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Unformatted text preview: 14.6-Directional Derivatives and the Gradient VectorThe definition of the directional derivative at (,)x0 yoin the direction of <a,b>o,=Dufx0 y0lim+,+- (,)h 0fxo ha y0 hb f xo yo hThe Gradient Vector of a function of x and y is Defined as =<, ,, >ffxx y fyx yThe directional derivative in the direction of a unit vector u = <a,b>o, =Dufx yo, = , * +, *fx y dot ufxx y a fyx y bIn a function of three variables, F(x,y,z) with a unit vector v=<a,b,c>o, =< , , >< , , > Dvfx yf x f y f z dot a b cNotice the directional derivative is a scalar not a vectorAlso we can determine the direction of maximum change by using the geometric interpretation of the dot product, = a dot babcos, look at pg946You can construct tangent planes using the gradient vector and a point on the plane. Take a look at page 948.Extra Exercises p950 #5, #7, #21, #39, #41, supero14.7 Maximum and Minimum valuesYou find critical points the same way you did in calc 1, now we just have two partial derivatives to set equal to zero instead of one. After we do that, we get two equations with two unknowns, so we can solve them to find out where on the function there is a critical point....
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This test prep was uploaded on 04/19/2008 for the course MATH 123 taught by Professor Cox during the Spring '08 term at SUNY Fredonia.
- Spring '08