Question

Use a linear approximation to approximate cubic root of 26 using the fact the fact that cubic root of 27 equals 3

Answer 1

we know that linear approximation of f(x) at a is given by,

---------------------------------------------------1)

The function we want to approximate is f(x) = x^1/3 and we want to estimate (26)^1/3

we need a value for a that is near to 26 and for which we will able to find f(a) that is we need a value for a for which we will able to find cubic root of a

As given we have to assume 27 as cubic root of 27 is 3

we have f(x) = x^1/3 and we have a = 27 hence we can say that,

we can write,

Hence,

we have a = 27, f(a) = 3 and f'(a) = 1/27 put them in equation 1) we can say that,

we have approximate x = 26 hence out x = 26 we can say that,

Hence we can write cubic roots of 26 is equals to 80/27 or in decimal 2.96296296296