Water temperature affects the growth rate of brook trout. The table shows the amount of weight gained by brook trout after 24 days in various water temperatures.

Question

Temperature (°C) 15.5 17.7 20.0 22.4 24.4
Weight gained (g) 37.0 30.7 20.0 9.0 −9.9
If W(x) be the weight gain at temperature x, construct a table of estimated values for W′. (Use a one-sided difference quotient with an adjacent point for the first and last values, and the average of two difference quotients with adjacent points for all other values. Round your answers to two decimal places.)
x = 15.5, 17.7, 20.0, 22.4, 24.4

Answer 1

To find the estimated values for W′, we need to calculate the difference quotient using adjacent points.

For x = 15.5, we can use the one-sided difference quotient with the adjacent point x = 17.7.

W′(15.5) = (W(17.7) - W(15.5)) / (17.7 - 15.5)
= (30.7 - 37.0) / (17.7 - 15.5)
= -6.3 / 2.2
≈ -2.86

For x = 17.7, we can use an average of two difference quotients: one with the adjacent point x = 15.5 and one with the adjacent point x = 20.0.

W′(17.7) ≈ [(W(17.7) - W(15.5)) / (17.7 - 15.5) + (W(20.0) - W(17.7)) / (20.0 - 17.7)] / 2
≈ [(-6.3 / 2.2) + (20.0 - 30.7) / (20.0 - 17.7)] / 2
≈ (-2.86 + (-10.7)) / (2.3) / 2
≈ -13.56 / 2.3 / 2
≈ -2.94

For x = 20.0, we can use an average of two difference quotients: one with the adjacent point x = 17.7 and one with the adjacent point x = 22.4.

W′(20.0) ≈ [(W(20.0) - W(17.7)) / (20.0 - 17.7) + (W(22.4) - W(20.0)) / (22.4 - 20.0)] / 2
≈ [(20.0 - 30.7) / (20.0 - 17.7) + (9.0 - 20.0) / (22.4 - 20.0)] / 2
≈ (-10.7 / 2.3) + (-11.0 / 2.4)] / 2
≈ -4.65 + (-4.58)] / 2
≈ -9.23 / 2
≈ -4.62

For x = 22.4, we can use an average of two difference quotients: one with the adjacent point x = 20.0 and one with the adjacent point x = 24.4.

W′(22.4) ≈ [(W(22.4) - W(20.0)) / (22.4 - 20.0) + (W(24.4) - W(22.4)) / (24.4 - 22.4)] / 2
≈ [(9.0 - 20.0) / (22.4 - 20.0) + (-9.9 - 9.0) / (24.4 - 22.4)] / 2
≈ (-11.0 / 2.4) + (-18.9 / 2) / 2
≈ -4.58 + (-9.45) / 2
≈ -14.03 / 2
≈ -7.02

For x = 24.4, we can use the one-sided difference quotient with the adjacent point x = 22.4.

W′(24.4) = (W(24.4) - W(22.4)) / (24.4 - 22.4)
= (-9.9 - 9.0) / (24.4 - 22.4)
= -18.9 / 2
= -9.45

Therefore, the table of estimated values for W′ is as follows:

x | W′
---------------------
15.5 | -2.86
17.7 | -2.94
20.0 | -4.62
22.4 | -7.02
24.4 | -9.45