Use differentials (or, equivalently, a linear approximation) to estimate the given number.
sqrt100.4
I got 10.02 is that right?
yeh
To estimate the number √100.4 using differentials or a linear approximation, we can start by considering the function f(x) = √x.
The linear approximation of f(x) around a point a is given by the equation:
L(x) = f(a) + f'(a)(x - a)
In this case, we want to estimate √100.4, so a = 100 and x = 100.4.
First, we need to find the derivative of f(x) = √x. The derivative can be found using the power rule:
f'(x) = 1/2 * (x)^(-1/2)
Next, we evaluate f(a) and f'(a):
f(100) = √100 = 10
f'(100) = 1/2 * (100)^(-1/2) = 1/2 * (1/10) = 1/20
Now, we substitute these values into the linear approximation formula:
L(x) = f(a) + f'(a)(x - a)
L(100.4) = 10 + (1/20)(100.4 - 100)
Simplifying further:
L(100.4) = 10 + (1/20)(0.4)
= 10 + 0.02
= 10.02
Therefore, using differentials or linear approximation, we estimate that √100.4 is approximately equal to 10.02.