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iop360

  • 2 years ago

find the limit as h approaches 0: f(13+h) - f(13) divided by h if f(x) = ³√1695 - 8x^2

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  1. Algebraic!
    • 2 years ago
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    \[ ³√1695 - 8x^2\] ?

  2. Algebraic!
    • 2 years ago
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    seems unlikely... missing something?

  3. iop360
    • 2 years ago
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    \[\lim_{h \rightarrow 0}\frac{ \sqrt[3]{1695 - 8x^2} -7 }{ h } \]

  4. Algebraic!
    • 2 years ago
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    k

  5. satellite73
    • 2 years ago
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    same gimmick as with a square root, rationalize the numerator, but this time instead of using \[(a-b)(a+b)=a^2-b^2\] you have to use \[(a-b)(a^2+ab+b^2)=a^3-b^3\] so it is a pain in the arse

  6. iop360
    • 2 years ago
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    the answer is \[\frac{ -208 }{ 147 }\] i want to figure what to do to get it.

  7. iop360
    • 2 years ago
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    o i see

  8. satellite73
    • 2 years ago
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    have fun but you can do it use \(a=\sqrt[3]{1695-8x^2}\) and \(b=7\)

  9. iop360
    • 2 years ago
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    thanks

  10. iop360
    • 2 years ago
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    \[\lim_{h \rightarrow 0} \frac{ f(13 + h) - f(13) }{ h }\] this is the original question btw

  11. iop360
    • 2 years ago
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    if f(x) is \[\sqrt[3]{1695 - 8x^2}\]

  12. satellite73
    • 2 years ago
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    what is \(f(13)\)?

  13. iop360
    • 2 years ago
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    \[\sqrt[3]{1695 - 8x^2} \] with 13 plugged in, which equals 7

  14. satellite73
    • 2 years ago
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    ok then you can start with \[\frac{\sqrt[3]{1695-8(13+h)^2}-7}{h}\]\]

  15. iop360
    • 2 years ago
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    yep

  16. satellite73
    • 2 years ago
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    you can leave it in this form, or you can write \[\sqrt[3]{343-h^2-208h}-7\]

  17. satellite73
    • 2 years ago
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    who gave you this problem? this really sucks unless you are supposed to use a shortcut, namely recognize this as the derivative and evaluate

  18. iop360
    • 2 years ago
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    university online homework

  19. iop360
    • 2 years ago
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    mathXL

  20. satellite73
    • 2 years ago
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    do you know how to take a derivative? because then it it would be not so hard but if you do not, then there is a ton of work to be done

  21. iop360
    • 2 years ago
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    i do know, im not sure if it would give me the answer im supposed to get

  22. iop360
    • 2 years ago
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    ill try taking the derivative

  23. satellite73
    • 2 years ago
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    this is the derivative of \[\sqrt[3]{1695-8x^2}\] evaluated at \(x=13\)

  24. satellite73
    • 2 years ago
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    so if you can take the derivative, then plug in 13, you will get your answer

  25. satellite73
    • 2 years ago
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    just to finish quick derivative is \[\frac{-16x}{3(1695-8x^2)^{\frac{2}{3}}}\]

  26. satellite73
    • 2 years ago
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    by the chain rule and power rule replace \(x\) by 13 and you should get your answer this is a much snappier way then doing it by hand

  27. iop360
    • 2 years ago
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    ohh ok thanks!

  28. iop360
    • 2 years ago
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    yeah i think i got the same

  29. satellite73
    • 2 years ago
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    yw

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