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Math Help - Curve Analysis Question

  1. #1
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    Curve Analysis Question

    This is a curve sketching/analysis question I have with a graph involving e^x because I'm unsure about how the graph of this changes, etc.



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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by SportfreundeKeaneKent View Post
    This is a curve sketching/analysis question I have with a graph involving e^x because I'm unsure about how the graph of this changes, etc.



    Link to the image in case you can't see it:
    http://img243.imageshack.us/img243/3343/1812ge5.png
    What do you mean when you say "Fully analyze the graph"? Are you looking for the critical points, such as maximums/minimums, points of inflection, increasing/decreasing slopes, intervals of upward/downward concavity?

    Question 12 is clear, so I'll do that for now until you specify what you need for 10.

    h(x) = sqrt(lnx)
    h'(x) = 1/2*1/sqrt(lnx)*1/x = 1/(2x*lnx)
    h'(e) = 1/(2*e*lne) = 1/(2e)
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  3. #3
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    I got my teacher to tell me what fully analyze means.

    a. find symmetry (if f(-x) = f(x) or -f(x)
    b. find intercepts
    c. find intervals of increase or decrease
    d. find inflection points
    e. find concavity
    f. sketch the curve
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by SportfreundeKeaneKent View Post
    I got my teacher to tell me what fully analyze means.

    a. find symmetry (if f(-x) = f(x) or -f(x)
    b. find intercepts
    c. find intervals of increase or decrease
    d. find inflection points
    e. find concavity
    f. sketch the curve
    I leave "fully analyzing" the function to you, I'll go to the meat of the matter.

    We have the largest rectangle when the area of the rectangle is largest, so let's maximize the area.
    What is our formula for the area? length times width of course. so we have,

    A = 2x*e^(-2x^2)
    => A' = 2e^(-2x^2) + -8x^2*e^(-2x^2)

    For max A, set A' = 0
    => 2e^(-2x^2) + -8x^2*e^(-2x^2) = 0
    => 2e^(-2x^2)(1 - 4x^2) = 0
    => 2e^(-2x^2)(1 + 2x)(1 - 2x) = 0
    => 2e^(-2x^2) = 0 ......impossible
    or (1 + 2x) = 0 => x = -1/2
    or (1 - 2x) = 0 => x = 1/2

    So we have the largest area for the points (-1/2,0) and (1/2 , 0)

    (Do you need help with the "fully analyzing" part?)
    Attached Thumbnails Attached Thumbnails Curve Analysis Question-largestrec.jpg  
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Ah, i'm kinda bored. let's see how far we get with these guys.

    Quote Originally Posted by SportfreundeKeaneKent View Post
    I got my teacher to tell me what fully analyze means.

    a. find symmetry (if f(-x) = f(x) or -f(x)
    let y = f(x) = e^(-2x^2)

    now f(-x) = e^(-2(-x)^2) = e^(-2x^2) = f(x)

    since f(x) = f(-x), y is symmetric about the y-axis

    b. find intercepts
    For x-intercept, set y = 0
    => e^(-2x^2) = 0
    no solution, thus there is no x-intercept

    For y-intercept, set x = 0
    => y = e^(-2(0)^2) = e^0 = 1
    so we have one y-intercept, at y = 1

    c. find intervals of increase or decrease
    f(x) is increasing in an interval if f'(x)>= 0 for all x in that interval
    f(x) is decreasing in an interval if f'(x)<= 0 for all x in that interval

    now f'(x) = -4x*e^(-2x^2)

    e^(-2x^2) is always positive, so the sign of f'(x) depends on the -4x

    -4x > 0 if x < 0 .......so the function is increasing when this happens
    -4x < 0 if x > 0 .......so the function is decreasing when this happens
    note that where -4x = 0, f'(x) = 0 and so we have a critical point. further analysis will show us that this is a local maximum.

    so f(x) is increasing on (-infinity, 0) and decreasing on (0,infinity) it is at rest at x = 0

    d. find inflection points
    we have the possibility of inflection points where the second derivative = 0

    now f''(x) = 16x^2 *e^(-2x^2)
    f''(x) = 0 when x = 0 .......so this is a possible inflection point. let's make sure. if the sign of the derivative on both sides of the point is the same, it is an inflection point, otherwise, it is a max or min. take two numbers, one to the right of x = 0 and one to the left. say,
    x1 = -1
    x2 = 1

    f ' (-1) > 0
    f ' (1) < 0

    since we are increasing on the left and decreasing on the right, it means we have a maximum, not an inflection point. there are no inflection points, since the only possibility turned out to be a maximum.

    e. find concavity
    as you saw, there is one critical point, at x = 0, it is a maximum and hence we have concave down. so the function is concave down everywhere, that is on the interval (-infinity, infinity)

    f. sketch the curve
    i did that when i was solving the largest rectangle problem, see the post above (would you know how to draw a rough sketch if you had to do it by hand?)
    Last edited by Jhevon; April 24th 2007 at 02:48 PM.
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