use tangent line approximation (linear approximation) to estimate The cube root of 1234 to 3 decimal places. Hint: the equation should be y=f'(x0)(x-x0)+f(x0)
11^3=1331 can be easily computed using binomial theorem.
I used linear approximation and got 10.733, but it is not correct to 3 decimal places. Cube root of 1234 is 10.726
You are correct. It's just that at x=11, the linear approximation isn't very good when you get var away from 11.
You were not asked to get an approximation which agrees with the true value to 3 places, just to get a value out that far.
Extra credit: how far away from 11 can you go and still have the linear approximation good to three places?
To use the tangent line approximation (linear approximation) method to estimate the cube root of a number, follow these steps:
1. Choose a point near the number you want to find the cube root of. In this case, we'll choose x0 = 10 because it is close to the cube root of 1234.
2. Find the derivative of the function f(x) = x^(1/3). The derivative of this function can be found using the power rule, where you multiply the exponent by the coefficient and subtract 1 from the exponent: f'(x) = (1/3)x^(-2/3).
3. Evaluate f'(x) at x0. In this case, f'(10) = (1/3)(10)^(-2/3) = 1/(3*10^(2/3)).
4. Write the equation of the tangent line using the point-slope formula: y = f'(x0)(x - x0) + f(x0).
5. Plug in the values: y = 1/(3*10^(2/3))(x - 10) + 10^(1/3).
6. Estimate the cube root of 1234 by substituting x = 1234 into the equation. y = 1/(3*10^(2/3))(1234 - 10) + 10^(1/3).
Now, if you follow this calculation, you will get an estimated value close to 10.733, but it's not correct to 3 decimal places.
To find a more accurate estimate, you can use the binomial theorem to calculate the cube of 11, which equals 1331. Thus, you know the cube root of 1234 must be slightly less than 11.
A more accurate estimate would be to use x0 = 11 instead of 10 in the tangent line approximation.
By using the same steps as before with x0 = 11, you'll get an estimated value close to 10.726, which is correct to 3 decimal places.