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2051-quiz1
Course: MATH 2051, Fall 2009School: Missouri S&T
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x x F f r x x ut8p x g H xB yhw u v g 2 ift8p oA xv w uu r e "#SR a4s#1UrUq`Sph o4l5&G 52RWf6nafS lkihgf"d$ QGV7 6 # 4 4 64 m j F B e h 7 7 h st e h 4 F yxw y Cisftv u 8qifdbYa(SYXRB F w s w Xs5s r p h g ec 0 64 ` # W F F S 6 P F 0 F...
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 2051
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 309
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Missouri S&T - MATH - 309
Advanced Calculus I (Math 309)Fall 2002 Lecture NotesMartin BohnerVersion from December 11, 2002Author address: Department of Mathematics and Statistics, University of Missouri Rolla, Rolla, Missouri 65409-0020 E-mail address: bohner@umr.edu UR
Missouri S&T - MATH - 309
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Missouri S&T - MATH - 309
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Missouri S&T - MATH - 309
Problems #7, Math 309, Dr. M. Bohner.Oct 25, 2002. Due Nov 1, 1:30 pm.37. Where are the following f : R R continuous? x if x < 0 0 if x 1 f (x) = and f (x) = x2 if x 0 x if x > 1. 38. Prove that there exists an x > 0 with1 x+x2+ x2 2
Missouri S&T - MATH - 309
Problems #8, Math 309, Dr. M. Bohner.Nov 1, 2002. Due Nov 11, 1:30 pm.44. A function f : D R is called a Lipschitz function if there exists some c 0 such that |f (u) f (v)| c|u v| for all u, v D. Find a Lipschitz function that is not unifor
Michigan State University - LIB - 1925
August 15.1925UNITED STATES GOLF ASSOCIATION173greens when it is properly watered and otherwise cared for. The thickness of %-inch was found to be the minimum thickness practicable with bent sod.Fighting the June Beetle with CaddiesBy De'Vit
Missouri S&T - MATH - 309
Problems #10, Math 309, Dr. M. Bohner.Nov 18, 2002. Due Dec 2, 1:30 pm.59. Let f : [a, b] = I R be 3 times dierentiable and suppose y (a, b) with f (y) = 0 and f (y) = 0. Assume there is > 0 such that f (x) = 0 and | (x)| 1 f whenever x I w
Missouri S&T - MATH - 309
Final Exam, Math 309, Dr. M. Bohner, Dec 17, 2002. Each problem is worth 21 points. Only responses entered in the allocated space (no extra space allowed) for each problem will be graded. Present only the complete solution including all explanation (
Missouri S&T - MATH - 315
Problems #2, Math 315, Dr. M. Bohner.Jan 19, 2005. Due Jan 26, 2 pm. 12. Show that the Riemann-Stieltjes integral is unique, if it exists. 13. Prove that 14. Find 15. Find 16. Findb a b a b af dg is linear in f and g.f dg, where f is a constant
Missouri S&T - MATH - 315
Problems #3, Math 315, Dr. M. Bohner.Jan 26, 2005. Due Feb 4, 2 pm. 21. Let fn (x) = x x . Find the limit function f of {fn } on [0, 1] and n decide whether fn f . Does1 0nfn (x)dx 1 0f (x)dx hold?22. Let fn (x) = nx(1 x)n . Graph f1 , f2
Missouri S&T - MATH - 315
Problems #4, Math 315, Dr. M. Bohner.Feb 4, 2005. Due Feb 11, 2 pm. 33. Suppose that {fn } is a monotonic function sequence that converges pointwise on [a, b] to f . If f and every fn are continuous on [a, b], show that {fn } converges uniformly on [
Missouri S&T - MATH - 315
Problems #7, Math 315, Dr. M. Bohner.Mar 4, 2005. Due Mar 11, 2 pm. 60. For the following functions, calculate the integral between 0 and 1 once by the Riemann method and another time by the Lebesgue method (as illustrated in Example 4.1 in class): (
Missouri S&T - MATH - 315
Exam #2, Math 315, Dr. M. Bohner, Mar 14, 2005.Name: f (x) = . .A function f is said to be represented by a power series around a providedIf R is the radius of convergence of a power series, then it converges uniformly on f (n) (x) = k=1With
Missouri S&T - MATH - 315
Problems #8, Math 315, Dr. M. Bohner.Mar 16, 2005. Due Mar 25, 2 pm. 66. Prove Theorem 4.18 from the lecture notes. 67. Let A be a -algebra. Show: (a) A; (b) if Ak A for all k N, then (c) if A, B A, then A B A. 68. Let X = and be a set. Supp
Missouri S&T - MATH - 315
Exam #3, Math 315, Dr. M. Bohner, Apr 29, 2005.Name: |I|= . TheThe volume of an interval I = [a1 , b1 ] [aN , bN ] is dened by outer measure of any set A RN is dened by (A) = For A = {a} we have (A) =. (I) =. If I is an interval, the
Missouri S&T - MATH - 311
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Missouri S&T - MATH - 311
Problems #1, Math 311, Dr. M. Bohner.Jan 12, 2004. Due Jan 21, 1 pm.85. Prove Theorem 9.2 from the lecture notes, i.e., the set Rn is a vector space. 86. Prove the following statements, where x, y, z Rn . (a) x 0 and x = 0 i x = 0; (b) x = || x
Missouri S&T - MATH - 311
Exam #1, Math 311, Dr. M. Bohner, Feb 18, 2004.Name:A real vector space V equipped with an inner product , : V V R satisfying is called an space. A real vector space V equipped with a norm : V R satisfying is called a space. A set V equipped
Missouri S&T - MATH - 311
Problems #5, Math 311, Dr. M. Bohner.Feb 16, 2004. Due Feb 25, 1 pm.114. Determine whether following arcs are rectiable, and nd the arc lengths: the r cos t (a) x(t) = r sin t , t [0, 2] (r, h > 0 are xed); ht2 t , t (0, 1], x(0) = 0 .
Missouri S&T - MATH - 311
Problems #7, Math 311, Dr. M. Bohner.Mar 8, 2004. Due Mar 17, 1 pm.128. Use the denition of the derivative to nd f for the following functions: (a) f (x, y, z) =4x+y3z3 1+2yz, f : R3 R2 ;(b) f (x) = Ax b, where A is an m n-matrix and b R
Missouri S&T - MATH - 311
Problems #11, Math 311, Dr. M. Bohner.Apr 26, 2004. Due May 5, 1 pm.159. Using partitions with equal volume and norm tending to zero and assuming that the integral exists, nd S(f, Zn ), S(f, Zn ), and (a) f (x1 , x2 ) = x1 + x2 , I = [0, 1]2 ; (b
Missouri S&T - MATH - 309
Advanced Calculus I (Math 309)Fall 2005 Lecture NotesMartin BohnerVersion from December 4, 2005Author address:Department of Mathematics and Statistics, University of Missouri Rolla, Rolla, Missouri 65409-0020 E-mail address: bohner@umr.edu UR
Missouri S&T - MATH - 309
Exam #1, Math 309, Dr. M. Bohner, Oct 3, 2005. Let A and B be sets. Then AB =Name: and A B = .Now, let X and Y be sets, f : X Y a function, A X and B Y . Then f (A) = provided and f 1 (B) = , and it is called one-to-one if . Also, f is called
Missouri S&T - MATH - 309
Problems #5, Math 309, Dr. M. Bohner. Sep 28, 2005. Due Oct 10, 2 pm. 26. Give a direct /N -verication of the convergence of the following sequences: (a) an = (b) an = (c) an = (d) an =2 ; n 1 ; n+3 3 n+2 n+ 4;n2 . n2 +n n27. Let {an } be
Missouri S&T - MATH - 309
Problems #6, Math 309, Dr. M. Bohner. 35. Prove the following statements: (a) If an 0 as n , then 1 + (b) If x R, then lim 1 +n xn n an n nOct 14, 2005. Due Oct 21, 2 pm. 1 as n .n xn nexists (put e(x) := lim 1 +1 . e(x)for x R).(c)
Missouri S&T - MATH - 309
Problems #7, Math 309, Dr. M. Bohner.Oct 24, 2005. Due Oct 31, 2 pm. 6 if x > 1 cx1 36. Let c 1. Discuss the continuity of f , where f (x) = 2 2x + 1 if x 1. 37. Prove that there exists a positive solution of the equation1 x+x2= 2x x2 .38
Missouri S&T - MATH - 309
Problems #8, Math 309, Dr. M. Bohner.Nov 2, 2005. Due Nov 9, 2 pm.42. Find (if dierentiable) the derivatives of the following functions: (a) f (x) = x4 + 5x2 ; (b) f (x) = (c) f (x) = (d) f (x) =x2 2x+1 ; x2x2 +x;2 x ; x2 +1+1(e) f (x)
Missouri S&T - MATH - 309
Exam #2, Math 309, Dr. M. Bohner, Nov 14, 2005. The sequence {xn } is called bounded if then it also satises the If an R and bn R, then Condition: an + b n Name: . If {xn } converges, . , an bn , and (if )an bnas n , and if an bn n
Missouri S&T - MATH - 309
Problems #10, Math 309, Dr. M. Bohner. Nov 21, 2005. Due Nov 28, 2 pm. 55. Let a > 0, n N, and dene the partition Zn of [0, a] by xk = 0 k n. Furthermore, put k = xk1 , k = xk , k =n lim S(f, Zn , ), lim S(f, Zn , n ), and lim S(f, Zn , n
Missouri S&T - MATH - 309
Final Exam, Math 309, Dr. M. Bohner, Dec 16, 2005.Each problem is worth 20 points and entering your name correctly in the box is worth 10 points. Only responses entered in the allocated space (no extra space allowed) for each problem will be graded.
Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 204
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Missouri S&T - MATH - 325
Problems #2, Math 325, Dr. M. Bohner.Jan 27, 99. Due Feb 3, 9:30 am.10. Show that the PDE (ux )2 + (ut )2 = 0 is not linear. Find its general solution. 11. Consider aux + but = 0. Use the geometric method twice to nd the general solution (one tim
Missouri S&T - MATH - 325
Problems #4, Math 325, Dr. M. Bohner.Feb 10, 99. Due Feb 17, 9:30 am.24. Transform the equation uxx + 2uxt + utt = 2u into standard form. Solve the obtained standard PDE. Then use the transformation to obtain the solution of the original PDE. 25.
Missouri S&T - MATH - 325
Problems #5, Math 325, Dr. M. Bohner.Feb 17, 99. Due Mar 3, 9:30 am.31. Let u be a solution of the wave equation utt = c2 uxx . Show the following: (a) Let y I Then v with v(x, t) = u(x y, t) solves the wave equation. R. (b) ux , ut , and uxx s