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Math Help - 2 Questions Trig Calc Help!

  1. #1
    Junior Member
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    2 Questions Trig Calc Help!

    f(x) = 2x-tan(x)

    When is f concave down? I know that when the second derivative is negative, the graph is concave down, so f''(x) = -2sin(x)/(cos(x))^3, right?
    When is that negative?

    Also,

    lim as x->0

    sin(5x)/x

    where 5x = theta

    and how do I use that result to find the derivative of sin(5x)?

    Any help appreciated.
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by amma0913 View Post
    f(x) = 2x-tan(x)

    When is f concave down? I know that when the second derivative is negative, the graph is concave down, so f''(x) = -2sin(x)/(cos(x))^3, right?
    When is that negative?

    Also,

    lim as x->0

    sin(5x)/x

    where 5x = theta

    and how do I use that result to find the derivative of sin(5x)?

    Any help appreciated.
    For the first one try to construct a table where you study the sign of each factor -- you're probably looking at this from 0 to 2pi, so make each row to represent the sign of the factors. Your bottom row should be the sign of the overall second derivative. you could also find the intervals or verify by graphing.

    For the second one:

    \lim_{x \to 0}\frac{\sin(5x)}{x} = 5 \lim_{x \to 0}\frac{sin(5x)}{5x}.

    Note that as x goes to zero can be replaced by 5x goes to zero, and \lim_{x \to 0} \frac{\sin u}{u} =1

    Hope this helps!
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