{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Macroeconomics Exam Review 204

# Macroeconomics Exam Review 204 - c 2001 Michael Carter All...

This preview shows page 1. Sign up to view the full content.

4.16 The directional derivative 𝐷 x 𝑓 ( x 0 ) measures the rate of increase of 𝑓 in the di- rection x . Using Exercises 4.10, 4.14 and 3.61, assuming x has unit norm, 𝐷 x 𝑓 ( x 0 ) = 𝐷𝑓 [ x 0 ]( x ) = < 𝑓 ( x 0 ) , x > 𝑓 ( x 0 ) This bound is attained when x = 𝑓 ( x 0 ) / 𝑓 ( x 0 ) since 𝐷 x 𝑓 ( x 0 ) = < 𝑓 ( x 0 ) , 𝑓 ( x 0 ) ∥∇ 𝑓 ( x 0 ) > = 𝑓 ( x 0 ) 2 ∥∇ 𝑓 ( x 0 ) = 𝑓 ( x 0 ) The directional derivative is maximized when 𝑓 ( x 0 ) and x are aligned. 4.17 Using Exercise 4.14 𝐻 = { x 𝑋 : < 𝑓 [ x 0 ] , x > = 0 } 4.18 Assume each 𝑓 𝑗 is differentiable at x 0 and let 𝐷𝑓 [ x 0 ] = ( 𝐷𝑓 1 [ x 0 ] , 𝐷𝑓 2 [ x 0 ] , . . . , 𝐷𝑓 𝑚 [ x 0 ]) Then f ( x 0 + x ) f [ x 0 ] 𝐷 f [ x 0 ] x = 𝑓 1 ( x 0 + x ) 𝑓 1 [ x 0 ] 𝐷𝑓 1 [ x 0 ] x 𝑓 2 ( x 0 + x ) 𝑓 2 [ x 0 ] 𝐷𝑓 2 [ x 0 ] x . . . 𝑓 𝑚 ( x 0 + x ) 𝑓 𝑚 ( x 0 ) 𝐷𝑓 𝑚 [ x 0 ] x and 𝑓 𝑗 ( x 0 + x ) 𝑓 𝑗 ( x 0 ) 𝐷𝑓 𝑗 [ x 0 ] x x 0 as x ∥ → 0 for every 𝑗 implies f ( x 0 + x ) f ( x 0 ) 𝐷 f [ x 0 ]( x ) x 0 as x ∥ → 0 (4.43)
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern