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 , …,
• 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 :
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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