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Math 2224 Common Exam Fall 2003
FORM A
Instructions: Please enter your NAME, your ID NUMBER, the FORM DESIGNATION LETTER and your CRN NUMBER on the op-scan sheet. The index number should be written in the upper right-hand box labeled โCourseโ. Darken the appropriate circles below the ID number and form designation letter. Use a No. 2 pencil; machine grading may ignore faintly marked circles. Mark your answers to the test questions in rows 1- 14 of the op-scan sheet. Your score on this part of the test will be the number of correct answers. You have one hour to complete this part of the final exam.
[1] The linearization of f (x, y, z) = x^2 + exy^ + yz^3 at (0, 1 , 1) is
- 3x + 2z โ y + 1 2) โ2 + x + y + 3z 3) โ 2 โ x + y + 3z 4) 2 โ x โ y โ 3 z = 0
[2] The limit lim (x,y)โ(0,0)
2 x^2 โ y^2 2 x^2 + y^2 is
- โ 1 4) Does not exist
[3] The direction in which f (x, y) = 2xy โ y^2 + 3x^2 has a maximum rate of change at (2, โ1) is
- 10~i + 6~j 2) โ 10 ~i โ 6 ~j 3) 10~i โ 6 ~j 4) 2
[4] The contour map for f (x, y) = x^2 โ 4 y^2 consists of
- Ellipses only 2) Circles, and a single point
- Hyperbolas only 4) Hyperbolas, and a union of lines
[5] Let f (x, y) = g(u, v), where u and v are functions of x and y, where
u(1, 1) = 2 v(1, 1) = โ 3
ux(1, 1) = 5 vx(1, 1) = 1 gu(2, โ3) = โ 2 gv(2, โ3) = 6
The partial derivative fx(1, 1) is
- โ 4 2) 4 3) 12 4) โ^15
[6] A lamina occupies the planar region bounded below by the x-axis and bounded above by the circle x^2 + y^2 = 4. Its density is ฯ(x, y) = x^2 + y^2. Its mass is
- 2ฯ 2) 8 ฯ 3
- 4ฯ 4) 64 ฯ 3
[12] Which of the following is the Maclaurin series for f (x) = eโx 2 ?
โ^ โ n=
x^2 n (2n)!
โ^ โ n=
(โ1)nx^2 n (2n)!
โ^ โ n=
(โ1)nx^2 n n!
โ^ โ n=
x^2 n n!
[13] For the power series
โ^ โ n=
2 n(4x โ 1)n, which of the following is the open
interval of convergence?
)
)
)
( โ
)
[14] The series
โ^ โ k=
ak has partial sums sn =
โ^ n k=
ak =
( 1 +
n
)n
. Which of the
following is true?
- The series
โ^ โ k=
ak diverges because, for every n, sn โฅ 1.
- The series
โ^ โ k=
ak converges to 1.
- The series
โ^ โ k=
ak converges to e.
- There is not enough information to determine whether the series
โ^ โ k=
ak converges or diverges.
