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Unformatted text preview: Test #2 Math 241 — Form B —— 100 points total *No calculators, No cell phones, No PDAs *No questions during the test unless there is a typing error *Please begin each problem at the top of a new page in the blue book *Write your name and row on the outside of the blue book.
*Show all work. Mark out any work that you do not want graded. 1. Given the function f (x) = V9— x2
a) (6 pts) Setup only the deﬁnite integral to ﬁnd the average value of
function on the interval [-3, 3]. b) (10 pts) Setup only the deﬁnite integrals to ﬁnd the centroid (ff) of the lamina of uniform density p bound by f(x) and the x—axis.
2. (6 pts) A function y(x) satisﬁes the following differential equation ; d . . . .
E:- = xcos y 0 S y S 27: What are the equlllbrium solut1ons? 3. (10 pts) Given % = 2y — x with y(0) = 1, use Euler’s Method to 1
approximate y(l) using step size Ax = E 4. (28 pts) Solve the following differential equations a) (x2 +1)??? = xy y(0) = 1 (implicit solution)
b) 2—? = 3w +1 w(0) = 2 (explicit solution) 5. For the following 3 problems, do not integrate BUT be sure to completely set the
deﬁnite integrals. Draw a picture that clearly shows your set—up; label your
variables, label the direction of your x or y axis, label where x= or y= 0 is on your
ﬁgure , etc. a) (10 pts) A force of 10 lbs is required to stretch a spring from its natural
length of 1 ft. to 1.5 ft. How much work is done in stretching the spring
from its natural length to a length of 3ft. ? b) (10 pts) A gate in an irrigation canal is in the form of a trapezoid 4 ft wide
at the bottom, 8 ft. wide at the top, with height equal to 2 feet. It is placed
vertically in the canal with the water extending 10 feet above the top of the gate. The density of water is 62.4lb/ ﬁts. Find the hydrostatic force on
c) (10 pts) A tank full of oil (density is SOlb / ft3 ) has the shape of a paraboloid of revolution (that is, the shape is obtained by rotating a
parabola about a vertical axis). If the height is 6 ft and the radius at the
top is 6 ft, ﬁnd the work required to pump the oil over the rim of the tank. 6. (10 pts) Find the orthogonal trajectories of the family of curves x = kyz. Name
the conic section for each family. ...
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- Spring '08
- 1 W, 1 ft