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frx

I have the function \[f(x)=\frac{ \sin(x \pi ) }{ \sin(2x \pi) }\] which is visually presented in the attached file. I'm supposed, only by looking at the graph, to decide * Where in the interval the function is discontinuous? * Where does the curve have a horizontal tangent? * Where does the function have local maximum, local minimum? I can't see these things and would hence appreciate some guidance.

  • one year ago
  • one year ago

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  1. frx
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    • one year ago
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  2. A_clan
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    (1) By definition, An infinite discontinuity occurs when there is a vertical asymptote at the given x value. So, your x value will be ....

    • one year ago
  3. frx
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    There are multiple x values then, x=-2.5; x=-1.5; x=-0.5; x=0.5; x=1.5;x=2.5

    • one year ago
  4. frx
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    in the interval -3 to 3

    • one year ago
  5. A_clan
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    So, you will have a range of values and not just one value in the solution

    • one year ago
  6. A_clan
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    x={-2.5,-1.5,-0.5,0.5,1.5,2.5}

    • one year ago
  7. frx
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    I get it, thanks! Then how about a horizontal tangent, don't I have to calculate the lim as f(x) goes to infinity or is it possible to see it?

    • one year ago
  8. A_clan
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    It is possible to see it

    • one year ago
  9. A_clan
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    Is there any point on the curve where , on drawing a straight line, you get a horizontal line ?

    • one year ago
  10. frx
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    I can't see it, no. The x-axis itself?

    • one year ago
  11. A_clan
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    |dw:1365268526951:dw| This is one example

    • one year ago
  12. frx
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    Are there two horizontal tangents then? Going through the points of the curvatures?

    • one year ago
  13. A_clan
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    No, every point of curvature has its own tangent

    • one year ago
  14. frx
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    So there are seven horizontal lines which are tangents but the question is asked in singular which should indicate that there where only one answer?

    • one year ago
  15. A_clan
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    My experience is, In mathematics, the grammar of the question may not always indicate the quantity of solution.

    • one year ago
  16. frx
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    So the solutionset is {-3,-2,-1,0,1,2,3} then ?

    • one year ago
  17. A_clan
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    yes, that looks right

    • one year ago
  18. frx
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    Great! What about the local maximum and minimum, there seems to be muliple max and mins too?

    • one year ago
  19. A_clan
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    The solution to last part is situational. Local maximum and minimum may change depending upon the interval you choose for calculating them.

    • one year ago
  20. frx
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    The interval is -3 to 3, like the graph shows

    • one year ago
  21. A_clan
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    We say 'a' is local maximum if the height of the function at 'a' is greater than (or equal to) the height anywhere else in that interval.

    • one year ago
  22. A_clan
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    In simple words, we can say, local maxima is the maximum height in certain part of the graph but if we consider the whole graph ,one of these has to be the 'global maximum'

    • one year ago
  23. A_clan
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    |dw:1365269863585:dw|

    • one year ago
  24. A_clan
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    A,B and C are the local maxima but out of these, B is the global maxima

    • one year ago
  25. frx
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    Sure i understand , so the local maximas are at -3, -1, 1 and 3 and the local mins are -2,0,2

    • one year ago
  26. A_clan
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    That seems right

    • one year ago
  27. frx
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    Should I include the endpoints -3 and 3 according to WA they are not included http://www.wolframalpha.com/input/?i=Local+max+sin%28x+pi%29%2Fsin%282x+pi%29+from+-3+to+3

    • one year ago
  28. A_clan
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    It seems the local maxima and local minima are left out at the boundary because we do not have the values of y from both left side and right side,when x is approaching -3 or 3 .

    • one year ago
  29. A_clan
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    Then correct solution should be -1 and 1

    • one year ago
  30. frx
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    Ok, that makes sense :)

    • one year ago
  31. frx
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    Thanks you so much for your help, really appreciate it!

    • one year ago
  32. A_clan
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    no problem

    • one year ago
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