Questions asked by martin23239

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• Let # f(x) # be the function # f(x) = 5^x - 5^{-x}. # Is # f(x) # even, odd, or neither ? Prove your result.
  
• Suppose # s(x) # and # c(x) # are 2 functions where: 1) # s'(x) = c(x) # and # c'(x) = -s(x); # 2) # s(0) = 0 # and # c(0) = 1. # What can you say about the quantity: # \qquad [ s(x) ]^2 + [ c(x) ]^2 # ?
  
• Let # f(x) = |x-1|. # 1) Verify that # f(x) # is neither even nor odd. 2) Can # f(x) # be written as the sum of an even function and an odd function ? a) If so, exhibit a solution. Are there more solutions ? b) If not, prove that it is impossible.
  
• A linear chain is made of 20 identical links. Each link can be made in 7 different colors. How many # physically # different chains are there?
  
• Is it possible for a finitely-generated group to contain subgroups that are not finitely-generated ? True or False. Prove your conclusion.
  
• Which is bigger: # ( 1 + \sqrt{2} )^{ 1 + \sqrt{2} + 10^{-9,000} } # or # ( 1 + \sqrt{2} + 10^{-9,000} )^{ 1 + \sqrt{2} } # ? If your calculator could actually handle this -- please put it away !! :)
  
• Can you simplify # cos(x)cos(2x)cos(4x)cos(8x)cos(16x) ... cos(2^n x) # ?,
  
• What can you say about the shape of the curve # f(x) = 7 cos( 1/3 x ) + \sqrt{19} sin( 1/3 x ) # ?
  
• #| ( (x-1)^4, (x-1)^3, (x-1)^2, (x-1), 1 ), ( (x-2)^4, (x-2)^3, (x-2)^2, (x-2), 1 ), ( (x-3)^4, (x-3)^3, (x-3)^2, (x-3), 1 ), ( (x-4)^4, (x-4)^3, (x-4)^2, (x-4), 1 ), ( (x-5)^4, (x-5)^3, (x-5)^2, (x-5), 1 ) | = # ?
  
• #| ( 1, 1, 1, 1, 1, 1, 1), ( 2^6, 2^5, 2^4, 2^3, 2^2, 2, 1 ), ( 3^6, 3^5, 3^4, 3^3, 3^2, 3, 1 ), ( 4^6, 4^5, 4^4, 4^3, 4^2, 4, 1 ), ( 5^6, 5^5, 5^4, 5^3, 5^2, 5, 1 ), ( 6^6, 6^5, 6^4, 6^3, 6^2, 6, 1 ), ( 7^6, 7^5, 7^4, 7^3, 7^2, 7, 1 ) | = #?
  
• Can you determine the determinant below ? (Would have put it here, but system wouldn't take it -- determinant just a tiny bit too big.) ...
  
• In my profile, how do I get to a page that shows all my Notifications at once, if possible ? So far, I've found the drop down menu for this, but this doesn't allow for prolonged viewing of these, or working with them as a whole. Thanks !!
  
• Are there polynomial functions whose graphs have: 11 points of inflection, but no max or min ?
  
• Can you calculate #\qquad \qquad e^{ ( ( ln(2), 1, 1, 1 ), ( 0, ln(2), 1, 1), ( 0, 0, ln(2), 1 ), ( 0, 0, 0, ln(2) ) ) } \qquad # ?
  
• Let #theta# be an angle where: #"1)" theta in "Quadrant III"# and #"2)" sin( theta ) = - 15/17#. What Quadrant does #12theta# belong to ? No Calculators !!
  
• Suppose #G # is a group where all non-identity elements are of order 2. Is #G# abelian ?
  
• # "Is there a group of order 48 in the set of" \ \ 3 xx 3 \ \ "matrices of integers ?" # # "If so, can you exhibit one ? If not, prove its impossibility." #
  
• Can you find the solutions of the equation: # \qquad qquad \qquad x^2 + i x - i \ = \ 0 \ "?" # Make sure to give your answers in standard complex form ( a + bi form).