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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.1, Problem 9E

(a)

To determine

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

Expert Solution

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

x | |

2 | 0 |

1.5 | 1.7321 |

1.4 | −1.0847 |

1.3 | −2.7433 |

1.2 | −4.3301 |

1.1 | −2.8173 |

0.5 | 0 |

0.6 | −2.1651 |

0.7 | −2.6061 |

0.8 | −5 |

0.9 | 3.4202 |

**Given:**

The equation of the curve is

The point *P*(1, 0) lies on the curve *y*.

The *Q* is the point

**Calculation:**

The slope of the secant lines between the points, *P*(1, 0) and *Q*

Obtain the slope of the secant line *PQ* for the value of

Substitute 2 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 1.5 for *x* in

Substitute *Q*

Thus, the slope of the secant line *P*Q for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 1.4 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 1.3 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 1.2 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 1.1 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 0.5 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 0.5 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 0.7 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ * for the value of

Obtain the slope of the secant line *PQ * for the value of

Substitute 0.8 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 0.9 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

**Conclusion:**

The slope does not appear to be approaching a limit. Suppose *x* approaches 1, then the slope

(b)

To determine

**To explain:** The slopes of the secant lines in part (a) are not close to the slope of the tangent line at *P* by using a graph.

Expert Solution

The graph of the curve

From Figure 1, it is observed that there seems to be the frequent oscillations of the graph. Moreover, the tangent line is so steep at the point *P*(1, 0). Thus, the slopes of the secant lines are not closer to the slope of the tangent line at *P*. Therefore, it is necessary to consider the values of *x* much closer to 1 for better accurate estimates of the slope.

(c)

To determine

**To estimate:** The slope of the tangent line to the curve at *P*(1, 0).

Expert Solution

The estimated value of the slope of the tangent line to the curve at *P* (1, 0) is −31.4.

The graph of the curve

The secant line is close to the tangent line at *P*(1, 0) when

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

Obtain the slope of the secant line *PQ* for the value of

Substitute 1.001 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

Obtain the slope of the secant line *PQ* for the value of

Substitute 0.999 for *x* in

Substitute *Q*

Thus, the slope of the secant line *PQ* for the value of

The slope of the secant line *PQ* for the value of *PQ* for the value of

Take the average of two slopes of the secant lines,

Thus, the estimated value of the slope of the tangent line to the curve at *P* (1, 0) is −31.4.