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Math Help - calculus project

  1. #1
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    Exclamation calculus project

    i have this project due and ive been stumped on it for awhile

    It goes like this The parabola y=x^2 has a circle with a radius 1 centered on the y axis and is tangent to the parabola.

    1. let (0,b) be the the center of the circle on the y axis that is tangent to the parabola. state the equation of the circle.

    answer: x^2-(y-b)^2=1

    2. Find the derivative of the parabola y=x^2 at any x.

    answer: f'(x)= 2x

    3. Find the derivative of the circle at any arbitrary point on the graph. use implicit differentiation.

    answer dy/dx= x/(y-b)

    This is where i need help!!


    4. since the derivative in step 2 and step 3 are equal where they are tangent, set the two derivatives equal and get an equation involving y and b.


    5. Since the two graphs intersect, solve for one variable in one equation and substitute into the other one to get a second equation involving y and b.(substitution method)


    6. Use the two equations from steps 4 and 5 to solve for b.


    7. Now write the equation of the circle.


    8. Find the points of intersection of the circle and parabola.


    So if anyone on here can please help me with 4-8 i would greatly appreciate it! im stumped and im just stressed because this is due tomorrow.
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  2. #2
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    Quote Originally Posted by zachmartin83 View Post
    i have this project due and ive been stumped on it for awhile

    It goes like this The parabola y=x^2 has a circle with a radius 1 centered on the y axis and is tangent to the parabola.

    1. let (0,b) be the the center of the circle on the y axis that is tangent to the parabola. state the equation of the circle.

    answer: x^2-(y-b)^2=1 Should be x^2+(y-b)^2=1

    2. Find the derivative of the parabola y=x^2 at any x.

    answer: f'(x)= 2x

    3. Find the derivative of the circle at any arbitrary point on the graph. use implicit differentiation.

    answer dy/dx= x/(y-b) dy/dx= -x/(y-b)

    This is where i need help!!


    4. since the derivative in step 2 and step 3 are equal where they are tangent, set the two derivatives equal and get an equation involving y and b. So put those two derivatives equal to each other: 2x = -x/(y-b). Notice how the x's cancel, to give you an equation involving only y and b.


    5. Since the two graphs intersect, solve for one variable in one equation and substitute into the other one to get a second equation involving y and b.(substitution method) If y=x^2 and x^2+(y-b)^2=1 then y+(y-b)^2=1.


    6. Use the two equations from steps 4 and 5 to solve for b. Now you do the rest!


    7. Now write the equation of the circle.


    8. Find the points of intersection of the circle and parabola.
    .
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