gradient vector If f is a function of two variables, the gradient of f is defined as f ( x , y ) = h f x ( x , y ) , f y ( x , y ) i = f x ˆ i + f y ˆ j We can then write the directional derivative as D u f ( x 0 , y o ) = f ( x , y ) · u Lecture 9: Directional Derivatives and the Gradient
Image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern