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Maximum and Range of the Equation (Calc.)

  1. Sep 15, 2011 #1
    1. The problem statement, all variables and given/known data

    Hello,

    I ran into this problem in the middle of my physics homework:

    Using calculus, you can find a function’s maximum or minimum by differentiating and setting the result to zero. Do this for equation y = (u^2 / g)*sin(2x), differentiating with respect to x, and thus find the maximum range for x.

    u = initial velocity
    g= acceleration of gravity
    x = theta

    2. Relevant equations

    Possibly x = ut for range?

    3. The attempt at a solution

    It's been a several months since I've done this type of problem, but I tried to differentiate it through the quotient rule (and product rule) and got:

    (2*g*u^2*cos(2x) + 2*g*u*sin(2x) - u^2*sin(2x)) / g^2

    I tried to set this to zero in order to find the maximum:

    0 = (2*g*u^2*cos(2x) + 2*g*u*sin(2x) - u^2*sin(2x)) / g^2

    but I couldn't figure out what to do and I had a suspicion I was doing everything wrong.

    Can anyone point out my mistakes / what to do next?
    Thanks!
     
    Last edited: Sep 15, 2011
  2. jcsd
  3. Sep 15, 2011 #2

    gneill

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    Staff: Mentor

    It would appear that u and g are constants, so that (u2/g) is also a constant. So you only have to worry about differentiating the sin(2x).

    What's the derivative of sin(2x) with respect to x?
     
  4. Sep 16, 2011 #3
    Thanks for the input, treating the u and g variables as constants made the problem make more sense and I was able to solve it.

    y = range
    x = theta

    Solution (for reference):

    If I ignore the other variables and differentiate the equation, y = sin(2x), I get:

    y' = 2cos(2x)

    Then set it to 0 to find the critical point:

    0 = 2cos(2x)

    Divide by 2:

    0 = cos(2x)

    Take the inverse cosine of both sides:

    cos^-1(0) = 2x

    Divide both sides by 2:

    90 / 2 = x

    45 degrees = x

    Therefore the maximum range given by x (theta) is 45.
     
  5. Sep 16, 2011 #4

    SammyS

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    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    That gives the value of x that makes y a maximum.

    The question asks for the maximum range, which is the y value when x = 45 degrees.
     
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