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HH_Sec_11_5

# HH_Sec_11_5 - Math 10B Lecture Examples Section 11.5 Growth...

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(9/1/08) Math 10B. Lecture Examples. Section 11.5. Growth and Decay Example 1 Match problems (I) through (IV) below to differential equations (a) through (d) and to the slope fields in Figures 1 through 4. (I) The thickness of the ice on a lake grows at a rate that is proportional to the reciprocal of its thickness. Find the thickness y = y ( t ) as a function of the time t . (II) A population grows at a rate proportional to its size. Find the population y = y ( t ) as a function of the time t . (III) A hot potato is taken out of the oven at time t = 0 into a kitchen that is at 20 Celsius. The rate of change of the potato’s temperature is proportional to the difference between its temperature and that of the kitchen. Find the temperature y = y ( t ) of the potato as a function of t . (IV) Find a function y = y ( t ) whose rate of change with respect to t is - 2 t . (a) dy dt = 0 . 2 y (b) dy dt = 20 y (c) dy dt = - 2 t (d) dy dt = - 2( y - 20) t 5 10 y 100 200 300 400 t 500 1000 y 50 100 150 200 FIGURE 1 FIGURE 2 t 1 2 3 4 5 6 y 10 20 30 40 50 t 1 2 3 y 10 20 30 40 50 60 FIGURE 3 FIGURE 4 Answer: Problem I goes with equation (b) and the slope field in Figure 2. Problem II goes with equation (a) and Figure 1. Problem III goes with equation (d) and Figure 4. Problem IV goes with equation (c) and Figure 3. Lecture notes to accompany Section 11.5 of Calculus by Hughes-Hallett et al. 1

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Math 10B. Lecture Examples. (9/1/08) Section 11.5, p. 2 Example 2 (a) Solve the differential equation (a) dy dt = 0 . 2 y from Example 1 (a population) with the initial condition y (0) = 50 and draw its graph with the corresponding slope field.
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