. . Errors: Absolute Error, Relative Error, and Percentage Error When we are approximating a value, there occurs an error. In this section, we consider the error, which occurs by linear approximation, given by ( ).
We shall consider different types of errors. Taking h , we get x h , then ( ) becomes E h ( ) = f x h x h ′ ... ( ) Note that E ( ) and as we have already observed lim h E h follows from the continuity of f at x . In addition, if f is differentiable, then from ( ), it follows that Fig.
. Linear Approximation by Tangent Line } ∆ y Tangent line )( )) x O L and and lim h E h h → = lim h h h ′ . Thus when f is differentiable at x , then the above equation shows that E h ( ) actually approaches zero faster than h approaching zero. Now, we define Definition .
Suppose that certain quantity is to be determined. It’s exact value is called the actual value . Some times we obtain its approximate value through some approximation process. In this case, we define Absolute error = |Actual value − Approximate value.| So ( ) gives the absolute error that occurs by a linear approximation.
Let us look at an example illustrating the use of linear approximation. Example . Use linear approximation to find an approximate value of . without using a calculator.
We need to find an approximate value of . using linear approximation. Now by ( ), we have + ∆ ≈ ′ ∆ . To do this, we have to identify an appropriate function f , a point x and D x .
Our choice should be such that the right side of the above approximate equality, should be computable without the help of a calculator. So, we choose f x x x and ∆= . . Then, ′