6. A potato is placed in an oven, and the potato\'s temperature F (in degrees Fa

Question

6. A potato is placed in an oven, and the potato's temperature F (in degrees Fahrenheit) at various points in time is taken and recorded in the following table. Time t is measured in minutes. t F(t) 0 70 15 180.5 30-251 45 296 60 324.5 75 342.8 90 354.5 Table 1.8.6: Temperature data for the potato. a. Use a central difference to estimate F (60). Use this estimate as needed in subsequent questions. b. Find the local linearization y L(t) to the function y = F(t) at the point where a = 60. c. Determine an estimate for F(63) by employing the local linearization d. Do you think your estimate in (c) is too large or too small? Why?

Explanation / Answer

a)

F'(60) =[F(75)-F(45)]/(75-45)

F'(60) =(342.8 - 296)/(75-45)

F'(60) = 46.8/30

F'(60) = 1.56

b)

L(t)=F(60) +(F'(60))(t-60)

L(t)=324.5 +(1.56)(t-60)

L(t)=324.5 +1.56t - 93.6

L(t)=230.9 +1.56t

c)

F(63) L(63)

F(63) 230.9 +(1.56*63)

F(63) 329.18

d)

estimate in (c) is a bit large

between t =60 and 75

average rate of change = [342.8-324.5]/(75-60) = 1.22

average rate of change is less than the cental diffference estimate F'(60)

so estimate in (c) is bit large .

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