Answer:
- reasonable M: 1/24
- lower bound: 12
- upper bound: 12 + 1/24
Explanation:
You want a reasonable value for the maximum rate of change on the interval [144, 145] for f(x) = √x, and corresponding values of lower and upper bounds on f(145).
Slope
The square root function f(x) has a graph that is concave downward, so has a positive slope that is decreasing everywhere.
This means for a < b, f'(a) > f'(b). Thus a reasonable choice of M can be f'(a).
Derivative
One way we can determine f'(a) is using the definition of the derivative:
Reasonable M
Then for a=144, we have ...
f'(144) ≤ 1/(2√144) = 1/24
A reasonable value for M is M = 1/24.
Limits
Since we already know that f(a) < f(b) for a < b, the lower limit for f(145) can be f(144) = 12.
Using the given inequality for f(b), we find the upper limit to be ...
f(145) ≤ f(144) +(1/24)(145 -144) = 12 1/24
Then the limits are 12 ≤ f(145) ≤ 12+1/24.
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Additional comment
If we write √n = √(s² +r), where s is floor(√n), then the root can be written ...
√n = √(s² +r) = s +x . . . . . for some value of x between 0 and 1
We can rearrange this to an iterative expression for the value of x, the fractional part of the root:
s² +r = (s +x)² = s² +2sx +x² = s² +x(2s +x)
x = r/(2s +x) . . . . . . . . subtract s² and divide by (2x+x)
In the current problem, n = 145, so s=12, r=1, and we have ...
x = 1/(24 +x)
Knowing the initial bounds on x are [0, 1], one iteration tells us the new bounds on x are [1/25, 1/24]. That is, the bounds on √145 can be refined to ...
12+1/25 < √145 < 12+1/24
Iterating one more time puts √145 in the interval (12+24/577, 12+25/601), between 12.041594 and 12.041597.
This "continued fraction" approach to finding the square root converges more or less linearly, adding about the same number of good decimal digits with each iteration. On the other hand, Newton's Method iteration doubles the number of good decimal digits of a square root with each iteration.