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Unformatted text preview: Math 251 Section 4: Review for Test4 Test4 covers Section11.4 through Section12.2 (except for Section11.8). Here is the highlight of each section. Section11.4 Find the equation of the tangent plane to a given surface (the formula will be given in the test); nd the dierential of a function. Section11.5 Find the partial derivatives by the Chain Rule; write out the Chain Rule for a given composite function. Section11.6 Find the directional derivative; nd the gradient of a given function and be aware of that f gives the direction along which the maximum rate of change occurs; nd the maximum rate of change i.e.  f . Section11.7 Find the local maximum and minimum values and saddle points of a function (The Second Derivative Test will be given.); nd extreme values for practical problems (Example 4, 5). Section12.1 Evaluate iterated integrals; evaluate double integrals over rectangle by iterated integrals. Section12.2 Evaluate double integrals over general regions (type I and type II). Here is a list formulas and facts that you need to memorize. (a) The dierential of z = f (x, y ) is dz = fx (x, y )dx + fy (x, y )dy . (b) The gradient of f (x, y ) is
f = fx , fy . (c) The directional derivative of f in the direction of a unit vector u = a, b is
Du f = f · u = fx a + fy b.
1 (d) A point (a,b) is called a critical point of f (x, y ) if fx (a, b) = 0 and fy (a, b) = 0. Finding critical points of some function is usually the rst step to nd the extreme values. (e) Fubini's Theorem If f is continuous on the rectangle R = {(x, y )a ≤ x ≤ b, c ≤ y ≤ d}, then
¨ ˆ
d ˆ b ˆ bˆ
a d f (x, y )dA =
R c a f (x, y )dxdy =
c f (x, y )dydx. (f) If f is continous on a type I region D = {(x, y )a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x)} then
¨ ˆ bˆ
a g2 (x) f (x, y )dA =
D g1 (x) f (x, y )dydx. If f is continous on a type II region D = {(x, y )c ≤ y ≤ d, h1 (y ) ≤ x ≤ h2 (y )} then
¨ ˆ
d ˆ h2 (y ) f (x, y )dA =
D c h1 (y ) f (x, y )dxdy. 2 ...
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 Spring '09
 CGHEN
 Math, Calculus, Derivative, Evaluate double integrals

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