Module 4 Using TI-89 or Voyage 200 to Explore the Concept of Limit
Module 4 Using TI-89 or Voyage 200 to Explore the Concept of Limit
Module 4 Using TI-89 or Voyage 200 to Explore the Concept of Limit
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Lesson 4.1: Tolerances<br />
A function can be described as a black box with an input and a c<strong>or</strong>responding output. Each input<br />
value x enters <strong>the</strong> box and <strong>the</strong>n undergoes a transf<strong>or</strong>mation that produces a c<strong>or</strong>responding<br />
output value y. F<strong>or</strong> continuous functions you can ensure that <strong>the</strong> output values will be close <strong>to</strong> a<br />
particular value <strong>of</strong> y if <strong>the</strong> input values are close enough <strong>to</strong> <strong>the</strong> c<strong>or</strong>responding value <strong>of</strong> x. In this<br />
lesson you will quantify <strong>the</strong> concept <strong>of</strong> closeness f<strong>or</strong> a specific function.<br />
Bef<strong>or</strong>e you begin this lesson, perf<strong>or</strong>m <strong>the</strong> "New Problem" command and clear all graphs and<br />
plots in <strong>the</strong> Y= Edit<strong>or</strong>.<br />
Investigating Tolerances <strong>of</strong> y =<br />
F<strong>or</strong> y = , y is close <strong>to</strong> 2 when x is close <strong>to</strong> 3.<br />
The key question is<br />
How close should x be <strong>to</strong> 3 <strong>to</strong> ensure that y is within 0.1 <strong>of</strong> 2<br />
Begin <strong>the</strong> investigation by graphing <strong>the</strong> function and <strong>the</strong> two h<strong>or</strong>izontal lines that represent <strong>the</strong> y-<br />
<strong>to</strong>lerances: one line that is 0.1 below y = 2 and <strong>the</strong> o<strong>the</strong>r line that is 0.1 above y = 2.<br />
Graphing <strong>the</strong> Function and Output Bounds<br />
In <strong>the</strong> Y= Edit<strong>or</strong> enter<br />
• The function: y1 =<br />
• The h<strong>or</strong>izontal line 0.1 below 2: y2 = 2 – 0.1 = 1.9<br />
• The h<strong>or</strong>izontal line 0.1 above 2: y3 = 2 + 0.1 = 2.1<br />
Display <strong>the</strong> graph <strong>of</strong> <strong>the</strong> function <strong>to</strong> see its basic shape and <strong>the</strong> graphs <strong>of</strong> <strong>the</strong> two h<strong>or</strong>izontal lines<br />
that represent <strong>the</strong> output <strong>to</strong>lerance. Display <strong>the</strong> window with xmin = 0, xmax = 10, xscl = 1, ymin<br />
= 0, ymax = 5, and yscl = 1.
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