108年 - 108東吳大學_碩士班招生考試_企業管理學系C 組：微積分#108282

科目：研究所、轉學考(插大)-微積分 | 年份：108年 | 選擇題數：0 | 申論題數：10

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所屬科目：研究所、轉學考(插大)-微積分

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申論題 (10)

1. City and Country Cycles finds that if it sells x racing bicycles per month, it’s costs will be C(x) = 20x+72,000 and its revenue will be R( x ) = −3 x 2 + 1,800 x (both in dollars). a. Find the store’s break-even points. b. Find the number of bicycles that will maximize profit, and the maximum profit.
2. If a person’s temperature after x hours of strenuous exercise is T (x ) = x 3 (4 − x 2 ) + 98.6 degrees Fahrenheit (for 0 ≤ x ≤ 2 ), find the rate of change of the temperature after 1 hour.
3. Based on a recent study, the “happiness” of people who live in a country whose average temperature is t degrees Fahrenheit is given by h(t ) = 8.2 − (0.01t − 2.8) 2 , for 35 ≤ t ≤ 72 (“Happiness” was rated from 1 = “not at all happy” to 4 “very happy”.) Find h (40) and h ' (40) . Interpret your answer.

4. A politician estimates that by campaigning in a country for x days, she will gain 2x (thousand) votes, but her campaign expenses will be 5 x 2 + 500 dollars. She wants to campaign for the number of days 2x that maximizes the number of votes per dollar, f ( x) =62a2b0a6b5298.jpg . For how many days should shcampaign?
5. A liquor distributor wants to increase its revenues by discounting its bestselling liquor. If the demand function for this liquor is D(p) = 60-3p, where p is the price per bottle, and if the current price is $15, will the discount succeed?
6. In 2013, Amazon’s annual revenue was 74 billion dollars and growing at the rate of 3.2x+17.4 billion dollars per year, where x is the number of years since 2013. Find a formula to predict Amazon’s annual revenue at any time and use your formula to predict their revenues in 2020.

7. An epidemic is spreading at the rate of 62a2b0db98168.jpg new cases per day, where t is the number of days since the epidemic began. Find the total number of new cases in the first 10 days of the epidemic.

8. A subject can perform a task at the rate of62a2b10772801.jpg tasks per minute at time t minutes. Find the total number of tasks performed from time t = 0 to time t = 12.

9. A company’s output is given by the Cobb-Douglas production function 62a2b136bce90.jpg, where L and K are the numbers of units of labor and capital. Each unit of labor costs $50 and each unit of capital costs $60, and $1500 is available to pay for labor and capital. How many units of labor and of capital should be used to maximize production?
10. A clothing designer’s sales S depends on x, the amount spent on TV advertising, and y, the amount spent on print advertising (both in ten thousands of dollars), according to the function S(x,y) = 60x2 + 90y2 - 6xy + 200. Find S x (2,3) and S y (2,3), and interpret these numbers.