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ASU MAT 210 - Rates of Change Practice

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Mat 210 - Rates of Change Practice.Quick summary:"Average Rate of Change" is the slope over an interval of a function. The line thatconnects the two points is called a secant line."Instantaneous Rate of Change" is the slope at a point of a function. The line that passesthrough this point (and just "nicks" the graph) is called a tangent line. For a tangent lineto exist at a point, the function's graph must be smooth and continuous.The units of a slope value is always: inputofunitsoutputofunitsProblem 1: Consider the given table, which is a city's population as a function of years.Year: 1980 1985 1990 1995 2000Population (thousands): 21 24 30 38 50a) Find the average rate of change of the city's population between the years 1980and 1995.b) Find a model for this population growth (use x = 0 for 1980).c) Carefully approximate the instantaneous rate of change of the city's population in1992. Explain your methods.Problem 2: The given table shows the profit (in $millions) of a company as a function ofmoney spent (in $thousands) in advertising promotion of their product.Money spent on advertising: 50 100 150 200 250 300 350Profit: 2.4 5.1 6.4 6.5 5.8 4.3 1.9a) Determine the rate of change in the company's profits as they increase advertisingspending from $150,000 to $300,000.b) Find a model for this set of data.c) Determine the average rate of change in the company's profits as they increasespending from $75,000 to $220,000.d) Approximate the instantaneous rate of change in the company's profits if theyspend $175,000. What does this mean in the context of the problem?e) Approximate the instantaneous rate of change in the company's profits if theyspend $300,000. What does this mean in the context of the problem? How doesthis compare to part d?f) Using your model, determine the optimal amount that the company should spendon advertising. Explain your methods. What would be the company's profits?Explain the significance of the tangent line at this point.Problem 3: (Problem 20, section 3.1) The percentage of high school seniors who haveused a graphing calculator can be modeled by the function:tetp725.0991100)(Where t is the number of years since 1986 and p is percent.a) On your own sheet of paper, sketch a graph of this function over the interval0  t  14. According to the model, what percent of high school seniors haveused a graphing calculator as of 1999?b) Determine the average rate of change of calculator use between the years 1993and 1999.c) During what one-year period (e.g. 1986-1987, etc.) did calculator usage increasethe fastest? By how fast? d) Is there any time during 1986-2000 when calculator usage is decreasing?Explain.e) On your graph, carefully determine the point at which the rate of change incalculator usage was growing the fastest. Sketch in a tangent line at this point,and carefully determine its slope.f) What do you notice about this tangent line compared to other possible tangentlines on this graph? Is there a relationship between this point and concavity?Explain.Problem 4: (Curve sketching) The population of a city is modeled by a function P(t),where t is time in years since 1980, and P is the population. The following information isgiven: P(0) = 25,000 P(17) = 45,000dtdP is negative for t > 12, and positive for t < 12.a) From this data, sketch a possible graph of P(t). Use 0  t  20 as your interval.b) What has occurred in the city's growth rate in 1992?c) The information given is minimal, but some assumptions can be made. Considerthe following assumptions and explain if they are valid assumptions (supported bythe information) or not valid.1) The population in 1994 was greater than the population in 1997.2) The population in 1991 was greater than the population in 1997.3) The average rate of change in the city's population from 1992 to 2000 wasdecreasing.4) In 1985, the city was experiencing growth.5) It is possible to determine the population of the city in 1992.6) There were more people in the city in 2000 than in


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ASU MAT 210 - Rates of Change Practice

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