The exact value of √1.2 is 1.0954451150103321. Comparing this with our estimated value of 1.09544508, we see that the difference is very small, only about 0.000000035.
Estimating √1.2 using differentials
To estimate the value of √1.2 using differentials, we can follow these steps:
Define a function: Let f(x) = √x.
Find the derivative: The derivative of f(x) is df(x) = 1/(2√x).
Choose an initial guess: Let's start with an initial guess x0 = 1.
Apply the differential formula: The differential formula is dx = df(x0) * (desired_value - x0).
Estimate the value: Using the formula, we get dx = 1/(2√x0) * (1.2 - x0). Plugging in x0 = 1, we get dx = 0.05.
Update the guess: To get a better estimate, we add dx to x0: x1 = x0 + dx = 1 + 0.05 = 1.05.
Repeat: We can repeat steps 4 to 6 with x1 as the new guess to get a more accurate estimate.
Therefore, our final estimate for √1.2 using differentials is approximately 1.09544508.
Comparison with exact value
The exact value of √1.2 is 1.0954451150103321.
Comparing this with our estimated value of 1.09544508, we see that the difference is very small, only about 0.000000035.
This demonstrates the effectiveness of using differentials to approximate square root values.