Activity Overview
In this Derive™ activity, students use the Remainder Theorem to find a tangent line to the graph of a polynomial function.
Before the Activity
See the attached DFW file for detailed instructions for this activity
Print pages from the attached DFW file for your class
During the Activity
Distribute the pages to the class.
Follow the Activity Procedures:
Set up a 2D plot window
Graph the given polynomial
Plot the equation whose right side is formed by constant term and the linear term of the polynomial in the same window
Note that the graph of the line is tangent to the graph of the polynomial at x = 0
When x is close to 0, the constant term plus the linear term dominate the value of the polynomial
Subtract these terms from the polynomial, set the window range, and graph the result
Note that the graph of the resulting polynomial lies entirely within the set window
Understand that the sum of the constant and linear terms will be almost equal to the value of the polynomial for a given x-value in an interval
Understand that lower degree terms dominate the polynomial
Realize that when x lies between -1 and 1, the dominance of x increases with its powers
Study the statement of the Remainder Theorem
Use the Derive steps to simplify the expression for the polynomial
Observe that the result is another constant polynomial
Evaluate it to obtain the right hand side of the given expression in the statement of the Remainder Theorem
Recognize the fact that if P(x) is a polynomial, then the tangent line to the graph of P(x) at x = a is given by the expression Q(a)(x - a) + P(a), where Q(a) is the polynomial (P(x) - P(a))/(x - a)
After the Activity
Students answer questions and solve problems listed on the activity sheet.
Review student results:
As a class, discuss questions that appeared to be more challenging
Re-teach concepts as necessary