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NathanJHW

  • one year ago

In this problem, you will investigate the family of functions h(x) = x + cos(ax) , where a is a positive constant such that 0 < a < 4.

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  1. NathanJHW
    • one year ago
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    A. Graph the curves y(x) = x + cos(¼x) and y(x) = x + cos(4x)

  2. NathanJHW
    • one year ago
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    B. For what values of a will h(x) have a relative maximum at x = 1?

  3. NathanJHW
    • one year ago
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    C. Which values of a make h(x) strictly decreasing? Justify your answer.

  4. NathanJHW
    • one year ago
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    @perl

  5. NathanJHW
    • one year ago
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    @phi

  6. NathanJHW
    • one year ago
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    I need help with A and C. Already got B.

  7. phi
    • one year ago
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    How far did you get ? Can you use Geogebra or other plotting tool?

  8. phi
    • one year ago
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    or do you have to do it by hand ?

  9. NathanJHW
    • one year ago
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    Didn't even think about using a graphing tool for part A. I'll do that now.

  10. NathanJHW
    • one year ago
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    Alright I got part A now, can you help me with C?

  11. NathanJHW
    • one year ago
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    I think we might need to use the work I did for part B so i'll post it.

  12. NathanJHW
    • one year ago
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    h'(x) = 1- a sin (ax); h'(1) = 0 h'(1)= 1 - a sin (a) = 0 1- a sin (a) = 0 1 = a sin a a is approx: +- 9.3 , +- 6.4, +- 2.8 , +- 1.1

  13. phi
    • one year ago
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    strictly decreasing means as x increases f(x) gets smaller.

  14. phi
    • one year ago
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    I don't see how to be decreasing when we are adding x

  15. NathanJHW
    • one year ago
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    That's what I thought too, but how should I justify?

  16. phi
    • one year ago
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    strictly decreasing means that at every x the slope of the function is never 0 or positive. i.e. the slope is always negative. f'(x) = 1 - a sin(ax) and because the sin(ax) will be negative for 1/2 of its period, the -a sin(ax) adds to 1 and that is clearly not negative.

  17. phi
    • one year ago
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    btw, how did you solve a sin a = 1 for a ? (and you should restrict your a's to the interval: a is a positive constant such that 0 < a < 4. )

  18. NathanJHW
    • one year ago
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    So then it would only be 1.1 and 2.8?

  19. NathanJHW
    • one year ago
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    and + not -

  20. phi
    • one year ago
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    yes. But how did you find those values?

  21. NathanJHW
    • one year ago
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    http://www.wolframalpha.com/input/?i=1%3D+a+sin%28a%29

  22. phi
    • one year ago
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    oh, because it looks hard to solve (without the wolf)

  23. NathanJHW
    • one year ago
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    exactly, that's why I used it

  24. NathanJHW
    • one year ago
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    Alright, well thanks for your help!

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