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Math Help - Slope Field and Differential Equations

  1. #1
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    Slope Field and Differential Equations

    Ok so I have a calc exam tmrw and I am having a hard time with slope fields. My professor told us that on the exam he is going to give us the slope field (graph) and we are going to have to pick one differential equation from a set as the one that matches the graph.
    SO my question is: how can you tell which slope field pretains to what graph? What are the rules in figuring that out?
    Please help ASAP!
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  2. #2
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    Well, what do you know about derivatives? If a derivative is positive, what is the original function doing? If a derivative is negative, what is the original function doing? Or if the derivative is 0?
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  3. #3
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    Quote Originally Posted by coolguy99 View Post
    Well, what do you know about derivatives? If a derivative is positive, what is the original function doing? If a derivative is negative, what is the original function doing? Or if the derivative is 0?
    A positive derivative means that the function is increasing
    A negative derivative means that the function is decreasing
    A zero derivative means that the function has some special behaviour at the given point. (max/min)

    for the second derivative:
    if it is positive then it is concave up
    if it negative then it is concave down

    Even when I consider the above rules...I cant seem to figure it out.
    For example: y'=1+y^2 the graph-answer given in the book is the same that my calculator gives but if I had to say why I chose that graph in words ( without saying "thats what the calculator had!") I would be lost.
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  4. #4
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    If they give you a slope field, then you just need to correlate the positive/negative values of the slope to their coordinates on the real equation. If the derivative is negative from x=1 to x=7, find an equation that is decreasing on those values. etc

    if you have a more specific question or example I could help you out better haha.
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  5. #5
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    Quote Originally Posted by coolguy99 View Post
    If they give you a slope field, then you just need to correlate the positive/negative values of the slope to their coordinates on the real equation. If the derivative is negative from x=1 to x=7, find an equation that is decreasing on those values. etc

    if you have a more specific question or example I could help you out better haha.
    lol ok.
    So lets say I have y'=sin(x) and I am able to tell what graph it corresponds to by using my calculator. Then if I had to explain why I chose that graph what would I say?

    Same thing for y'=1+y^2.

    I just want to get an idea of how you would reason your answer.
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  6. #6
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    You would explain the interval on which the derivative is positive/negative, then correlate it to the graph you chose by saying that it is increasing/decreasing on those same intervals, as well as noting relative maximums and relative minimums appearing at their respective places in relation to the derivative's cross-over points.
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  7. #7
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    Quote Originally Posted by coolguy99 View Post
    You would explain the interval on which the derivative is positive/negative, then correlate it to the graph you chose by saying that it is increasing/decreasing on those same intervals, as well as noting relative maximums and relative minimums appearing at their respective places in relation to the derivative's cross-over points.
    Ok I totally understand what you are saying about the y'=sin(x) graph. I checked the slope field and it was just like you described it. But when I tried to do the other equations once again I failed. lol I dont even know why such a small thing is confusing me, I have an A in the class!

    So it if was something like y'=4-y how would you explain it?
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