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F03_First_Midterm-K.Hare

# F03_First_Midterm-K.Hare - 14:37 FAX 510 642 9454.001 Math...

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Unformatted text preview: 03/02/2004 14:37 FAX 510 642 9454 .001 Math H53 Midterm Exam 1 October 3rd, 2003 Dr. K. G. Hare 1 a: (5 pts) Let f(t) :‘ (3H, 25‘). Find the length of f(t) between 0 g t g 2. b: (5 pts) Find all solutions to .24 : —4. 2: (5 pts) Let f be a harmonic function with continuous partial derivatives of any order. Further let fm(:r,y) = 2:3 + y. Find fy(m,y). 3: (5 pts) Let r09) = 4sin(36). Find the area of the curve in one loop 4 a: (5 pts) Consider the function f(sc,y) = \$4 + y4 + 9323,? ~ my + 3. Find the direction of steepest descent from the point (1, 2) b: (5 pts) Given 22+4x+4y2—24y=x2+22 Convert this to standard form. What sort of quadratic surface is this. (If you can’t remember the name, just draw a picture) 5: (5 pts) Let V be the vector Space of polynomials. Deﬁne the dot product (inner product) between two vectors f and g as fol f(93)9(95)d\$ Find 9(a) orthogonal to f(:1:) : m. 6: (5 pts) Let f : R2 —> R be a continuous functions, with continuous partial deriva— tives. Let {(331, 741)}?20 be a sequence of local maximums. Further let limp,OCJ 9:.- = c and liminmyi : 0!. Show that f has a critical point at (c, d). Bonus: (2 pts) Give an example to show that (12,03) can be a local minimum. ...
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