Use linear approximation, i.e. the tangent line, to approximate 999^4 as follows:
Let f(x)=x^4. The equation of the tangent line to f(x) at x=10^3 is best written in the form y=f(a)+f'(a)*(x−a) where a=10^3, f(a)=(10^3)^4, and f'(a)=4(10^3)^3.
Using this, we find our approximation: 999^4 approximately equals? ______
I'm stuck at the last part and can't figure it out. Can someone walk me through it without giving me the answer?
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Straight substitution of your formula gives:
compare with exact value of 999^4
OOOHHH!! So I plug 999^4 into x in the equation!
You probably made a miscalculation for f'(a)=4(10^3)^3 with a ^3 too many.
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The only time you use 999 is (999-1000).
You do not need 999 for f(a) nor f'(a), since a=1000.
Sorry, I had a typo above for the equation, but the calculation that followed is correct.
Why do I use 999 instead of 999^4 at the end of the equation?
I did not, I put in 999^4 as x in 1000^4 +4(1000^3)(x-1000) and got the wrong answer. I put in 999 for x and got the right answer.
I think it's because 999 is x and f(x)=x^4, which is where the rest of my equations come from.
I put that in and the program counted it as wrong, make 999^4 just 999 and it counts it right.
Dude, I don't know, I copied and pasted the question and put in the correct symbols where they needed to be. I was surprised I made it to the end without help. If the program says 999 is x, instead of 999^4, and I get the problem right I won't argue. I'm sure it was just the wording of the problem that threw us off.
Sorry I was offline.
(999-1000) represents delta-X in the differential, that's why it is not raised to the fourth power.
The equation that you correctly stated was:
where a=1000, (x-a) = dx, so the linear approximation becomes:
which geometrically is shown in the figure below:|dw:1326861483536:dw|