By George W. Bluman
The e-book comprises the entire major difficulties for a primary 12 months calculus direction. engaged on those difficulties may dramatically assist you achieve excellent grades out of your calculus classse. (At least that occurred to me and my buddies who use it.) irrespective of you're a math, physics, biology, or economics or maybe chemistry significant, you will locate it super invaluable. It additionally comprises a few very demanding difficulties as much as the extent of the world's so much tricky undergraduate contest: The Putnam!
Read Online or Download Problem Book for First Year Calculus (Problem Books in Mathematics) PDF
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Extra info for Problem Book for First Year Calculus (Problem Books in Mathematics)
X lnx Jo dx. dx. 1)(eibX — 1) dx (a, b ER). 26. 27. 31. 33. q > 0). 32. 16a). prove that +o(1). With the help of the Stirling formula find the value of the Euler-Poisson integral dx. §2. 34. 35. k=O,1,2,3,4,5. 36. 37. 38. f°° ir(x)/(x3 — x) dx, where ir(x) is the number of prime numbers not exceeding x. §2. 7 it is required to compute the given integrals. 1. 2. — a) x)dx1 . . 3. 4. b) (e>O). dx. fR a) ffR2 lax + 0"2dx (a E R'1, fR l(x, > —1). lix — yll1 dy1 dy2 dy3 (x E R3). 5. dx dy du dv. 6.
And < +oo f(x) = cosnx g(x) E C([0, irj). Prove that the series cos nx sin nx can be integrated termwise on [0, irj. 20. a) b) < +oo if and Suppose that sin nx, = sinnx. Prove that (ni E N). Prove that for XE (0. it): > 0; —1. v. 21. is — 45 §6. 22. Let = IG = (the Gibbs phenomenon). 23. 24. Let 1(x) = = f(+0) dU> > 0. Prove that a (x,y€R). 25. 26. where A E (1, 3). Prove that I E sin but f Let 1(x) = Lips for /3 > Prove that I E Sin for any E (0, 1). Is it true that f E Lip1? 27. x). Prove that: a) I for any 1J; b) f is nondifferentiable at every point.
7. Prove that as p —, +oo: ir/2p; sin(x")dx a) b) f°°cos(x")dx—'l. 8. as e—'+O. 9. 10. 11. c,/e 0. Prove that f11°° ço(t)dt b) Isinxllcosxl dx (p > —1); d) j Suppose that the function I decreases (p> 1). to zero on [a, +oo). the dt =0. is continuous on R and has period T> 0, and fj function for A a. 12. Suppose that I decreases to zero on [a, +oo) , and I on R and has period T> 0. Prove that JA JA dt = f(t)dt + I + 0(1(A)), where = fj dt and I = f is nonnegative and monotone on [a, +oo), is continuous on R and has period T> 0, and C, = + f' ço(t)dt 0.