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Definitions and Theorems - Midterm

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

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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 ...
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