linear approximation please help as soon as possible?

Question

Linear approximation?
(a)Find the linear approximation of the function at cube root(1+x) at a=0

I found the linear approximation but I dont know what to do next please show me

the linear approximation is ( 1+ x / 3 )

(b)Use the linear approximation from part (a) to approximate cube root(0.96)

(c)Use the linear approximation from part (a) to approximate cube root(1.07)

Answer 1

(a)
f(x) = cbrt(1 + x)

f'(x) = (1/3)(1 + x)^(-2/3)

f(0) = 1
f'(0) = (1/3)

Linear Approximation ---> y = 1 + (x/3)
as you got.

(b)
cbrt(.96) = cbrt( 1 - .04)
---> y = 1 + (-.04/3) = 0.9867

(c)
cbrt(1.07) = cbrt( 1 + .07)
---> y = 1 + (.07/3) = 1.0233

Answer 2

(a) is correct. Therefore:
(b) approximation cube root(1 + x) = | 1 + x = 0.96 => x = - 0.04 |
= 1 - 0.04/3 = 0.986667, while the exact root = 0.986485 (0.02 % error).
(c) 1 + x = 1.07 => x = 0.07 therefore:
1 + 0.07/3 = 1.023333 while the root is 1.022809 (0.05 % error).

Answer 3

I assume you have y = (1+x)^(1/3) and you are approximating with z = 1+x/3

(a) at x=0, z = 1
(b) at x = -0.04, z = 1 - 0.04/3 = .98667
(c) at x = +0.07, z = 1 + 0.07/3 = 1.02333

Answer 4

I think maybe your asking people to evaluate an expression containing no 'a' at a=0 has confused them ( it certainly put me off).
Maybe you could expand on what you really want to do?