Estimate 1/(sqrt(82)) using linear approximation and the exact value for 1/(sqrt(81))
Your estimate:
To estimate 1/(sqrt(82)) using linear approximation, we can start by finding the linear approximation of the function 1/sqrt(x) at x = 81.
The linear approximation of a function f(x) at x = a can be represented as:
L(x) = f(a) + f'(a) * (x - a)
Here, f(x) = 1/sqrt(x), a = 81, and we need to find the value of L(82), which is the linear approximation of 1/(sqrt(82)).
To find the exact value of 1/(sqrt(81)), we can plug 81 into the function 1/sqrt(x):
1 / sqrt(81) = 1 / 9 = 0.1111...
Now, let's find the linear approximation:
First, find f(81) (the value of the function at x = 81):
f(81) = 1/sqrt(81) = 1/9 = 0.1111...
Next, find f'(x) (the derivative of the function f(x)):
f'(x) = -1 / (2 * sqrt(x^3))
Now, find f'(81) (the value of the derivative at x = 81):
f'(81) = -1 / (2 * sqrt(81^3)) = -1 / (2 * sqrt(531441)) = -1 / (2 * 729) = -1 / 1458
Finally, substitute the values into the linear approximation formula:
L(82) = f(81) + f'(81) * (82 - 81)
= 0.1111... + (-1 / 1458) * (82 - 81)
= 0.1111... + (-1 / 1458) * 1
= 0.1111... - 1 / 1458
= 0.1111... - 0.000686...
So, your estimate for 1/(sqrt(82)) using linear approximation is approximately 0.1104.