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Unformatted text preview: Spring 2007 M408D Test #3 McAdam
SHOW YOUR WORK
Points: 1) 40 2) 5 3)5 4)1S 5) 10 6) 15 7) 10
WRITE YOUR DISCUSSION SECTION TIME (OR ESP) ON THE COVER 1) Let f(x, y) = x + (y/X). (Note that X cannot be zero.) a) Note that f(2, Z) = 3‘ Sketch the level curve le, y) = 3.
Make your sketch at least 1/2 page in siZe. Show your scale.
Make one unit be at least as long as W . 0 1 b) Find Vf at the point (2, 2). Sketch that vector into the
sketch you made in part (a). Make the vector start at (2, ZI c) Find the directional derivative of f(X, y) at the point (2, Z),
in the direction going from the point (2, 2) towards the point (6, 5). 01) My pet ant is walking on the surface given by f(X, y). At
time t, she is on the surface directly over the point (ZtZ, 2t4).
Therefore, at time t = 1, she is directly over the point (2, 2). How fast is she going uphill at time t = 1? (Here, time is in minutes, distance is in inches.) 2) Let le, y) = xyZI Suppose that X = t3. Suppose that y: ylt)
is also some function of t, but you do not have a formula for it. Since Xlt), and ylt) are both functions of t, f can be thought of
as a function of t, as well. D. f
1 Suppose when t: 2, you know that (Z) = 176 and y(2) = 2. Q. . dv . dY
- = -— = ?
Find dt when t 2. That 18, Cit(2) . 3) I am thinking of a function f(X, y). I tell you that vf(l, 2) = (4, S), and that f(1, Z) = 3. Therefore,
the point (1, 2) is on the level curve flx, y) = 3. That level
curve has a tangent line at (1, 2). Find the slope of that
tangent line. 3 x
4) /[XY2 clde = ? I 0 5) Ixyzcosmzy) dX = ? 6) Suppose you wanted to integrate some function f(X, y) over
the shaded region shown below. Set this integral up in the dxdy order. 2 7) Suppose you wanted to integrate XV over the shaded region shown here. Set—up 1., the integral, using polar coordinates. (Do not calculate the final answer. just set the integral up) awn/t“ 80/0" ...
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