MATH 1351 TI-85 EXERCISE V

The sandwich theorem

Name: SID:

We'll begin this exercise with a hypothetical situation. Suppose you're in your VW bug driving in the center lane down a three lane, one-way street. You notice there is a flat bed truck carrying a D-10 Caterpillar in the left lane just beside you, and a dump truck loaded with gravel in the right lane. Suddenly, without warning, the road becomes a two lane. What happens to you? (Ouch)

With your TI-85 in RADIAN MODE, SimulG format, graph the three functions

y1 = 1	y2 = (sin x)/x	&	y3 = cos x

on the same screen using the ZTRIG viewing window. We are interested in the relation between these three functions near the value x = 0. Thus choose the following values for the RANGE:

xMin = -3.15, xMax = 3.15, yMin = -1.5, and yMax = 1.5.

In this viewing window what is the relation between the three graphs?

(This is the relation that is "proved" geometrically section 2.3 of the text.)

Use TRACE and move to the right (right arrow) until the value of x is 1. What is the corresponding value of y? Thus we are tracing which graph? Now try the arrow down button. Where does the cursor go? Notice the value of x remains the same but the value of y is now determined by the corresponding value of y2. What do you think will happen if you use the arrow down button again? Were you right? As you trace along the graph of y3 towards the value x = 0, what happens to the corresponding values of y3? Since the values of y2 are "sandwiched between" those of y3 and y1, what must be happening to the values of y2 as x approaches 0?

Trace along the graph of y3 until the cursor is at x = 0, y = 1. Now arrow up once to the graph of y2. For x = 0, what is the corresponding value indicated for y on the graph of y2?

Explain.

How close to 0 must we choose x if we want the value of y2 to be within .001 of 1?

Explain.