Derivative and functions Two graphs are shown in Figure 2.70 and 2.71. One is te

Question

Derivative and functions Two graphs are shown in Figure 2.70 and 2.71. One is temperature as a function of distance and one is illumination as a function of distance. Which is which? What is the average rate at which the temperature is changing when the illumination drops from 75% to 56%? Write the following two statements using Leibniz notation. Suppose you know that at 6 feet the instantaneous rate of change of the temperature is -4.5 degree F per foot. Suppose you also know that at 6 feet the instantaneous rate of change of illumination is -3% per foot. Using the information from part c, estimate the temperature and the illumination at 7.5 ft.

Explanation / Answer

a)

The graph of illumination function is concave up and the graph of Temperature function is concave down.

b)

When the illumination drops from 75% to 56% we need to find the rate of change in temperature during this period of time.

From the table/data given above we can see that illumination changes from 75% to 56% at d = 2 to 5 feet, so

the average rate of change in temperature = (53.5 - 47)/(2 - 5)
= 6.5/(-3) = - 2.17° F per ft

c)

Let Illumination as a function of the distance be I(d) and Temperature as a function of the distance be T(d), then

i) T(d) = T(6) + [dT/dd]d=6 *(d 6)

T(d) = T(6) - 4.5 *(d 6)

ii) Similarly as above

I(d) = I(6) - 3 *(d 6)

d)

Temperature & Illumination at 7.5 feet (using the notations of part (c) are :

T(7.5) = T(6) - 4.5 *(7.5 6) = 43.5 - 4.5*1.5

= 43.5 - 6.75 = - 36.75° F

I(7.5) = I(6) - 3 *(7.5 6) = 53 - 3*1.5

= 53 - 4.5 = 48.5%

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