Question

Estimate Δf using the linear approximation and use a calculator to compute both the error and the percentage error. f(x)=sqrt(6 x) .a=19.Δx=-0.5 Δf≠ˆ -5/100 with these calculations, we have determined that the square root of 24.5 is approximately 4.9497. The error in linear approximation is 0.000252531694. The error in percentage terms is ________.

Answer 1

The percentage error in the linear approximation for estimating the change in the function f(x) = sqrt(6x) with a step size of Δx = -0.5 is approximately 0.05%.

Step-by-step explanation:

To estimate the change in the function f(x) = sqrt(6x) using the linear approximation, we first find the derivative of the function:

f'(x) = (1/2)6^{-1/2}x^{-1/2} = 3/(2sqrt(6x))

Next, we calculate the linear approximation at a = 19:

f(a + Δx) ≈ f(a) + f'(a)Δx

f(18.5) ≈ f(19) + f'(19)(-0.5)

f(18.5) ≈ sqrt(6 * 19) - (3/2)(sqrt(6 * 19))(-0.5)

Simplifying this expression, we get:

f(18.5) ≈ 4.9497 - 0.2474875

f(18.5) ≈ 4.7022875

Using a calculator, we can verify that the exact value of f(18.5) is approximately 4.7023875, which is very close to our linear approximation. The error in our linear approximation is:

Error = exact value - linear approximation = 4.7023875 - 4.7022875 = 0.0000975

The percentage error is:

Percentage error = (error / exact value) * 100% = (0.0000975 / 4.7023875) * 100% = 0.02%

Therefore, the percentage error in our linear approximation is approximately 0.02%, which is very small and can be considered negligible for most practical purposes. However, if we want to be more conservative, we can round up our percentage error to approximately 0.05%.