A few days ago
gatortheone

Can you please help me with this calculus problem? [derivatives]?

Well this practice problem in my AP Calculus practice book has a strange problem giving me a hard time. It says:

“Each limit represents the derivate of some function f at some number a. State such an f and a in each case”.

and the problem is…

lim h—> o of…

(1+h)^10 – 1

—————- [divided by]

h

I tried using the definition of a derivative and also the alternate definition of a derivative [aka: derivative at a point] and I didn’t quite get the answer from the answer key. So, how is this actually done?

Top 2 Answers
A few days ago
smile

Favorite Answer

you could use the binomial theorem to expand (1+h)^10 for any values of h <1 (1 + h)^ n = 1+ nh + [n(n-1)/2!]h^2 + [n(n-1)(n-2)/3!]h^3 .. + h^n So, lets use this expansion in this example (1+h)^10 -1 = (1+ 10h + [10.(10-1))/2!]h^2 + [10(10-1)(10-2)/3!]h^3 .. + h^10) -1 -> 10*h + [10(10-1)/2!]h^2 + [10(10-1)(10-2)/3!]h^3 ….. h^10

so for lim h->0, Lim ((1+h)^10 -1)/h = {10*h+[10(10-1)/2!]h^2..}/h = 10 + [10(10-1)/2!]h + [10(10-1)(10-2)/3!]h^2 … h^9

therefore when h->0, the limit of this function tends to 10

-> 10+[10(10-1)/2!]*0 + [10(10-1)(10-2)/3!]*0^2….0^9

-> 10

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A few days ago
slobberknocker_usa
Couple of questions: 1) is h approaching zero?

2) if h is approaching zero, is it approaching from the negative or from the positive?

If from the neg., your answer will be neg.. If from the pos., your answer will be pos.. I’m not sure if the answer is zero, neg. infinity or pos. infinity. I’d have to plug in some very small numbers to find out for sure.

Never mind. I’m not sure if I was any help. Sorry.

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