lINEAR APPROXIMATION;
find a linearization at a suitably chosen interger near X at which the given function and its derivative are easy to evaluate;
f(x)=1+X
x=8.1
To find the linear approximation of a function at a specific point, you need to use the first-degree Taylor polynomial. The linear approximation helps estimate the value of the function near that point.
To start, we will find the first derivative of the given function f(x) = 1 + x:
f'(x) = 1
Since the derivative is a constant, it is easy to evaluate.
Now, let's choose an integer near x = 8.1. For simplicity, we will use x = 8.
To find the linear approximation, we will use the formula:
L(x) = f(a) + f'(a)(x - a)
where
L(x) is the linear approximation of f(x),
f(a) is the value of the function at point a,
f'(a) is the value of the derivative at point a, and
(x - a) is the difference between x and a.
Now, let's calculate the linear approximation:
a = 8
f(a) = f(8) = 1 + 8 = 9
f'(a) = f'(8) = 1
L(x) = 9 + 1(x - 8)
Simplifying the equation gives us the final linear approximation:
L(x) = 1 + x
Therefore, the linear approximation of the function f(x) = 1 + x at x = 8.1 is L(x) = 1 + x.