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How to interpret integrals of graphs

  1. Feb 7, 2007 #1


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    1. The problem statement, all variables and given/known data

    no specific question but generally they give you a graph of f(x) (with semicircles, lines, above/below x axis - could be anything) and say that g(x) is the integral from 0 to x of f(x).

    then they will ask things like:

    1) g(2)

    2) find relative min/max for g(x)

    3) find points of inflection of g(x) or where its concave up/down

    4) find g'(4)

    5) fine the equation of a line tangent to the graph of g at x=3

    2. Relevant equations


    3. The attempt at a solution

    1) g(2): ok this one is pretty easy. simply the area below the curve from 0 to whatever point is selected (in this case 2). just add all the areas above the x axis and subtract all the negative ones below it.

    2) find relative min/max for g(x): for this the only way i know how to is to logically see where the area is the largest (for maximums). in other words, as long as its above the x axis the area gets bigger and bigger. so i look for where it hits the x axis and those points are your possibilities for extrema.

    what else am i missing here? how do you justify your answer? is there a formula to use?

    3) find points of inflection of g(x) or where its concave up/down: i know points of inflection are where it changes concavity but how do you determine this? if g(x) is the integral of f(x) (the graph) then g'(x) should be f(x)? then g''(x) is f'(x)? so the points of inflection are where the slope changes from pos to neg or neg to pos?

    is that right?

    4) find g'(4): again since g'(x) is the same as f(x), if the question asks for g'(4) is that the same as f(4)? meaning i can simply look at the function value on the graph at x=4?

    5) fine the equation of a line tangent to the graph of g at x=3: this one im a little lost at. i know i need the slope of g at x=3 but other than that, im confused.

    again, there is no specific problem. im trying to learn the rules and problem solving strategies for these general types of problems. any help is appreciated. and am i on the right track with my explanations?
  2. jcsd
  3. Feb 7, 2007 #2
    Sounds to me that they're not necessarily asking you to use integrals, but asking you to use differentials, so what you in fact want is the differentials of the curve not the integration of that which lies under the curve. I could be wrong but it sounds very much like that, unless I'm misinterpreting the questions?
  4. Feb 7, 2007 #3

    Tom Mattson

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    It's actually quite simple. If you want relative extrema for g(x), you should be looking at the points on the x axis at which g'(x) is zero, right? But what is g'(x)? That's right, it's f(x). So g has an extremum wherever f has a root.


    To write down the equation of the tangent line, you'll need a point and a slope, right? OK, so the slope at x=3 is g'(3)=f(3). So just evaluate f at 3, and you've got the slope of the line. To get the point, plug x=3 into g itself.
  5. Feb 8, 2007 #4


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    The only way to do that is to integrate- unless these are simple enough geometric figures ("semicircles, lines" etc.) that you can use the fact that the integral is the area.

    The relative min/max will occur where the derivative is 0- and. by the "fundamental theorem of calculus", that is at f(x)= 0.

    Since this involves g"(x)= 0, you are looking for f'(x)= 0.

    Again, this is f(4).

    You will need to find the value of g(3) either by integrating or using geometric area. Of course, the slope of the tangent line is g'(3)= f(3).

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