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Math Help - Prove that the equation has at least one real root IN AN INTERVAL OF 0.01

  1. #1
    s3a
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    Prove that the equation has at least one real root IN AN INTERVAL OF 0.01

    Question:

    (a) Prove that the equation has at least one real root.
    (b) Use your calculator to find an interval of length 0.01 that contains a root.

    41. cos x = x^3

    If I were given the interval, I would know what to do but how do I know which interval of length 0.01 I should use??

    Any help would be greatly appreciated!
    Thanks in advance!
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  2. #2
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    Quote Originally Posted by s3a View Post
    Question:

    (a) Prove that the equation has at least one real root.
    (b) Use your calculator to find an interval of length 0.01 that contains a root.

    41. cos x = x^3

    If I were given the interval, I would know what to do but how do I know which interval of length 0.01 I should use??

    Any help would be greatly appreciated!
    Thanks in advance!
    Consider the function f(x) = \cos x - x^3. Note that f(0) > 0 and f(\pi) < 0. Therefore .... (why?)

    Now start an iterative process of subdividing the above interval and use your calculator to test each subdivision until you get one that satisfies the stated requirements.
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  3. #3
    s3a
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    So is a good answer:

    f(0) < 0 < f(0.85) ?
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    Quote Originally Posted by s3a View Post
    question:

    (a) prove that the equation has at least one real root.
    (b) use your calculator to find an interval of length 0.01 that contains a root.

    41. Cos x = x^3

    if i were given the interval, i would know what to do but how do i know which interval of length 0.01 i should use??

    Any help would be greatly appreciated!
    Thanks in advance!
    [.86,.87]
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  5. #5
    s3a
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    Do they expect us to plug in a million numbers manually until we find that restriction??
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    Quote Originally Posted by s3a View Post
    Do they expect us to plug in a million numbers manually until we find that restriction??
    Use your calculator to find an interval of length 0.01 that contains a root.
    \cos{x} = x^3 at x = 0.86547403...

    what's an interval of length 0.01 that includes that solution?
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  7. #7
    s3a
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    Did you plug in y = 0 to find x = 0.8655 ? If so, could you please explain the steps involved in solving for x with that rule?
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    Quote Originally Posted by s3a View Post
    Did you plug in y = 0 to find x = 0.8655 ? If so, could you please explain the steps involved in solving for x with that rule?
    I graphed the equation y = \cos{x} - x^3 and looked for the zero.
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  9. #9
    s3a
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    Do you have some sort of trick to do that fast because the way I do it, it takes so much time I would not be able to answer this on a test in time.
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  10. #10
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    Quote Originally Posted by s3a View Post
    Do you have some sort of trick to do that fast because the way I do it, it takes so much time I would not be able to answer this on a test in time.
    what kind of calculator are you using?
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