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Unformatted text preview: Math 16C Spring 1993 Final Exam 1. [7 points] Find and sketch the domain of f(x, y) = ‘/ x2 + y —— 9 . 2. [10 points] Let {(x, y) = cos(1 + xy2). Compute the following: (a) [4 points] fy (b) [6 points] fyx. 3. [15 points] Let {(x, y) = 5 + x2 — x2y _ y2 _ l y3 (a) [8 points] Find all critical points f.
(b) [7 points] Classify each critical point of f as a relative maximum, relative
minimum, or saddle point . 4. [10 points] Use Newton’s method with initial guess x1 = O to compute two successive approximations to the solution of the equation x3 + 3x = 1 . 5. [12 points] Determine whether the following sequences converge or diverge. Find the
limit of the convergent ones. (a) (b) an=(—1>ncos(§) 6. [18 points] Determine whether the following series converge or diverge. Clearly explain why.
(I) (D 1
(a) 2 n b 2
n::()50011 + jg ( ) n=1n§JH (c) I: (2n)! ( é )n 7. [12 points] Find the radius and interval of convergence of the power series
(I) —n
3 nEln+ 1 (x + 1)11 . (Do not check end points.) (I)
. . . 1
8. [10 pomts] Find the sum of the series )3 ——§ .
n=l 3H2“— Math 16C Final Exam S95 9. 10. 11. 12. 13. 14. [26 points] Evaluate the following double integrals. 2 J; 4 2
2 3  5
(a)ff y(x—y) dydx (b)ffxsm(l+y)dydx
0 0 0 J)?
[26 points] Solve the following differential equations. (3.) ex _xy’ + y = 2xy (b) xy’ — 2y = x In x, y(1) = 0
[12 points] John is supposed to learn 1,000 French vocabulary words, of which he
initially knows none. Suppose that he learns these words at a rate proportional to
the number of words that he has not yet learned, and that he learns 150 words in
the ﬁrst 5 days. How many days does it take him to learn half the words? (Let N be the number of words learned after t days.) [13 points] Use any method to ﬁnd the 3rd—degree Taylor polynomial centered at
1
= 1 for =m. 1
3
[13 points] Approximate the deﬁnite integral f ice"x dx using a 7th—degree 0
3
Taylor polynomial for f(x) = ire—x . Express your answer as a fraction. [12 points] Use the method of Lagrange multipliers to minimize 2 f(x:yaz)=x2+y +22 subjectto x—y+2z=3 and 3x+y_z=0. ...
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 Spring '10
 Kouba
 Calculus, Critical Point, Taylor Series

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