Math 251 Final Review Pack

Keplers laws astronomers tycho brache and johannes

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ary motion based on astronomical observation. i. ii. iii. Sun- it lies on the focus of the path of the planet, the planet travels in an ellipse Line Sun- planet sweeps out constant area through constant time Period of Revolution- Isaac Newton derived mathematically these laws as consequences of the second law of motion: 14 Chapter 14: Partial Derivatives 15 14.1 Functions of Several Values With a one variable function, there is one input to one output, however for an n variable function; there are n inputs but only one output. Definition (Function of n Variables): An assignment of a real value, Z, to any point in a set The set D is called the domain of the function. If the function is f then: We often write: Independent variables: Dependent variable: Z Note: A function of n variables may be considered as: A function of n inputs A function of a vector with n coordinates: As a function of a point: Definition (Linear Function): Any function of the form: Consider a two variable function Definition (Level or Contour Curves): The curves called level or contour curves of f. 16 ,where k is a constant, are 14.2 Limits and Continuity Recall (1 Variable Limit): means that such that Definition (Limit): that , suppose there exists a disk U around a point Then means such such that: Independence of Path The definition of the limit takes care of the fact that if approaches L along any path that approaches then Non-existence of a Limit If path as with along a path then and does not exist. as along a Note: All limit laws and squeeze theorem hold true for both single variable limits and multivariable limits Definition (Continuity): we say is called continuous on a point if is continuous on if it is continuous on every point . Note: All continuous laws hold for both single variable and multivariable limits. i.e.: sums/product/etc of continuous functions is also continuous Polynomial function of two variables is the sum of terms of the form: Where Rational function of two variables is the ratio of two polynomial functions in All rational functions are continuous under the domain Functions of 3 (or more) Variables For limits and continuity, all of the generalizations in single variables work in multivariable 17 14.3 Partial Derivatives Recall (1 Variable Derivative): Interpretation: Slope of the line tangent to at a point Two variables: Derivative?? Definition (Partial Derivative): , That is, where where We may consider as functions Other notation: For second and higher order derivatives: 18 or the rate of change of at Rules: To compute function of we consider Clairaut’s Theorem: continuous on then: to be a constant in a disk in that is to consider . If This can be generalized for functions of 3 or more variables 19 and as a are both 14.4 Tangent Planes and Linear Approximation Tangent Planes : the curve of intersection of the surface with the plane : the curve of intersection of the surface with the plane : the tangent vector to at The tangent plane to S at P is the plane determined by and approximates most closely the surface S near the point P. In order to find the equation of the tangent plane we need The plane will be of the form: We have: We then parameterize: We then evaluate at to get : Sim...
View Full Document

This note was uploaded on 02/03/2014 for the course CMPT 150 taught by Professor Dr.anthonydixon during the Spring '08 term at Simon Fraser.

Ask a homework question - tutors are online