Use a linear approximation (or differentials) to estimate the given number.
radical (99.4)
√99.4 = √(100-.6)
The slope of y=√x is 1/2√x
at x=10, slope is .05
so, a linear approximation is
∆y/.6 = .05
∆y = .03
y-∆y = 10-.03 = 9.97
check: √99.4 = 9.96995
To estimate the given number, radical (99.4), using linear approximation or differentials, we will start by using a known value to find an approximate value.
Let's use the fact that the square root of a number can be expressed as a power of that number raised to 1/2. So, we can rewrite the given number as:
radical (99.4) = 99.4^(1/2)
Now, we need to find a convenient value, preferably a whole number, close to 99.4. One such value is 100. So, let's express 99.4 as the sum of 100 and the difference between 99.4 and 100:
99.4 = 100 + (-0.6)
Now, we can rewrite the expression using differentials:
radical (99.4) = radical (100 + (-0.6))
The linear approximation tells us that when we have a function f(x) that is well-behaved around a point (a, f(a)), we can approximate f(x) using the tangent line to the graph of f at x = a. In this case, our function is f(x) = radical (x), and our point of interest is a = 100.
First, let's find the derivative of the function f(x) = radical (x):
f'(x) = (1/2) * x^(-1/2)
Next, we'll evaluate the derivative at x = 100:
f'(100) = (1/2) * 100^(-1/2) = 1/20
Now, we can use the linear approximation formula:
f(x) ≈ f(a) + f'(a) * (x - a)
Plugging in our values:
radical (99.4) ≈ radical (100) + (1/20) * (-0.6)
We can evaluate radical (100) as 10 since the square root of 100 is 10. Substituting the values, we get:
radical (99.4) ≈ 10 + (1/20) * (-0.6) = 10 - 0.03 = 9.97
Therefore, using linear approximation, the estimated value of radical (99.4) is approximately 9.97.