Use linear approximation to estimate the number 1.999^4

Question

Use linear approximation to estimate the number (1.999)⁴

 

That means:

a = 2

f(2) = 16

f(1.999) ≈ ?

 

 

16 + 8(2)³ * .001

f(1.999) ≈ 16.064

 

However, this is not the right answer according to my book (see below). Where did I go wrong?

 

#23, p255

ANSWER: 15.968

Answer 1

You know the function is x⁴.  You can assume this from the original function if f(2)=16 then it has to be x⁴

 

So f(a)=16

f'(a) = 4x³

f'(a)=4(2)³= 32

 

The formula is

 

f(a)+f'(a)(x-a)

 

16+32(x-2) You said 1.999 was what you were trying to evaluate so

 

16+32(1.999-2)

 

= 16 +32(-.001)

 

This gives me a value of 15.968