What is lim h->0 ((8 (1/2+h)^8) - 8 (1/2)^8)/h ?
or,
they might want you to recognize that this is the
First Principles approach to finding the derivative of
y = 8x^8 at x = 1/2
dy/dx = 64x^7 = 64(1/2)^7 = 1/2
assuming that they are at the stage when they know how to differentiate 8x^8
you could just use L'Hopital's rule
lim h->0 ((8 (1/2+h)^8) - 8 (1/2)^8)/h
= lim 64(1/2 + h)^7 / 1 , h ---> 0
= 64(1/2)^7
= 64/128
= 1/2
or, use the binomial expansion ...
(8 (1/2+h)^8) - 8 (1/2)^8)
= 8[ 1/2+h)^8) - (1/2)^8 ]
= 8[ (1/2)^8 + 8(1/2)^7 h + ... + h^8 - (1/2)^8 ]
= 8[ 8(1/2)^7 h + ... + h^8 ]
= 8[ h(8(1/2)^7 + ... + h^7 ]
then lim h->0 ((8 (1/2+h)^8) - 8 (1/2)^8)/h
= lim h --> 0 ( 8[ h(8(1/2)^7 + ... + h^7 ] )/h
= lim h --> 0 ( 8[ (8(1/2)^7 + ... + h^7 ] )
= 8[8(1/128)]
= 64/128
= 1/2
I expect the second solution is the one intended here. It looks like a problem before calculus.
Well, well, well, it seems like you've stumbled upon a calculus merry-go-round! Let's hop on and take a spin. So, we have this fun-looking expression:
lim(h → 0) [(8(1/2 + h)^8 - 8(1/2)^8)/h]
Now, to tackle this beast, let's break it down into smaller chunks and have some mathematical amusement along the way.
Step 1: Distribute those 8s on the top, because why not?
lim(h → 0) (8(1/2 + h)^8 - 8(1/2)^8)/h
= lim(h → 0) (8(1/2)^8 + 8(8C1(1/2)^7h + 8C2(1/2)^6h^2 + ... + 8C8h^8) - 8(1/2)^8)/h
Step 2: Cancel out those twinkling 1/2^8 terms, because they're feeling a bit redundant.
= lim(h → 0) (8(8C1(1/2)^7h + 8C2(1/2)^6h^2 + ... + 8C8h^8))/h
Step 3: Alright, party people, let's simplify things further. How about dividing each term by h, because that's our cool goal here?
= lim(h → 0) (8C1(1/2)^7h + 8C2(1/2)^6h^2 + ... + 8C8h^8)
Step 4: Ah, the final exhilarating step. As h approaches 0, all those h^2, h^3, and h^4 terms vanish into thin air, leaving us with only the first term.
= 8C1(1/2)^7
Now, let's calculate that sweet little expression: 8C1 is just 8, and (1/2)^7 is equal to 1/128. Let's put on our virtual calculator hat and crunch those numbers.
8 * (1/2)^7 = 8 * 1/128 = 1/16
So, my math-loving friend, the answer is 1/16. Isn't calculus a roller coaster of laughter and delight? Enjoy your newfound knowledge, and remember, math can be amusing too!
To find the limit of the given expression, we can simplify it step by step. Let's go through the process:
Step 1: Expand the numerator
The numerator is in the form a^8 - b^8, which can be factored as (a - b)(a^7 + a^6b + a^5b^2 + ... + ab^6 + b^7). Let's apply this to our expression:
((8 (1/2+h)^8) - 8 (1/2)^8)
= (8 (1/2+h) - 8 (1/2))((1/2+h)^7 + (1/2+h)^6(1/2) + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)^7)
Step 2: Simplify the numerator
We can now simplify the numerator by distributing the 8 across the terms:
((8 (1/2+h) - 8 (1/2))((1/2+h)^7 + (1/2+h)^6(1/2) + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)^7)
= (4 + 4h)(1/2+h)^7 + (1/2+h)^6 + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)^7
Step 3: Cancel out common factors
Now, we can cancel out common factors in the numerator:
= (4 + 4h)(1/2+h)^7 + (1/2+h)^6(1/2 + 1/2) + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)^7
= (4 + 4h)(1/2+h)^7 + 2(1/2+h)^6 + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)^7
Step 4: Simplify the expression
Let's further simplify the expression by removing the common factor of (1/2+h)^6:
= (4 + 4h)(1/2+h)^7 + 2(1/2+h)^6 + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)^7
= (4 + 4h + 2)(1/2+h)^6 + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)^7
Step 5: Simplify the expression further
We now have a common factor of (1/2+h)^6 in all terms except the first and last. Let's factor it out:
= [(4 + 4h + 2)(1/2+h) + (1/2+h)(1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)](1/2+h)^6
Step 6: Cancel out the common factor of (1/2+h)
Now, we can cancel out the common factor of (1/2+h) in the numerator:
= [(4 + 4h + 2) + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)](1/2+h)^6
Step 7: Simplify the expression further
Simplifying the expression, we get:
= [6 + (1/2+h)^5(1/2)^2 + ... + (1/2)(1/2+h)^6 + (1/2+h)](1/2+h)^6
Step 8: Take the limit as h approaches 0
Now, we can evaluate the limit as h approaches 0. Substituting h = 0 into the expression, we get:
= [6 + (1/2)^5(1/2)^2 + ... + (1/2)(1/2)^6 + (1/2)](1/2)^6
= [6 + (1/2)^7 + (1/2)^8 + (1/2)](1/2)^6
= [6 + 1/128 + 1/256 + 1/2](1/2)^6
= [6 + 9/256 + 1/2](1/2)^6
= [839/256](1/64)
= 839/16384
Therefore, the limit of the given expression as h approaches 0 is 839/16384.