Chapter 3 | Derivatives
275
Years since 1800 Population (millions)
Time after dropping (s)
Position (m)
1
0.8795
0
0
11
1.040
1
−1
21
1.264
2
−2
31
1.516
3
−5
41
1.661
4
−7
51
2.000
5
−14
61
2.634
169. [T] a. Using a calculator or computer program, find the best-fit quadratic curve to the data. b. Find the derivative of the position function and explain its physical meaning. c. Find the second derivative of the position function and explain its physical meaning. 170. [T] a. Using a calculator or computer program, find the best-fit cubic curve to the data. b. Find the derivative of the position function and explain its physical meaning. c. Find the second derivative of the position function and explain its physical meaning. d. Using the result from c. explain why a cubic function is not a good choice for this problem. The following problems deal with the Holling type I, II, and III equations. These equations describe the ecological event of growth of a predator population given the amount of prey available for consumption. 171. [T] The Holling type I equation is described by f ( x ) = ax , where x is the amount of prey available and a >0 is the rate at which the predator meets the prey for consumption. a. Graph the Holling type I equation, given a =0.5. b. Determine the first derivative of the Holling type I equation and explain physically what the derivative implies. c. Determine the second derivative of the Holling type I equation and explain physically what the derivative implies. d. Using the interpretations from b. and c. explain why the Holling type I equation may not be realistic.
71
3.272
81
3.911
91
4.422
Table 3.4 Population of London Source: http://en.wikipedia.org/wiki/ Demographics_of_London.
167. [T] a. Using a calculator or a computer program, find the best-fit linear function to measure the population. b. Find the derivative of the equation in a. and explain its physical meaning. c. Find the second derivative of the equation and explain its physical meaning. 168. [T] a. Using a calculator or a computer program, find the best-fit quadratic curve through the data. b. Find the derivative of the equation and explain its physical meaning. c. Find the second derivative of the equation and explain its physical meaning. For the following exercises, consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table.
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