   Chapter 2.1, Problem 3E

Chapter
Section
Textbook Problem

The point P(2, –1) lies on the curve y = 1/(1 – x).(a) If Q is the point (x, 1/(1 – x)), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x :(i) 1.5(ii) 1.9(iii) 1.99(iv) 1.999(v) 2.5(vi) 2.1(vii) 2.01(viii) 2.001(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at P(2, –1).(c) Using the slope from part (b), find an equation of the tangent line to the curve at P(2, –1).

(a)

(i)

To determine

To find: The slope of the secant line PQ for x=1.5.

Explanation

Given:

The equation of the curve is y=1(1x).

The point P(2,1) lies on the curve y.

Calculation:

The slope of the secant line between the points, P(2,1)  and Q(x,11x) is

m=11x(1)x2 (1)

Obtain the slope of the secant line PQ when x=1.5.

Substitute 1.5 for x in 1(1x).

1(1x)=1(11

(i)

To determine

To find: The slope of the secant line PQ for x=1.5.

(ii)

To determine

To find: The slope of the secant line PQ for x=1.9.

(iii)

To determine

To find: The slope of the secant line PQ for x=1.99.

(iv)

To determine

To find: The slope of the secant line PQ for x=1.99.

(v)

To determine

To find: The slope of the secant line PQ for x=2.5.

(vi)

To determine

To find: The slope of the secant line PQ for x=2.1.

(vii)

To determine

To find: The slope of the secant line PQ for x=2.01.

(viii)

To determine

To find: The slope of the secant line PQ for x=2.001.

(b)

To determine

To guess: The value of the slope of the tangent line to the curve at P(2,1).

(c)

To determine

To find: The equation of the tangent line to the curve at P(2,1).

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