#### To determine

**To construct:** The table of estimated values of W′ and sketch its graph.

#### Answer

The table of the estimated values of W′(x) will be:

x |
15.5 |
17.7 |
20 |
22.4 |
24.4 |

W′(x) |
−2.82 |
−3.845 |
−4.52 |
−6.98 |
−9.75 |

The unit of W′(x) is gram per degree (g/∘C).

#### Explanation

**Given:**

The table represents the weight gained at temperature *x.*

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

limh→0W(x+h)−W(x)h≈W(x+h)−W(x)h

**Calculation:**

Calculation of W′(15.5) is as follows,

Substitute h=2.2 in the formula

W′(15.5)=W(15.5+2.2)−W(15.5)2.2=W(17.7)−W(15.5)2.2=31−37.22.2=−2.82

Thus, W′(15.5)=−2.82.

Calculation of W′(17.7) is as follows,

Substitute h=−2.2 in the formula

W′(17.7)=W(17.7−2.2)−W(17.7)−2.2=W(15.5)−W(17.7)−2.2=37.2−31−2.2=−2.82

Substitute h=2.3 in the formula

W′(17.7)=W(17.7+2.3)−W(17.7)2.3=W(20)−W(17.7)2.3=19.8−312.3=−4.87

Take the average of above two calculations of W′(17.7)

W′(17.7)=12(−2.82−4.87)=−3.845

Thus, W′(17.7)=−3.845.

Calculation of W′(20) is as follows,

Substitute h=−2.3 in the formula,

W′(20)=W(20−2.3)−W(20)−2.3=W(17.7)−W(20)−2.3=31−19.8−2.3=−4.87

Substitute h=2.4 in the formula,

W′(20)=W(20+2.4)−W(20)2.4=W(22.4)−W(20)2.4=9.7−19.82.4=−4.17

Take the average of above two calculations of W′(20),

W′(20)=12(−4.17−4.87)=−4.52

Thus, W′(20)=−4.17.

Calculation of W′(22.4) is as follows,

Substitute h=−2.4 in the formula,

W′(22.4)=W(22.4−2.4)−W(22.4)−2.4=W(20)−W(22.4)−2.4=19.8−9.7−2.4=−4.2083

Substitute h=2 in the formula,

W′(22.4)=W(22.4+2)−W(22.4)2=W(24.4)−W(22.4)2=−9.8−9.72=−9.75

Take the average of above two calculations of W′(22.4)

W′(22.4)=12(−4.2083−9.75)=−6.97915

Thus, W′(22.4)=−6.97915.

Calculation of W′(24.4) is as follows,

Substitute h=−2.4 in the formula

W′(24.4)=W(24.4−2)−W(24.4)−2=W(22.4−2)−W(24.4)−2=9.7+9.8−2=−9.75

Thus, W′(24.4)=−9.75

Thus, the required table will be:

x |
15.5 |
17.7 |
20 |
22.4 |
24.4 |

W′(x) |
−2.82 |
−3.845 |
−4.52 |
−6.98 |
−9.75 |

**Graph:**

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

From Figure 1, it is observed that W′(x) is negative.

The unit of W′(x) is gram per degree (g/∘C).