slides0-handout

slides0-handout - Econ407 J. Tessada Spring 2009 Intro...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Econ407 J. Tessada Spring 2009 Intro Derivatives Growth Rates Optimization Constrained Optimization Econ407 Advanced Macroeconomics Lecture 0: A Quick Math Refresher Jos´e Tessada University of Maryland - College Park January 29, 2009 Econ407 J. Tessada Spring 2009 Intro Derivatives Growth Rates Optimization Constrained Optimization Announcements Web: http://sites.google.com/site/econ407 Syllabus updated with some links to readings Next two lectures: We will review what the evidence tells us about economic growth No readings assigned After that we will start studying the Solow growth model
Background image of page 1

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

View Full DocumentRight Arrow Icon
Econ407 J. Tessada Spring 2009 Intro Derivatives Simple Derivatives Partial Derivatives Growth Rates Optimization Constrained Optimization Derivatives: A Refresher The derivative of a function y = f ( x ) at the point P = ( x 1 , f ( x 1 )) is the slope of the tangent line at that point The derivative is also a “function” The derivative of a function represents the ratio of the change in the function when the argument variables changes an infinitesimal amount Let y = f ( x ) be a function, then the derivative of y with respect to x , f 0 ( x ) is defined as f 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h if the limit exists and is finite. Econ407 J. Tessada Spring 2009 Intro Derivatives Simple Derivatives Partial Derivatives Growth Rates Optimization Constrained Optimization Derivatives: Some Notation A derivative can be denoted in many different ways Let us again use the example of y = f ( x ) , its derivative can be written as any of the following f 0 ( x ) y 0 dy dx df dx d dx [ f ( x )] If we evaluate the derivative at a particular point x = a we write f 0 ( a ) or dy dx ± ± ± ± a Note: not every continuous function is differentiable (i.e., they are not the same concept)
Background image of page 2
Econ407 J. Tessada Spring 2009 Intro Derivatives Simple Derivatives Partial Derivatives Growth Rates Optimization Constrained Optimization Rules of Differentiation I Some usual rules ( a , b , n are constants), all are derivatives with respect to x Constant functions: f ( x ) = a f 0 ( x ) = 0 Additive and multiplicative functions f ( x ) = a + bx f 0 ( x ) = b Powers of x f ( x ) = x n f 0 ( x ) = nx n - 1 Natural logs f ( x ) = ln ( x ) f 0 ( x ) = 1/ x Exponential f ( x ) = e ax f 0 ( x ) = ae ax where e is the exponential function Functions of the form a x f ( x ) = a x f 0 ( x ) = a x ln ( a ) Econ407 J. Tessada Spring 2009 Intro Derivatives Simple Derivatives Partial Derivatives Growth Rates Optimization Constrained Optimization Rules of Differentiation II Sums f ( x ) = g ( x ) + h ( x ) f 0 ( x ) = g 0 ( x ) + h 0 ( x ) Products f ( x ) = g ( x ) × h ( x ) f 0 ( x ) = g 0 ( x ) h ( x ) + g ( x ) h 0 ( x ) Quotient (assume h ( x ) 6 = 0) f ( x ) = g ( x ) h ( x ) f 0 ( x ) = h ( x ) g 0 ( x ) - g ( x ) h 0 ( x ) [ h ( x )] 2 Chain rule: consider the composite function h ( x ) = g f ( x ) = g ( f ( x )) , we can obtain the derivative of h ( x ) with respect to x using the chain rule h ( x ) = g f ( x ) = g ( f (
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/08/2009 for the course ECON 407 taught by Professor Josétessada during the Spring '09 term at Maryland.

Page1 / 17

slides0-handout - Econ407 J. Tessada Spring 2009 Intro...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online