chapter 4 section 4

Excel Companion Chapter 4 section 4
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Secant Lines and Tangent Lines	*Secant.xls*

Open the workbook Secants.xls.

The slope of the tangent line to y = f(x) at the point P(a, b) is found by a limit process. In the last column of the table the expression

is referred to as the ‘difference quotient’. It gives the slope of the secant line through two points

(x, f(x)) and Q(x + h, f(x+h)).

When these two points are very close to each other, the difference, h, between their x coordinates will be small.

Secant.xls - Secant-lines

This graph shows a portion of the curve around the point P(1,8). The second column in the table displays values of h in which each successive value is 1/10 of the previous value. For the three secant lines shown on the graph, the values of h are 1.0 , 0.1, and 0.01. [Note: In order to clearly see the pink and light blue secants as separate lines it may be necessary to change the value in the zoom control box to 150% or higher]. From the last column in the table we see that the slopes of the secant lines get closer and closer to a limiting value as h gets smaller and smaller. The secant lines themselves approach the tangent line to the curve at point P. For this particular cubic function at the point (1,8), the slopes of successive secant lines seem to approach the value 7.000.

Using modern mathematical notation, the slope of the tangent line at the point (x, f(x))

is denoted by f ¢ (x) and is defined as

Both the function definition in cell C11 and the specific point under investigation can be changed. To change the point of interest, (a, f(a)), simply enter its first coordinate a in cell H6.

QUESTIONS:

1.	Keep the same cubic function but change the point P so that its x coordinate is 0.5.
	Make a conjecture for the value of f¢ (0.5).
2.	For the function make conjectures for the slope of the tangent line at:
	a) x=1
	b) x=2
3.	For the natural log function f(x) =LN(x) make a conjecture for the slope of the tangent line at
	a) x=1
	b) x=2

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