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Unformatted text preview: MATH 138  FINAL EXAM. Spring Term 2006 Page 2 of 15 1. Evaluate the following integrals. a) f 23 cos z2dz b) f xexdz oo 1
c) f0 5511:: MATH 138 _ FINAL EXAM. Spring Term 2006 Page 3 of 15 2. Let S be the solid obtained by rotating the shaded region around the x axis. 3.) Indicate how you would “slice” the solid to ﬁnd its volume by indicating the shape
and volume of a generic slice. b) Using your answer to part a), write down the integral which gives the volume of S. 0) Evaluate the integral from part b). MATH 138 — FINAL EXAM. Spring Term 2006 Page 4 of 15 3. Solve the initial value problem y' = ycosz; y(0) = 1. MATH 138 — FINAL EXAM. Spring Term 2006 Page 5 of 15 4. Let P(t) be the population of a. species at time t, and suppose that P(t) is modeled by
the following differential equation P’ = 2P(P — 100)(500 — P). 9.) Sketch the family of general solutions to this differential equation‘ b) Identify the equilibrium solutions and classify these as being either stable, semistable
or unstable. 0) Describe, in words, the eventual fate of the species starting with an initial population
of P(O) = 50. d) Describe, in words, the eventual fate of the species starting with an initial population
of P(0) = 110. MATH 138  FINAL EXAM. Spring Term 2006 Page 6 of 15 5. 3.) Deﬁne precisely, in terms of e and 6, what the following statement means lim f(x) = L I—m. b) Deﬁne precisely, in terms of the limit of a sequence, what the following statement means
00
2 an = L
n=l MATH 138  FINAL EXAM. Spring Term 2006 ' Page 7 of 15 6. For each series below, circle whether the series converges or civerges and state the conver—
gence test you are using to make the conclusion. Use the space below each question to show your work. ﬂ . I
a) °° 5 Circle one: converges diverges "=1 Convergence test used: on 1 . .
b) "=2 m Clrcle one: converges diverges Convergence test used: 00 7. . .
0) Circle one: converges diverges n=l e" Convergence test used: ...
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