## 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

1. Algebraic!

$³√1695 - 8x^2$ ?

2. Algebraic!

seems unlikely... missing something?

3. iop360

$\lim_{h \rightarrow 0}\frac{ \sqrt[3]{1695 - 8x^2} -7 }{ h }$

4. Algebraic!

k

5. satellite73

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

the answer is $\frac{ -208 }{ 147 }$ i want to figure what to do to get it.

7. iop360

o i see

8. satellite73

have fun but you can do it use $$a=\sqrt[3]{1695-8x^2}$$ and $$b=7$$

9. iop360

thanks

10. iop360

$\lim_{h \rightarrow 0} \frac{ f(13 + h) - f(13) }{ h }$ this is the original question btw

11. iop360

if f(x) is $\sqrt[3]{1695 - 8x^2}$

12. satellite73

what is $$f(13)$$?

13. iop360

$\sqrt[3]{1695 - 8x^2}$ with 13 plugged in, which equals 7

14. satellite73

ok then you can start with $\frac{\sqrt[3]{1695-8(13+h)^2}-7}{h}$\]

15. iop360

yep

16. satellite73

you can leave it in this form, or you can write $\sqrt[3]{343-h^2-208h}-7$

17. satellite73

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

university online homework

19. iop360

mathXL

20. satellite73

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

i do know, im not sure if it would give me the answer im supposed to get

22. iop360

ill try taking the derivative

23. satellite73

this is the derivative of $\sqrt[3]{1695-8x^2}$ evaluated at $$x=13$$

24. satellite73

so if you can take the derivative, then plug in 13, you will get your answer

25. satellite73

just to finish quick derivative is $\frac{-16x}{3(1695-8x^2)^{\frac{2}{3}}}$

26. satellite73

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

ohh ok thanks!

28. iop360

yeah i think i got the same

29. satellite73

yw