estimate cube root of 25 by linear approximation

f(x+delta(x))= f(x) + (df(x)/x) * deltax
3 + 1/3x^(-2/3) * -2
3 - 2/3x^(-2/3)
3 - (2/3* ((cube root(9)))

is this right? thanks

y = x^3

I want x for y = 25
dy = 3 x^2 dx

start with x = 3
then y1 = 3^3 = 27

y2 - y1 = 3 * 9 (x2 - x1)
25 - 27 = 27 (x2 -3)
X2 -3 = -2/27
x2 = 2.93

do it again with x1 = 2.93 etc
right answer is
2.924017738

Thank you! :-)

To estimate the cube root of 25 using linear approximation, you can use the formula:

f(x + ∆x) = f(x) + (df(x)/dx) * ∆x

Let's break down the steps for a more accurate answer:

1. Choose a starting point, which is close to the value for which you want to find the cube root. In this case, let's choose x = 27 because it is close to 25.

2. Calculate f(x) using the cube root function f(x) = ∛x, where x = 27. In this case, f(27) = ∛27 = 3.

3. Find the derivative of f(x) with respect to x. In this case, df(x)/dx = 1/(3 * x^(2/3)).

4. Determine the difference between the initial point and the point where you want to find the cube root. In this case, the difference, or ∆x, is -2 (since 27 - 25 = -2).

5. Plug in the values into the linear approximation formula:

f(x + ∆x) = f(x) + (df(x)/dx) * ∆x
f(27 + (-2)) = 3 + (1/(3 * 27^(2/3))) * (-2)
f(25) = 3 + (1/(3 * 27^(2/3))) * (-2)

6. Simplify the equation:

f(25) ≈ 3 - (2/(3 * ∛27))

7. Calculate the approximation:

f(25) ≈ 3 - (2/(3 * 3))
f(25) ≈ 3 - (2/9)
f(25) ≈ 3 - 2/9
f(25) ≈ (27 - 2)/9
f(25) ≈ 25/9

Therefore, the cube root of 25, estimated using linear approximation, is approximately 25/9.