- #1
JasonJo
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1) For all positive functions f and g of the real variable x, let ~ be the relation defined by:
f ~ g iff (if and only if) lim x --> (infinity) [tex]\frac{f(x)}{g(x)}[/tex] = 1. Which of the following is NOT a consequence of f~g?
A. f^2 ~ g^2
B. sqrt(f) ~ sqrt(g)
C. e^f ~ e^g (e as in the natural logarithm)
D. f + g ~ 2g
E. g ~f
So I am really curious why the relationship in C DOES NOT hold. Since
if e^f ~ e^g, then
lim x --> infinity [tex]\frac{e^f(x)}{e^g(x)}[/tex] = lim e^(f - g) = lim e^(0) since f and g have a common limit as x approaches infinity, and hence e^0 = 1.
2) What is the minimum value of the expression x+4z as a function defined on R^3, subject to the constraint that x^2 + y^2 + z^2 [tex]\leq[/tex] 2?
A. 0
B. -2
C. -sqrt(34)
D. -sqrt(35)
E. -5sqrt(2)
- For this one I tried using Lagrange multiplier method to solve it. I switched to an equality since we want the minimum value, and that will occur on the surface of the sphere of radius sqrt(2). So I get (1, 0, 4) = L(2x, 2x, 2z), but the numbers don't quite work out. I get something like -17*sqrt(2), not quite the right answer C, but close. I don't know why I'm not getting the right value.
3) For how many positive integers k does the ordinary decimal representation of the integer k! end in exactly 99 zeroes?
A. None
B. One
C. Four
D. Five
E. Twenty four
- I was a little confused by this one. I didn't quite know which way I was supposed to approach it. I tried working from the fact that 100! ends in 24 zeroes, but I didn't get very far.
Thanks
f ~ g iff (if and only if) lim x --> (infinity) [tex]\frac{f(x)}{g(x)}[/tex] = 1. Which of the following is NOT a consequence of f~g?
A. f^2 ~ g^2
B. sqrt(f) ~ sqrt(g)
C. e^f ~ e^g (e as in the natural logarithm)
D. f + g ~ 2g
E. g ~f
So I am really curious why the relationship in C DOES NOT hold. Since
if e^f ~ e^g, then
lim x --> infinity [tex]\frac{e^f(x)}{e^g(x)}[/tex] = lim e^(f - g) = lim e^(0) since f and g have a common limit as x approaches infinity, and hence e^0 = 1.
2) What is the minimum value of the expression x+4z as a function defined on R^3, subject to the constraint that x^2 + y^2 + z^2 [tex]\leq[/tex] 2?
A. 0
B. -2
C. -sqrt(34)
D. -sqrt(35)
E. -5sqrt(2)
- For this one I tried using Lagrange multiplier method to solve it. I switched to an equality since we want the minimum value, and that will occur on the surface of the sphere of radius sqrt(2). So I get (1, 0, 4) = L(2x, 2x, 2z), but the numbers don't quite work out. I get something like -17*sqrt(2), not quite the right answer C, but close. I don't know why I'm not getting the right value.
3) For how many positive integers k does the ordinary decimal representation of the integer k! end in exactly 99 zeroes?
A. None
B. One
C. Four
D. Five
E. Twenty four
- I was a little confused by this one. I didn't quite know which way I was supposed to approach it. I tried working from the fact that 100! ends in 24 zeroes, but I didn't get very far.
Thanks
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