Approximate the square root of 64.3, y=x^(1/2). Starting point is the square root of 64.
y = x^(1/2)
dy/dx = (1/2)x^(-1/2)
dy = 1/(2√x) dx
when x = 64, dx = .3
dy = (1/2√64)(.3) = (1/16)(.3) = appr .01875
so √64.3 = 8.01875
actual √64.3 by calculator = 8.018729
error is .000022
64.31
To approximate the square root of 64.3 using the formula y = x^(1/2), we can start by plugging in the starting point, which is the square root of 64. We know that the square root of 64 is 8 because 8 x 8 equals 64.
Now, let's proceed with the approximation:
1. Calculate the derivative of the function y = x^(1/2):
The derivative of y with respect to x is 1/2 * x^(-1/2), which simplifies to 1/(2√x).
2. Use the derivative to find the equation of the tangent line:
The equation of the tangent line is given by y = mx + b, where m is the slope and b is the y-intercept.
We can substitute the value of x (64) into the derivative to find the slope:
m = 1/(2√64) = 1/16 = 0.0625
Now, we need to find the y-intercept. Since the tangent line passes through the point (√64, √64), we can substitute these values into the equation:
√64 = 0.0625(64) + b
8 = 4 + b
b = 4
Therefore, the equation of the tangent line is y = 0.0625x + 4.
3. Use the tangent line to approximate the square root of 64.3:
Substitute 64.3 for x in the equation of the tangent line:
y = 0.0625(64.3) + 4
y ≈ 4 + 4.01875
y ≈ 8.01875
Therefore, the square root of 64.3, approximated using the tangent line, is approximately 8.01875.