Linear Approximations Accuracy

Applications of Linear Approximation and Accuracy

Note, x - a is often symbolized by Δx

Dealing with Linear Approximation, we are dealing with calculations that are… approximate. In this there is a margin for error, and some results are more accurate than others.

Definition - Linear Approximation Error

If the variable x-value is measured as x = a, with the error as Δx (or x - a) units, then the Δf can be the error in estimating our f(x),

Δf = f(x) - f(a) =roughly= f’(a)Δx

Linear Approximation Cube

part 1 of the solution to calculating the volume of a 12cm-sided cube and determining the error in the volume calculation if the uncertainty of the side measurements is 0.3cm.
part 2 of the solution to calculating the volume of a 12cm-sided cube and determining the error in the volume calculation if the uncertainty of the side measurements is 0.3cm.

Linear Approximation Error

part 1 of the solution to findin the maximum possible error in a calculation involving a right triangle with an internal angle of 16 degrees with a related uncertainty of 1 degree.
part 2 of the solution to finding the maximum possible error in a calculation involving a right triangle with an internal angle of 16 degrees with a related uncertainty of 1 degree.

part 3 of the solution to finding the maximum possible error in a calculation involving a right triangle with an internal angle of 16 degrees with a related uncertainty of 1 degree.
part 4 of the solution to finding the maximum possible error in a calculation involving a right triangle with an internal angle of 16 degrees with a related uncertainty of 1 degree.

Linear Approximation Error

part 1 of the solution to calculating the maximum allowable error in the length the radius of a sphere (a basketball in this example) based on a given allowed error in terms of the sphere's radius.
part 2 of the solution to calculating the maximum allowable error in the length the radius of a sphere (a basketball in this example) based on a given allowed error in terms of the sphere's radius.

Relative Error and Percentage Error

What we have been working with above is the absolute error. This gives, as the name suggests, the pure value of the error. However, it is not an error value comparable between different types of measurements. For example, an error when measuring the distance between two cities of 1 meter, is not the same as a an error of measuring shoe size of 1 meter. This is where we introduce Relative Error and Percentage error, to give us a better understanding of the relevance of error.

Definition - Relative Error

Relative Error = 
error in f
value of f
 = 
Δ f
f

Definition - Percentage Error

Percentage Error is simply Relative Error as a Percentage

Δ f
f
  x 100

Relative Error

part 1 of the solution to calculating the error possibility and relative error in the volume of a 9cm-sided cube if the error in the side length is 0.03cm.
part 2 of the solution to calculating the error possibility and relative error in the volume of a 9cm-sided cube if the error in the side length is 0.03cm.