Estimate delta(x) using the Linear Approximation and use a calculator to compute both the error and the percentage error.
f(x)=cosx a=(pi/4) delta(x)=.06
delta(x)= ______
i got this part and it's -.04242
The error in Linear Approximation is:
and
The error in percentage terms is:
My book says that the error= the absolute value of [(delta(x))- [(f'(a))(delta(x))]].
I do not know if this formula is incorrect, or if i am not inputting the values correctly, but it seems that the error would equal 0, which is incorrect.... HELP!!
To estimate delta(x) using the linear approximation, you need to use the formula:
delta(x) ≈ f'(a) * delta(x),
where f'(a) is the derivative of the function f(x) at the point a.
In this case, f(x) = cos(x) and a = pi/4. The derivative of cos(x) is -sin(x), so f'(a) = -sin(a) = -sin(pi/4) = -sqrt(2)/2.
Plug in the values into the formula:
delta(x) ≈ (-sqrt(2)/2) * 0.06,
which simplifies to approximately -0.04242.
Now let's calculate the error in the linear approximation. The formula you mentioned, error = |(delta(x)) - [(f'(a))(delta(x))] |, is correct.
In this case, the error is:
error = | (-0.04242) - [(-sqrt(2)/2)(0.06)] |.
Evaluate the expression inside the absolute value brackets:
error = |-0.04242 - (-sqrt(2)/2)*(0.06)|
= |-0.04242 + sqrt(2)/2*0.06|
= |-0.04242 + 0.0424264|
= 0.0000064.
Now let's calculate the percentage error. The formula for the percentage error is:
percentage error = (error / delta(x)) * 100.
In this case, the percentage error is:
percentage error = (0.0000064 / 0.06) * 100
= 0.0106%.
Therefore, the error in the linear approximation is approximately 0.0000064 and the percentage error is approximately 0.0106%.