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anonymous

  • one year ago

The slope of the tangent to a curve at any point (x, y) on the curve is -x/y. Find the equation of the curve if the point (2, −2) is on the curve.

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  1. anonymous
    • one year ago
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    not sure how to do this one

  2. ganeshie8
    • one year ago
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    what do you know about the relation between `slope of tangent line` and `first derivative` ?

  3. anonymous
    • one year ago
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    well the first derivative represents the slope of the tangent line on any point of the function you derive

  4. ganeshie8
    • one year ago
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    You're given that the `slope of tangent line` equals -x/y, so setup an equation using that info

  5. anonymous
    • one year ago
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    i thought that -x/y was the derivative

  6. ganeshie8
    • one year ago
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    Yes \[\dfrac{dy}{dx}=\dfrac{-x}{y}\]

  7. anonymous
    • one year ago
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    what do i solve for though ?

  8. ganeshie8
    • one year ago
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    thats the equation ^

  9. ganeshie8
    • one year ago
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    you need to solve the curve \(y\)

  10. anonymous
    • one year ago
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    so implicit differentiation ?

  11. ganeshie8
    • one year ago
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    An usual algebraic equation involves isolating a "variable", but a differential equation involves solving for a "curve"

  12. ganeshie8
    • one year ago
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    familiar with variable separation ?

  13. anonymous
    • one year ago
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    yep, give me a sec

  14. anonymous
    • one year ago
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    okay im getting \[\sqrt{C - x^2}\]

  15. anonymous
    • one year ago
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    is that right?

  16. ganeshie8
    • one year ago
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    looks partially correct, show me ur work

  17. anonymous
    • one year ago
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    |dw:1439277292623:dw|

  18. anonymous
    • one year ago
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    this drawing tool is dreadful

  19. anonymous
    • one year ago
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    then i isolated y

  20. ganeshie8
    • one year ago
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    leave it as \[y^2=-x^2+C\]

  21. anonymous
    • one year ago
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    |dw:1439277527370:dw|

  22. ganeshie8
    • one year ago
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    Now plugin the initial condition, \((2, -2)\), and solve \(C\)

  23. ganeshie8
    • one year ago
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    |dw:1439277677058:dw|

  24. anonymous
    • one year ago
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    |dw:1439277671100:dw|

  25. ganeshie8
    • one year ago
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    |dw:1439277771874:dw|

  26. anonymous
    • one year ago
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    oh woops , my mind was slipping

  27. anonymous
    • one year ago
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    okay so know we just plug that into |dw:1439277880677:dw| right?

  28. anonymous
    • one year ago
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    these area my answer choices by the way x + y = 0 x2 − y2 = −2 x2 + y2 = 16 x2 + y2 = 8 i assume its D

  29. ganeshie8
    • one year ago
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    Yes

  30. anonymous
    • one year ago
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    thanks for the help

  31. ganeshie8
    • one year ago
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    \[y^2=-x^2+C\] plugin \((2, -2)\) \[(-2)^2=-2^2+C\implies C = 8\] so the solution is \[y^2=-x^2+8\] which is same as \[x^2+y^2=8\]

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