Department of Mathematics


William Paterson University of New Jersey

Jan Paxton

Wood-Ridge High School

TOPIC:

Secant and Tangent Lines

LEVEL:

Mathematica: Calculus: Grade 12

PROFICIENCY:

Students should be familiar with opening a notebook in Mathematica. They should also be familiar with tangent and secant lines, as well as the two standard definitions for slope of a tangent line.

CLASS TIME:

This lesson is designed to be done in a single class period by small groups of students working together, although they may need to complete the writing portion of the assignment for homework.

NOTES TO TEACHER:

Objective: The student will recognize the use of the slope of the secant line and connect it with the graph of the secant line, as the secant line approaches the tangent line. The student will also recognize the two forms of the slope formula used and practice finding the equation of the tangent line.

Student Work Sheet

A.P. Calculus
Secant and Tangent Lines

Open a new notebook in Mathematica.
  1. Enter the following sequence:
    • f[x_]:= x^2 + 1
    • c:=1
    • d1:= 1
    • d2:= .5
    • d3:= .2
    • y1[x_]:=((f[c+d1]-f[c])/(d1))(x-c)+f[c]
    • y2[x_]:=((f[c+d2]-f[c])/(d2))(x-c)+f[c]
    • y3[x_]:=((f[c+d3]-f[c])/(d3))(x-c)+f[c]
    • Plot[{f[x],y1[x],y2[x],y3[x]},{x,-1,3},PlotRange->{-2,8}, GridLines->Automatic]

    Remember that Mathematica is case-sensitive, so type exactly what you see. After the last line, press [Shift] [Enter].

    Study the graph and the sequence of commands you entered. In the space below, write an explanation of what your commands represent in the graph and what is being illustrated by this sequence. Now, with paper and pencil, compute the slope of the tangent line at the appropriate point and write an equation of that tangent line. Print out your graph and add the graph of the tangent line in by hand.

  2. For this problem, enter the following sequence, ending with [Shift] [Enter]:

    • g[x_]:=x^3-x-1
    • a:=2
    • a1:=1
    • a2:=1.5
    • a3:=1.8
    • k1[x_]:=((g[a]-g[a1])/(a-a1))(x-a)+g[a]
    • k2[x_]:=((g[a]-g[a2])/(a-a2))(x-a)+g[a]
    • k3[x_]:=((g[a]-g[a3])/(a-a3))(x-a)+g[a]
    • Plot[{g[x],k1[x],k2[x],k3[x]},{x,-1,2.5},PlotRange->{-8,8}, GridLines->Automatic]

    Study this graph and this sequence of commands. In the space below, compare and contrast the two graphs and sequences of commands. Now, with paper and pencil, compute the slope of the tangent line at the appropriate point and write an equation of that tangent line. Print out your graph and add the graph of the tangent line in by hand.

  3. Choose one of the two sequences as a model and repeat the process for h(x) = x^3 - x^2 at x = -1.