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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.1, Problem 3E

(a)

(i)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

**Given:**

The equation of the curve is

The point *y*.

**Calculation:**

The slope of the secant line between the points,

Obtain the slope of the secant line *PQ* when

Substitute 1.5 for *x* in

Substitute 1.5 for *x* and −2 for

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

(i)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

**Given:**

The equation of the curve is

The point *y*.

**Calculation:**

The slope of the secant line between the points,

Obtain the slope of the secant line *PQ* when

Substitute 1.5 for *x* in

Substitute 1.5 for *x* and −2 for

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

(ii)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

Substitute 1.9 for *x* in

Substitute 1.9 for *x* and

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

(iii)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

Obtain the slope of the secant line *PQ* when

Substitute 1.99 for *x* in

Substitute 1.99 for *x* and

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

(iv)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

Obtain the slope of the secant line *PQ* when

Substitute 1.999 for *x* in

Substitute 1.999 for *x* and

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

(v)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

Obtain the slope of the secant line *PQ* when

Substitute 2.5 for *x* in

Substitute 2.5 for *x* and

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

(vi)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

Obtain the slope of the secant line *PQ* when

Substitute 2.1 for *x* in

Substitute 2.1 for *x* and

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

(vii)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

Obtain the slope of the secant line *PQ* when

Substitute 2.01 for *x* in

Substitute 2.01 for *x* and

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

(viii)

To determine

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

Expert Solution

The slope of the secant line *PQ* when

Obtain the slope of the secant line *PQ* when

Substitute 2.001 for *x* in

Substitute 2.001 for *x* and

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

(b)

To determine

**To guess:** The value of the slope of the tangent line to the curve at

Expert Solution

The estimated value of the slope of the tangent line to the curve at

The secant line is closer to the tangent line at

From part (a), the slope of the secant line *PQ* when *PQ* when

The value of the slope of the tangent line to the curve at *P.*

Take the average of the two secant lines when

Thus, the estimated value of the slope of the tangent line to the curve at

(c)

To determine

**To find:** The equation of the tangent line to the curve at

Expert Solution

The equation of the tangent line is

**Formula used:**

The equation of the tangent line to the curve

**Calculation:**

From part (b), the slope of the tangent line to the curve is 1.

Substitute

Isolate *y*.

Thus, the equation of the tangent line to the curve at