1 / 18

Readings for those who have problems with calculus.

Readings for those who have problems with calculus. Mathematics for economists: Sydsæter , K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis", 3rd ed., Prentice Hall. Fundamental Methods of Mathematical Economics, Kevin Wainwright, Alpha Chiang.

nyla
Download Presentation

Readings for those who have problems with calculus.

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. Readings for those who have problems with calculus. Mathematics for economists: Sydsæter, K., P. Hammond, 2008, "Essential Mathematics for Economic Analysis", 3rd ed., Prentice Hall. Fundamental Methods of Mathematical Economics, Kevin Wainwright, Alpha Chiang. Mathematics for Economists, Carl P. Simon, Lawrence E. Blume

  2. Derivatives - introduction • Suppose a car is accelerating from 30mph to 50mph. • At some point it hits the speed of 40mph, but when? • Speed = (distance travelled)/(time passed) • How is it possible to define the speed at a single point of time?

  3. Approximations • Idea: find distance/time for smaller and smaller time intervals. • Here f shows how the distance is changing with the time, x. • The time difference is h while the distance travelled is f(x+h) – f(x).

  4. The derivative • Our approximation of the speed is • As h gets smaller, the approximation of the speed gets better. • When h is infinitesimally small, the calculation of the speed is exact. • Leibniz used the notation dy/dx for the exact speed.

  5. Example • f(x) = x2 • f(x+h) – f(x) = (x + h)2 – x2 = (x2 + 2xh + h2)- x2 = 2xh + h2 • Speed = (2xh + h2)/h = 2x + h • When h is infinitesimally small, this is just 2x.

  6. Rules for Differentiation The derivative of a constant is zero. If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example:

  7. Rules for Differentiation We saw that if , . power rule This is part of a pattern. examples:

  8. Rules for Differentiation Proof:

  9. Rules for Differentiation constant multiple rule: examples:

  10. Rules for Differentiation constant multiple rule: sum and difference rules: (Each term is treated separately)

  11. Rules for Differentiation product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:

  12. Rules for Differentiation quotient rule: or

  13. Formulas you should learn (Cxa)’=Cax(a-1); C, a – a real number (ex)’=ex (ax)’=axlna; a>0

  14. Derivatives rules - summary , , , , c is a constant , for

  15. Chain Rule Consider a simple composite function:

  16. Chain Rule If is the composite of and , then: Find: example: Chain Rule:

  17. Differentiation of Multivariate Functions • The partial derivative of a multivariate function f(x,y) with respect to x is defined as

  18. Differentiation of Multivariate Functions f(x1,x2)= Cx1ax2b

More Related