Approximate the value on a curve using the equation of a tangent line
I can use a tangent line to approximate the value on a curve.
I can determine if my tangent line approximation is an underestimate or overestimate.
Quick Lesson Plan
Linearization is a complicated name for a familiar task: writing an equation for the tangent line at a point on a graph. Subsequently, students will use the tangent line to approximate other values on the graph and use visual confirmation to test their conjectures about underestimating, overestimating, and preliminary thoughts about concavity.
Stress the three main components of today’s work: writing an equation for a tangent line (they’re in familiar territory here with a direct application of derivatives), using the linearization to approximate other values on the graph (students easily comprehend that approximations are best near the point of tangency), and deciding if their approximation is an over-or underestimate of the desired value. While we are not yet defining concavity by using second derivative values (this appears in Unit 5), our students remembered the terms concave up and concave down from other courses (physics, Algebra 2, or Precalculus).
Our students were able to conjecture about tangent lines to a graph that is concave up (the tangent line is “below” the graph and will underestimate values on the graph) or concave down (the tangent line is “above” the graph and will overestimate values on the graph).
You may want to extend the conversation to consider how far off--- or how much error--- is acceptable in our work or the work of engineers, financial planners, or carpenters. This being, of course, a precursor to a conversation about series error and error bound calculations.
Question 3 in the section to Check Your Understanding asks students to draw an original graph to meet the given condition. Students generally do well on this task. Question 4 requires careful interpretation; the students in your class who like a challenge will want to try this one.
The concept of linearization permeates the AB and BC curricula. Linearization is a characteristic of a differentiable function and tangent lines are a useful method for approximating values (Euler’s method uses repeated linearization techniques and any study of infinite series will encounter a study of error):
2012 BC 4 is an excellent question for BC students, but AB students can navigate parts a) and c) if their instructor is willing to take a side trip into Euler’s method.
The use of equal signs is not appropriate when we are finding an approximation. In order to help students use notation correctly, we need to model correct and precise communication by using signs for approximately equal to when a tangent line is approximating a function value.