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bmelyk

  • 2 years ago

Looking for someone to check my answer: Find an equation of the tangent line through the given point. x 2 y 3 + 15y = 34x, (3, 2)

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  1. bmelyk
    • 2 years ago
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    \[\frac{ 34-2x }{ 2y+15 }\]

  2. bmelyk
    • 2 years ago
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    thats the answer i got.

  3. bmelyk
    • 2 years ago
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    well that's what i had for y'

  4. asnaseer
    • 2 years ago
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    You first need to use implicit differentiation to find an expression for \(\displaystyle\frac{dy}{dx}\). Then substitute x=3 and y=2 into that expression to get the slope of the tangent line at that point. So then you will know the slope of the tangent line and you also know it passes through the point (3,2) - use this information to calculate the equation of the tangent line.

  5. bmelyk
    • 2 years ago
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    was my y' equation correct?

  6. asnaseer
    • 2 years ago
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    it doesn't look correct to me - can you please list your steps so that I can help spot where you may have made a mistake?

  7. bmelyk
    • 2 years ago
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    \[x ^{2}y ^{2}+15y=34x\] \[2x*2yy'+15y'=34\] \[y'(2y+15)=34-2x\] \[y'=\frac{ 34-2x }{ 2y+15 }\]

  8. asnaseer
    • 2 years ago
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    I thought you had \(y^3\) in the equation listed in your question?

  9. bmelyk
    • 2 years ago
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    lol so it is.

  10. bmelyk
    • 2 years ago
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    \[y'=\frac{ 34-2x }{ 3y ^{2}+15 }\]

  11. bmelyk
    • 2 years ago
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    did i have it right the second time (right above what you just wrote)

  12. asnaseer
    • 2 years ago
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    sorry I meant:\[\frac{d}{dx}(x^2y^3)=(x^2)\frac{d}{dx}(y^3)+y^3\frac{d}{dx}(x^2)\]

  13. asnaseer
    • 2 years ago
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    you haven't used the chain rule correctly

  14. asnaseer
    • 2 years ago
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    I mean "product rule"

  15. asnaseer
    • 2 years ago
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    have a look here: http://www.1728.org/chainrul.htm

  16. bmelyk
    • 2 years ago
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    \[x ^{2}*3y ^{2}y'+y ^{3}2x+15y'=34\]

  17. asnaseer
    • 2 years ago
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    that is correct :)

  18. bmelyk
    • 2 years ago
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    okay so now i isolate y'

  19. asnaseer
    • 2 years ago
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    exactly

  20. bmelyk
    • 2 years ago
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    \[y'=\frac{ 34-2xy ^{3} }{ x ^{2}3y ^{2}+15 }\]

  21. asnaseer
    • 2 years ago
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    yup - now follow the other steps that I had listed above.

  22. asnaseer
    • 2 years ago
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    NOTE: we don't usually write an expression as \(x^23y^2\) - it is better to write it as \(3x^2y^2\)

  23. bmelyk
    • 2 years ago
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    ok.

  24. asnaseer
    • 2 years ago
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    the general rule of thumb is to write in this order: 1. Constants first 2. Then letters in alphabetical order

  25. bmelyk
    • 2 years ago
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    okay so now i sub in my points right?

  26. asnaseer
    • 2 years ago
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    correct

  27. bmelyk
    • 2 years ago
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    so i had: -14/123

  28. asnaseer
    • 2 years ago
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    perfect! just a couple of more steps to go now :)

  29. bmelyk
    • 2 years ago
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    is the equation: \[y-2=-14/123x+42/123\]

  30. asnaseer
    • 2 years ago
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    yes - that looks correct. I wouldn't have separated the two constants here (the -2 and the 42/123)

  31. asnaseer
    • 2 years ago
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    you may also want to multiply both sides by 123 to remove the fractions from the final equation.

  32. bmelyk
    • 2 years ago
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    so how would you make it look?

  33. asnaseer
    • 2 years ago
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    ok, you got to this equation:\[y-2=-14/123x+42/123\]first add 2 to both sides to get:\[y=-14x/123 + 288/123\]then multiply both sides by 123 - what will you get then?

  34. bmelyk
    • 2 years ago
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    y123=-14x+288

  35. asnaseer
    • 2 years ago
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    correct - but again, remember to write constants first - so 123y instead of y123

  36. bmelyk
    • 2 years ago
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    that's not really how you write the equation of a line though.

  37. asnaseer
    • 2 years ago
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    so I would write the final equation as:\[123y=288-14x\]

  38. asnaseer
    • 2 years ago
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    it is still an equation of a line. you can write it in "standard form" as follows:\[14x+123y=288\]

  39. asnaseer
    • 2 years ago
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    maybe you are only used to seeing it in the form: \(y=mx +c\)

  40. asnaseer
    • 2 years ago
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    One last advice before I leave - you wrote one of the terms in your original equation as: -14/123x this can sometimes be confused for: \[-\frac{14}{123x}\]so it is usually better to write it as: -14x/123

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