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?

Question

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?

mathmate · Answer

Straight substitution of your formula gives:
f(x)=x^4
f'(x)=4x^3
let a=1000, 
f(x)=f(a)+f'(x)(x-a)
=1000^4 +4(1000^3)(999-1000)
=996000000000
compare with exact value of 999^4
=996005996001

anonymous · Answer

OOOHHH!! So I plug 999^4 into x in the equation!

mathmate · Answer

You probably made a miscalculation for f'(a)=4(10^3)^3 with a ^3 too many.

mathmate · Answer

The only time you use 999 is (999-1000).
You do not need 999 for f(a) nor f'(a), since a=1000.

mathmate · Answer

Sorry, I had a typo above for the equation, but the calculation that followed is correct.
should read:
f(x)=f(a)+f'(a)(x-a)

anonymous · Answer

Why do I use 999 instead of 999^4 at the end of the equation?

anonymous · Answer

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.

anonymous · Answer

I think it's because 999 is x and f(x)=x^4, which is where the rest of my equations come from.

anonymous · Answer

I put that in and the program counted it as wrong, make 999^4 just 999 and it counts it right.

anonymous · Answer

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.

mathmate · Answer

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:
f(x)=f(a)+f'(a)(x-a)
where a=1000, (x-a) = dx, so the linear approximation becomes:
f(x)=f(a)+f'(a)[delta-x]
which geometrically is shown in the figure below:|dw:1326861483536:dw|