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Math 2224 Common Exam Spring 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 ”Course”. 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] To which of the following functions does this graph correspond?
x
y
z
x
- (x − y)^2 2) (x + y)^2 3) xy 4)
xy
[2] The table below gives function values of f as a function of x and y. The approximation
y x^0 2 4 6 0 3 7 14 20 27 2 11 15 20 25 40 4 15 26 30 39 46 6 20 35 40 50 50 8 25 40 48 50 50
of the partial derivative ∂f /∂y at the point (4, 4) equals
- 2 2) 3.25 3) 4.5 4) 5
[3] Let z = xy^2 sin(xy). The partial derivative with respect to x is
- y^2 cos(xy) 3) y^2 sin(xy) + x^2 y^2 cos(xy)
- y^3 cos(xy) 4) y^2 sin(xy) + xy^3 cos(xy)
[4] The directional derivative in the direction v = 〈 4 , − 3 〉 of f (x, y) = 3x^2 + xy at the point (1, 1) equals
- 1 2) -25 3) 5 4) 25
[5] Consider the function f (x, y) = x^2 − 4 x + y^2 on a closed bounded region. The region’s boundary includes the segment AB with A=(0, 0) and B=(4, 4). For finding absolute minima and maxima of f (x, y) you need to consider the following points on AB:
- (0, 0) and (4, 4) only. 3) (0, 0), (2, 2), and (4, 4).
- (0, 0) and (1, 1), and (4, 4). 4) No points on this boundary.
[6] Given the sequences an = 1/n and bn = 1 + 1/n^2. Which of the following statements is true?
- Both {an} and {bn} diverge. 3) {an} diverges and {bn} converges.
- {an} converges and {bn} diverges. 4) Both {an} and {bn} converge.
[7] For the series
∑^ ∞ n=
(−1)n^
n^2 6 + 10n^2
, which of the following statements is true:
- The series is convergent by the alternating series test.
- The series is divergent by the alternating series test.
- The series is convergent by the comparison test.
- The series is divergent by the test for divergence.
[8] The radius of convergence of
∑^ ∞
n=
(2x − 3)n n^2
equals
- 1/2 2) 1 3) 3/2 4) 2
[9] In the Taylor series for f (x) = e−^2 x^ centered at a = 1, the sum of the first 3 terms is:
- 1 − 2(x − 1) + 2(x − 1)^2 3) 1 − 2(x − 1) + 4(x − 1)^2
- 1 + 2(x − 1) + 4(x − 1)^2 4) None of (a), (b), or (c) is correct.
