estimate change in f using the linear approximation and compute both error and the % error
f(x)= (3+x)^1/2
a=1
change in x=0.5
estimate ln(e^2+0.1)-ln(e^2)
f(x) = ln x
df = (df/dx) dx
df = (1/x) dx
f(x+dx) = f(x) + (df/dx) dx
f( e^2 + 0.1) = ln(e^2) + (1/e^2)(0.1)
f(e^2 +0.1) - ln(e^2) = 0.1/e^2
= .0135335
no copy and paste for me any more I guess.
To estimate the change in f using linear approximation, we need to find the derivative of the function f(x) = (3 + x)^(1/2) and evaluate it at the point a = 1.
Step 1: Find the derivative of f(x):
Using the power rule, the derivative of (3 + x)^(1/2) is:
f'(x) = (1/2)(3 + x)^(-1/2)
Step 2: Evaluate the derivative at x = a:
Substitute x = 1 into the derivative function:
f'(1) = (1/2)(3 + 1)^(-1/2) = (1/2)(4)^(-1/2) = (1/2)(2)^(-1) = 1/4
Step 3: Estimate the change in f:
The change in f is estimated by multiplying the derivative by the change in x:
change in f ≈ f'(1) * (change in x) = (1/4) * 0.5 = 0.125
The estimated change in f is 0.125.
Now let's calculate the error and the percentage error:
Step 4: Calculate the actual change in f:
Evaluate the original function f(x) = (3 + x)^(1/2) at x = a and x = a + change in x:
f(1) = (3 + 1)^(1/2) = 2^(1/2) = √2
f(1.5) = (3 + 1.5)^(1/2) = 4.5^(1/2) = √4.5
Actual change in f = f(1.5) - f(1) = √4.5 - √2
Step 5: Calculate the error:
The error is the difference between the estimated change and the actual change in f:
Error = actual change in f - estimated change in f
Step 6: Calculate the percentage error:
The percentage error is the absolute value of the error divided by the actual change in f, multiplied by 100:
Percentage error = |Error / actual change in f| * 100
To find the error and percentage error, we need to calculate the actual change in f:
Actual change in f = √4.5 - √2 ≈ 1.5811 - 1.4142 ≈ 0.1669
Error = 0.1669 - 0.125 = 0.0419
Percentage error = |0.0419 / 0.1669| * 100 ≈ 25.08%
Therefore, the error in the linear approximation is approximately 0.0419 and the percentage error is approximately 25.08%.