#### To determine

**To construct:** The table of estimated value of H′(t) and sketch its graph.

#### Answer

The table of the estimated values of H′(t) will be:

t |
14 |
21 |
28 |
35 |
42 |
49 |

H′(t) |
137 |
2314 |
1814 |
1 |
1114 |
57 |

#### Explanation

**Given:**

The table represents the height of a typical pine tree grown for lumber at a managed site.

**Formula used:** For small *h*,

limh→0H(t+h)−H(t)h≈H(t+h)−H(t)h

**Calculation:**

The value of H′(14) is computed as follows,

Substitute h=7 in the formula,

H′(14)=H(14+7)−H(14)7=H(21)−H(14)7=54−417=137

Thus, H′(14)=137.

The value of H′(21) is computed as follows,

Substitute h=−7 in the formula,

H′(21)=H′(21−7)−H′(21)−7=H′(14)−H′(21)−7=41−54−7=137

Substitute h=7 in the formula,

H′(21)=H′(21+7)−H′(21)7=H′(28)−H′(21)7=64−547=107

Take the average of above two calculations of H′(21)

H′(21)=12(137+107)=2314

Thus, H′(21)=2314.

The value of H′(28) is computed as follows,

Substitute h=−7 in the formula,

H′(28)=H(28−7)−H(28)−7=H(21)−H′(28)−7=54−64−7=107

Substitute h=7 in the formula,

H′(28)=H(28+7)−H(28)7=H(35)−H(28)7=72−647=87

Take the average of above two calculations of H′(28)

H′(28)=12(107+87)=97

Thus, H′(28)=97.

The value of H′(35) is computed as follows,

Substitute h=−7 in the formula,

H′(35)=H(35−7)−H(35)−7=H(28)−H(35)−7=64−72−7=87

Substitute h=7 in the formula,

H′(35)=H(35+7)−H(35)7=H(42)−H(35)7=78−727=67

Take the average of above two calculations of H′(35)

H′(35)=12(87+67)=1

Thus, H′(35)=1.

The value of H′(42) is computed as follows,

Substitute h=−7 in the formula,

H′(42)=H(42−7)−H(42)−7=H(35)−H(42)−7=72−78−7=67

Substitute h=7 in the formula,

H′(42)=H(42+7)−H(42)7=H(49)−H(42)7=83−787=57

Take the average of above two calculations of H′(42)

H′(42)=12(67+57)=1114

Thus, H′(42)=1114.

The value of H′(49) is computed as follows,

Substitute h=−7 in the formula,

H′(49)=H(49−7)−H(49)−7=H(42)−H(49)−7=78−83−7=57

Thus, H′(49)=57.

Thus, the required table will be:

t |
14 |
21 |
28 |
35 |
42 |
49 |

H′(t) |
137 |
2314 |
1814 |
1 |
1114 |
57 |

**Graph:**

Use the above information and trace the graph of H′(t) as shown below in Figure 1.

From Figure 1 it is observed that the rate of height with respect to time is decreasing.