Guided Practice 1.8: Tangent line approximation

Overview

In this section we will be looking at a common application of the derivative to making accurate predictions about a function, when we don’t have complete information about the function. This is the basic idea behind such applications as weather forecasts, financial forecasting, laboratory estimates, and more. We know by now that the derivative $f’(a)$ at a point \(x=a\) gives the slope of the tangent line to the graph of \(f\) at \(x = a\). This tangent line is also called the local linearization of \(f\) at \(x = a\), and we will learn how to compute local linearizations and use them to estimate values of a function.

NOTE: You will need to be 100% fluent in finding the equation for a line using the point-slope form for this section. If you need a refresher, go here: https://www.youtube.com/watch?v=eHPTyYbNmx4.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• Given the value of the derivative of f at a point x = a (that is, given f’(a)), write the equation of the tangent line to the graph of f at x = a.
• Explain what is meant by the local linearization of a function f at the point x = a.
• Use a local linearization of a function at x = a to approximate values of f near x = a.

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Given a function f, find its local linearization at x = a.
• If L(x) is the local linearization of a function f(x) at x = a, and if b is some point near a, determine whether L(b) is greater than, less than, or equal to f(b) and explain.

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 1.8. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus).

• Quick Review: The tangent line approximation (2:18)
• Calculating a tangent line (5:42)
• Using a tangent line (3:27)
• Using the local linearization (7:07)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.