What Is a Linear Approximation?

Function

• Linear approximation depends on using a function to create approximation of the solution for x. A function is a mathematical expression in which the variable x always results in single y. For example, y = 5x + 3 is a function, because no matter what variable is plugged in for x, it results in a single y. Function notation is how the function is displayed mathematically. For y = 5x + 3, the function notation is f(x) = 5x + 3.

Derivation

• Derivation is a mathematical function of calculus and involves using mathematical rules to define the function on a range of x, called a limit. For example, a derivative would help solve a function from x = 1 to 15. Linear approximation requires having a remainder when deriving a function at different intervals.

Linear Approximation

• When the function has a remainder term, it is no longer a linear function, and that makes it difficult to solve. A function is considered linear when it uses real numbers that create an answer. In essence, in its simplest form, a function is linear if A + B = C. When a function does not result in a real number, linear approximation allows the removal of the remainder attribute to make the function linear and easier to solve.

Error

• Error estimation uses linear approximation by allowing the person doing the measurement to see how the remainder effects the outcome. For example, assume you measure the radius of a circle of a subject with an error of plus or minus .2 cm, and you want to know how that error changes the area. By dropping the remainder, the .2, you can solve for the true area and see how the error estimation deviates from it.