Find the linear approximation of the function
g(x) = fifth root(1 + x) at a = 0.
g(x)≈ 1+(x/5)
Correct: Your answer is correct.
Use it to approximate the numbers
fifth root (0.95) and fifth root (1.1)
(Round your answers to three decimal places.)
so, assuming that you have the linear function 1+x/5
root(0.95) = root(1-.05) ≈ 1+(-.05)/5 = 1-.01 = 0.99
root(1.1) = root(1+0.1) ≈ 1+0.1/5 = 1.02
Oh, I love playing with numbers! Let's use our linear approximation to approximate the fifth root of 0.95 and 1.1, shall we?
Using g(x) ≈ 1 + (x/5), when x = 0.95 - 0, we have:
g(0.95) ≈ 1 + (0.95/5) ≈ 1.19
And when x = 1.1 - 0, we have:
g(1.1) ≈ 1 + (1.1/5) ≈ 1.22
So, the approximate fifth root of 0.95 is 1.19, and the approximate fifth root of 1.1 is 1.22. Remember, these are just approximations, so handle them with care and don't let them take root in your heart too deeply!
To approximate the numbers fifth root(0.95) and fifth root(1.1) using the linear approximation of the function g(x) = 1 + (x/5) at a = 0, we substitute the values into the linear approximation equation.
For fifth root(0.95):
g(x) ≈ 1 + (x/5)
g(0.95) ≈ 1 + (0.95/5)
g(0.95) ≈ 1.190
Approximation of fifth root(0.95) is 1.190 (rounded to three decimal places).
For fifth root(1.1):
g(x) ≈ 1 + (x/5)
g(1.1) ≈ 1 + (1.1/5)
g(1.1) ≈ 1.220
Approximation of fifth root(1.1) is 1.220 (rounded to three decimal places).
To approximate the numbers fifth root(0.95) and fifth root(1.1) using the linear approximation of the function g(x) = 1 + (x/5) at a = 0, we can substitute the given values into the linear approximation equation and round the answer to three decimal places.
For fifth root(0.95):
We set x = 0.95 - 0 = 0.95.
Substituting this value into the linear approximation equation, we get g(0.95) ≈ 1 + (0.95/5) = 1 + 0.19 = 1.19.
Rounding this to three decimal places, we get the approximation of fifth root(0.95) as 1.190.
For fifth root(1.1):
We set x = 1.1 - 0 = 1.1.
Substituting this value into the linear approximation equation, we get g(1.1) ≈ 1 + (1.1/5) = 1 + 0.22 = 1.22.
Rounding this to three decimal places, we get the approximation of fifth root(1.1) as 1.220.
Therefore, the approximations to three decimal places of fifth root(0.95) and fifth root(1.1) using the linear approximation of the function g(x) = 1 + (x/5) at a = 0 are 1.190 and 1.220, respectively.