7) Explain how the authors estimated H, (10,20) and H, (10,20) in Example 4 on p
ID: 2890624 • Letter: 7
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7) Explain how the authors estimated H, (10,20) and H, (10,20) in Example 4 on page 761 14.1 THE PARTIAL DERIVATIVE 76 Figure 14.6 shows the contour diagram for the temperature H(x, t) (inC) in a room as a function of distance x (in meters) from a heater and time t (in minutes) after the heater has been turned on What are the signs of H (10, 20) and Ht (10, 20)? Estimate these partial derivatives and explain the answers in practical terms Example 4 t (minutes) 60 50 40 30 20 10 32 20 15 10 14 r (meters) 5 10 15 20 25 30 Figure 14.6: Temperature in a heated room. Heater at x = 0 is turned on at t 0 The point (10.20) is nearly on the H = 25 contour. As x increases, we move toward the H = 20 contour, so H is decreasing and H, (10.20) is negative. This makes sense because the H = 30 contour is to the left: As we move further from the heater, the temperature drops. On the other hand, as t increases, we move toward the H 30 contour, so H is increasing; as t decreases H decreases Thus, Ht(10, 20) is positive. This says that as time passes, the room warms up Solution To estimate the partial derivatives, use a difference quotient. Looking at the contour diagram, 20 contour about 14 units to the right of the point (10,20) we see there is a point on the H Hence, H decreases by 5 when r increases by 14, so we find Rate of change of H with respect to x = H1 (10, 20) ~-~:-0.30°C/meter 14 This means that near the point 10 m from the heater, after 20 minutes the temperature drops about 0.36, or one third, of a degree, for each meter we move away from the heater To estimate (10, 20), we notice that the H = 30 contour is about 32 units directly above the point (10, 20). So H increases by 5 when t increases by 32. Hence, Rate of change of H with respect to t = Ht (10.20) = 0.10°C/minute 32 This means that after 20 minutes the temperature is going up about 0.16, or 1/6, of a degree each minute at the point 10 m from the heater.Explanation / Answer
We need to understand what partial derivatives Hx(x,t) and Ht(x, t) means.
Hx(x,t) is rate of change of H with respect to x treating time as a constant.
Hx(x,t)=(change in H)/change in x)...... when time does not change.
Thus Hx(10,20)= (change in H at x=10)/(change in x)
We move in horizontal direction because time is constant in horizontal direction.
From the graph , when x changes from 10 to 24 , H changes from 25 to 20.
Thus change in x =24-10= 14. and change in H =(20-25)= -5.
Thus Hx(10,20)= -5/14 = -0.36.
Similarly, Ht(x,t) is rate of change of H with respect to time treating x as constant.
Since x has to constant , we will see change in H in vertical direction.
Ht(10,20) = (Change in H in vertical direction)/(change in time)
From the graph , change in H = 30-25= 5 degree celsius
change in time= 52-20 = 32 min
Hence ,Ht(10,20) = 5/32 = 0.16.
Hope this helped you.
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