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Hwang Review solns - Holy Cross College Fall Semester 2003...

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Holy Cross College, Fall Semester, 2003 Math 131, Course Review Sheet Answers Professor Hwang 1. What type of function (linear, exponential, power, trig) is the most appropriate for modeling short-term population growth (such as the US population on a scale of a few years, or of a bacterial culture on a scale of ten minutes)? What about modeling long-term population growth? Write a few sentences to justify your answers. Solution In the short term, a linear function is sufficient because the rate of growth is nearly constant over short time intervals. Over the long term, an exponential function is more suitable. Populations grow at a rate proportional to their size, and an exponential function’s derivative is proportional to the function itself. (In the very long term, additional considerations—lack of resources, epidemics, or medical advances, say—can make an exponential model inaccurate.) 2. Let f ( x ) = sin x x for x 6 = 0. (a) What value for f (0) makes f continuous at x = 0? (b) Show that lim x 0 f 0 ( x ) exists. Solution By l’Hˆ opital’s rule, the value f (0) = 1 makes f continuous at x = 0. For part (b), use the quotient rule to see that f 0 ( x ) = x cos x - sin x x 2 . L’Hˆ opital’s rule gives lim x 0 x cos x - sin x x 2 = lim x 0 - x sin x + cos x - cos x 2 x = - lim x 0 sin x 2 = 0 . (c) Answer the same two questions for g ( x ) = e x - 1 x Solution g (0) = 1, lim x 0 g 0 ( x ) = 1 2 . 3. A stone is dropped from a high bridge. After t seconds, the height of the stone is y ( t ) = 150 - 16 t 2 feet. (a) Without using a graphing calculator, sketch the graph of height as a function of time. What is the domain of the function y ? (b) Find the average speed of the stone (including units) over the time interval 0 . 5 t 2, and draw the secant line whose slope represents this average speed.
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Solution The stone hits when y ( t ) = 0, or t = 5 4 6 3 . 062 seconds, so the do- main is [0 , 5 4 6]. The average speed of the stone on the inteval [0 . 5 , 2] is - 40 ft/sec. 0 1 2 3 0 30 60 90 120 150 180 Slope = - 40 ft/sec Height = 146 ft Height = 86 ft 4. A savings account is opened on January 1 with $1000 at a constant interest rate of 2% per year, compounded continuously. (a) Find a formula for the account balance as a function of time, including units. Solution The balance is B ( t ) = 1000(1 . 02) t dollars after t years. (b) How many years does it take for the balance to double? To grow to $1,000,000? Solution About 35 years; about 349 years. (c) Find the average rate of earnings for the first quarter of the third year; express your answer in dollars per day. Solution The third year starts when t = 2 (!), so use t = 2 and t = 2 . 25 to compute the change in balance: $5.163. (I do not plan to ask questions whose wording could mislead you in this way.) (d) Use linear approximation to estimate the amount of interest that accrues on the first day of the tenth year. Solution First, B 0 ( t ) = 1000 ln(1 . 02)(1 . 02) t dollars per year after t years, or about 0 . 054254(1 . 02) t dollars per day after t years. The interest accrued on the first day of the tenth year is roughly 0 . 054254(1 . 02) 9 0 . 065
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5. Let f ( x ) = ( x 2 - 1) 2 . Find the critical point(s) and inflection point(s) of f by making sign diagrams for the first and second derivatives. Classify the critical point(s) as local
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