Use the linear 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.
Well, well, well, if it isn't our old friend linear approximation. Don't worry, I'm here to make this approximation business a little less... square-root-y for you.
First, let's take a look at our function: f(x) = 1/sqrt(4+x). We want to approximate it near zero, so let's start there.
Now, let's rewrite the function using our trusty linear approximation formula (1+x)^k ≈ 1 + kx. In our case, k is equal to -0.5 (since the square root is equivalent to raising to the power of 0.5). So we have:
f(x) = 1/sqrt(4+x) ≈ 1 + (-0.5)x
See what we did there? We simply replaced k with -0.5 and (1+x) with 1. Okay, maybe not that exciting, but bear with me.
Now, we can use this approximation to estimate the value of f(x) for values of x near zero. For example, if x = 0.1, we have:
f(0.1) ≈ 1 + (-0.5)(0.1) = 1 - 0.05 = 0.95
So, the approximation of f(0.1) is approximately 0.95. Not too shabby, huh?
Remember, though, that the linear approximation isn't perfect. It works best for values of x that are near our chosen point (in this case, x = 0). The farther we stray away from zero, the less accurate our approximation becomes. But for nearby values, it can give you a good enough estimate to get the job done.
Hope that approximation got your funny bone tickled!
To find an approximation for the function f(x) = 1/sqrt(4+x) using the linear approximation (1+x)^k ≈ 1 + kx, we need to first rewrite f(x) using the binomial form (1+x)^k.
Let's start by rewriting the denominator of f(x) = 1/sqrt(4+x) as (4+x)^(1/2).
Using the binomial form (1+x)^k, we can rewrite (4+x)^(1/2) as (4(1+x/4))^(1/2).
Now, let's identify the key values in the binomial form: k = 1/2 and x/4 = 1.
We can now apply the linear approximation, given as (1+x)^k ≈ 1 + kx.
Plugging in the values, we get:
(4(1+x/4))^(1/2) ≈ 1 + (1/2)(x/4)
≈ 1 + (1/8)x
≈ (8 + x)/8
Therefore, an approximation for f(x) = 1/sqrt(4+x) for values of x near zero is (8 + x)/8.
To find an approximation for the function f(x) = 1/sqrt(4+x) near x = 0 using the linear approximation, we can use the formula:
(1 + x)^(1/2) ≈ 1 + (1/2)x
Applying this formula, we have:
f(x) ≈ 1 + (1/2)x
Now let's substitute x = 0 into our approximation formula:
f(0) ≈ 1 + (1/2)(0)
f(0) ≈ 1
Therefore, the linear approximation for f(x) = 1/sqrt(4+x) near x = 0 is approximately equal to 1.
Let's go astep further and also work out the quadratic term using the general formula:
(1+x)^k=1+kx + k(k-1)/2 x^2 +
k(k-1)(k-2)/3! x^3 + ....
f(x)= (4 + x)^(-1/2) =
4^(-1/2) (1 + x/4)^(-1/2) =
1/2 [1 - x/8 + 3/128 x^2 + ...] =
1/2 - x/16 + 3/256 x^2 + ...