Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
Linear Approximations and Differentials- DM5
Objectives
- Describe the linear approximation to a function at a point.
- Write the linearization of a given function.
- Draw a graph that illustrates the use of differentials to approximate the change in a quantity.
- Calculate the relative error and percentage error in using a differential approximation.
Summary
The tangent line to a differentiable function at the point is given in point-slope form by the equation
The principle of local linearity tells us that if we zoom in on a point where a function is differentiable, the function will be indistinguishable from its tangent line. That is, a differentiable function looks linear when viewed up close. We rename the tangent line to be the function where Thus, for all near
If we know the tangent line approximation to a function then because and we also know the values of both the function and its derivative at the point where In other words, the linear approximation tells us the height and slope of the original function. If, in addition, we know the value of we then know whether the tangent line lies above or below the graph of depending on the concavity of