Use the linearization approximation (1+x)^k=1+kx to find an approximation for the function f(x)=(1/square root of 4+x) for values of x near zero.
To find an approximation for the function f(x) = (1/√(4+x)) for values of x near zero using the linearization approximation, we can follow these steps:
Step 1: Find the first derivative of the function.
The first derivative of f(x) is given by:
f'(x) = -1/2(4+x)^(-3/2)
Step 2: Evaluate the first derivative at x = 0 to find the slope of the tangent line.
f'(0) = -1/2(4+0)^(-3/2) = -1/2(4)^(-3/2) = -1/2(1/8) = -1/16
Step 3: Use the linearization approximation formula.
The linearization approximation is given by:
L(x) = f(a) + f'(a)(x - a)
In this case, a = 0, so the linearization approximation becomes:
L(x) = f(0) + f'(0)(x - 0)
Step 4: Evaluate f(0).
f(0) = 1/√(4+0) = 1/2
Step 5: Substitute the values into the linearization approximation formula.
L(x) = 1/2 + (-1/16)(x - 0)
Simplifying the equation:
L(x) = 1/2 - 1/16x
So, the linearization approximation for the function f(x) near x = 0 is:
L(x) = 1/2 - 1/16x
Therefore, the linearization approximation for the function f(x) = (1/√(4+x)) for values of x near zero is 1/2 - 1/16x.