If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License Therefore, the percentage error in the measurement of the cardboard is larger, even though 0.25 in. when the actual width is 8 in., our absolute error is 1 4 1 4 in., whereas the relative error is 0.25 8 = 1 32, 0.25 8 = 1 32, or 3.1 %. By comparison, if we measure the width of a piece of cardboard to be 8.25 in. when the actual height is 62 in., the absolute error is 1 in.

For example, if we measure the height of a ladder to be 63 in. The percentage error is the relative error expressed as a percentage.

Given an absolute error Δ q Δ q for a particular quantity, we define the relative error as Δ q q, Δ q q, where q q is the actual value of the quantity. We are typically interested in the size of an error relative to the size of the quantity being measured or calculated. The measurement error dx ( =Δ x ) ( =Δ x ) and the propagated error Δ y Δ y are absolute errors. Let dx be an independent variable that can be assigned any nonzero real number, and define the dependent variable d y d y byĮstimate the error in the computed volume of a cube if the side length is measured to be 6 cm with an accuracy of 0.2 cm. Suppose y = f ( x ) y = f ( x ) is a differentiable function. Here we see a meaning to the expressions dy and dx. Although we used the expressions dy and dx in this notation, they did not have meaning on their own. When we first looked at derivatives, we used the Leibniz notation d y / d x d y / d x to represent the derivative of y y with respect to x. Differentials provide us with a way of estimating the amount a function changes as a result of a small change in input values. To discuss this more formally, we define a related concept: differentials. They can also be used to estimate the amount a function value changes as a result of a small change in the input. We have seen that linear approximations can be used to estimate function values. Recall that the tangent line to the graph of f f at a a is given by the equationįind the linear approximation of f ( x ) = ( 1 + x ) 4 f ( x ) = ( 1 + x ) 4 at x = 0 x = 0 without using the result from the preceding example. Linear Approximation of a Function at a PointĬonsider a function f f that is differentiable at a point x = a. In addition, the ideas presented in this section are generalized later in the text when we study how to approximate functions by higher-degree polynomials Introduction to Power Series and Functions. Linear functions are the easiest functions with which to work, so they provide a useful tool for approximating function values. In this section, we examine another application of derivatives: the ability to approximate functions locally by linear functions. We have just seen how derivatives allow us to compare related quantities that are changing over time. 4.2.4 Calculate the relative error and percentage error in using a differential approximation.

4.2.3 Draw a graph that illustrates the use of differentials to approximate the change in a quantity.

4.2.2 Write the linearization of a given function.

4.2.1 Describe the linear approximation to a function at a point.

Linearization calculus examples