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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.1, Problem 4E

(a)

To determine

**To find:** The slope of the secant line *PQ* for the following values of *x.*

Expert Solution

The slope of the secant line *PQ* for the following values of *x* is given below:

(i) The slope of the secant line *PQ* when

(ii) The slope of the secant line *PQ* when

(iii) The slope of the secant line *PQ* when

(iv) The slope of the secant line *PQ* when

(v) The slope of the secant line *PQ* when

(vi) The slope of the secant line *PQ* when

(vii) The slope of the secant line *PQ* when

(viii) The slope of the secant line *PQ* when

**Given:**

The equation of the curve

The point *P*(0.5, 0) lies on the curve *y*.

The point *Q* is

**Calculation:**

The slope of the secant line between the points, *P*(0.5, 0) and *Q*

**Section (i):**

Obtain the slope of the secant line *PQ* when

Substitute 0 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* when

**Section-(ii):**

Obtain the slope of the secant line *PQ* when

Substitute 0.4 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* when

**Section-(iii):**

Obtain the slope of the secant line *PQ* when

Substitute 0.49 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* when

**Section-(iv):**

Obtain the slope of the secant line *PQ* when

Substitute 0.499 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* when

**Section-(v):**

Obtain the slope of the secant line *PQ* when

Substitute 1 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* when

**Section-(vi):**

Obtain the slope of the secant line *PQ* when

Substitute 0.6 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* when

**Section-(vii):**

Obtain the slope of the secant line *PQ* when

Substitute 0.49 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* when

**Section-(viii):**

Obtain the slope of the secant line *PQ* when

Substitute 0.501 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* when

(b)

To determine

**To guess:** The slope of the tangent line to the curve at *P* (0.5, 0).

Expert Solution

The estimated slope of the tangent line to the curve at *P* (0.5, 0) is

**Formula used:**

The slope of the tangent line is the limit of the slope of the secant line.

That is,

**Calculation:**

From part (a), the slope of the secant line for many values of *x* is closer to 1. Thus, the slope

Substitute

Since

Thus, the estimated slope of the tangent line to the curve at *P* (0.5, 0) is

(c)

To determine

**To find:** The equation of the tangent line to the curve at *P*(0.5, 0).

Expert Solution

The equation of the tangent line to the curve at *P*(0.5, 0) is

**Formula used:**

The equation of the tangent line to the curve

**Calculation:**

Substitute

Thus, the equation of the tangent line to the curve at *P*(0.5,0) is

(d)

To determine

**To sketch:** The curve, two of the secant line, and the tangent line.

Expert Solution

**Formula used:**

Slope-intercept formula:

**Calculation:**

Obtain the secant line at

The slope of the secant line *PQ* when

Use the slope-point intercept formula to find a equation of the secant line.

Substitute

Thus, the equation of the secant line at

Obtain the secant line at

The slope of the secant line *PQ* when

Use the slope-point intercept formula to find a equation of the secant line.

Substitute

Thus, the equation of the secant line at

Note that the equation of the secant line at

Draw the curve

Thus, the required sketch is obtained.