Definitions and Theorems - Midterm

Definitions and Theorems - Midterm - Math 237 Midterm...

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: Math 237 Midterm Review – Definitions and Theorems • A scalar function : , is a rule which assigns to each ordered pair , a value . • Domain is the set of , for which the function is defined. • Range is the set of , |, • The level curves of a function , are the curves which equation , where , is and a constant in the range of . • The cross-sections of a surface , • , are the curves defined by with , constant. A quadratic surface is the graph of a second degree equation in three variables. 0. They are implicitly defined surfaces. • If : is defined in some neighbourhood of , except possibly at , if for every 0 such that | there exists a | whenever 0 0 then we say the limit as x approaches a exists and equals L. lim • The Squeeze Theorem. If there exists a function such that | : for all x in some neighbourhood of a, except possibly at, and lim | 0 then lim • A function of two variables is said to be continuous at a point a if lim • A function of two variables is said to be continuous on a region at each point a • The Continuity Theorem. If , : if it is continuous , then If , are both continuous at , then o If , are both continuous at , and If : . . o • . , then and : , , 0, are continuous at is continuous at . is the composition of and . Note that cannot be done. • The Theorem of Continuity of Composition of Functions. Let : Then if • is continuous at Let : and . The partial derivatives of o , , lim , lim o , , The tangent plane to the surface , , • If : , Let : , then the linearization of , . The gradient of , , then are defined by and , at is given by the equation , is defined by is defined by , , . is continuous at . , • • is continuous at and : , , , • Let : . Suppose the partial derivatives of , o all exist at . Then , …, · o • Let : . If • Let : . o and • Let : • If • Let : 0 where one point , Let is continuous at . , and differentiable on , . Then there is at least such that A function : • then then it is not differentiable at . that is continuous on • 0 | , is differentiable at is not continuous at Let : , | both exist at . . If • then lim , is said to be differentiable at a point a if , lim o exists and we set . Then if and are continuous at then is differentiable at . is called a vector-valued function or parametric equation. , and lim . If lim then lim , . Also, , • If , is differentiable at then the derivative of • • , · , , • and exit, is at the point The directional derivative of : in the direction of the unit vector | If : · and , · is • , , has continuous partial derivatives at cos where then · 1 The greatest rate of increase is in the direction and the greatest rate of decrease is in the direction of – • Let : . Suppose has continuous partial derivatives and orthogonal to the level curve • and returns a vector in A normal line to a surface is that passes through . The gradient vector field of a function : points in • , 0. Then is the function : . It takes . passing through is the line passing through that is perpendicular to the tangent plane. • • , If and both exist and and , etc are both continuous at then . • , , , , , , , , , • Taylor’s Theorem. Let : neighbourhood on the line from , • , . If has continuous second partial derivatives in some of , then for all points to where such that , in that neighbourhood, there exists a point , , , , , If the hypothesis for Taylor’s Theorem is satisfied, then in the neighbourhood exists an such that , for all , . there ...
View Full Document

This note was uploaded on 03/17/2012 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

Ask a homework question - tutors are online