MATH 236 ELAC FALL 2017 CA 10 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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1 MATH 36 ELAC FALL 7 CA MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) In a certain country, the rate of increase of the population is proportional to the population P(t). In fact, P'(t) =.3P(t). Suppose that initially the country's population is,, and that years later there are, people. Which of the following equations expresses this information mathematically? A) = e.3() B), = e.3() C), =,e.3 D) = e.3t E) none of these ) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) A bacterial culture grows exponentially; that is, P(t) = e kt, where P(t) is the size of the culture at time t hours. Suppose that after hours the size of the culture is 4. What is k (approximately)? ) 3) A colony of bacteria is growing at a rate proportional to the number of bacteria present. At the beginning of an experiment there were about 3 bacteria present. In two hours, the count rose to 3 3 bacteria. At what time will there be 6 3 bacteria present? Enter just a real number rounded up to one decimal place (no units). 3) 4) Suppose that a school of fish in a pond grows according to the exponential law P(t) = Pe kt and suppose that the size of the colony triples in 4 days. If the initial size of the school was, when will the school contain fish? Enter your answer exactly in the form a ln b ln c. 4) Calculate. ) x - 3 dx Enter just a reduced fraction of form a b. ) 6) Initially, a population of rabbits was found to contain 89 rabbits. It was estimated that the population was growing exponentially at the rate of % per day. How long, to the nearest tenth of a day, will it take the population to double? 6) 7) The population of a certain region was million in 9. By 97, it had increased to 3. million. Assuming exponential growth, estimate the population in the year. Enter your answer exactly in the form ae c (no units). 7) Compute the net change of the function. 8) Given f (x) = 3x + 3, compute f(-) - f(-). 8)

2 9) Suppose that at time t, a bacteria culture is increasing at the rate of e.t bacteria per hour. Calculate the total increase in the number of bacteria from t = to t =. Enter your answer in the form a(e b - ) where b is a real number to one decimal place. 9) Find the integral. ) x x dx ) ) The marginal revenue from the sale of compact discs is given by R'(x) = 9-8x and R() =, where R(x) is the revenue in dollars. Find the price-demand equation. ) Find the integral. ) t - 7e t dt ) 3) ( + x 3 )(4 - x ) dx 3) 4) An rock's acceleration at time t is given by a(t) = 6t, and its initial velocity is 3. Find the velocity function v(t). 4) Provide an appropriate response. ) Find f(x) if f'(x) = 3 x and f =. ) Refer to the figure to evaluate the definite integral. 6) Evaluate f(x)dx - 6) f(x) - x + x - - Evaluate. 7) x dx 7)

3 Calculate. 8) e 4x - x dx Enter your answer as a(e b -e c ) + d. 8) 9) (3e 4 - x ) dx Enter your answer as a(b + e c ) with any fractions in reduced form e f. 9) ) A radioactive substance is observed to disintegrate at a rate such that 9 of the original ) amount remains after one year. What is the half-life of the substance? ) Potassium has a half-life of hours. How long will it take for a quantity of potassium to decay to its original size? ) Enter your answer as a real number rounded to one decimal place (no units). ) Plutonium has a decay rate of.3% per year. What is the half life? ) 3) 4C has a half life of 73 years. How old is a piece of charcoal which has lost 9% of its 4C? 3) 4) A certain radioactive substance is decaying at a rate proportional to the amount present. If grams decays to 3. grams in 4 years, how long will it take for 9 grams to decay to 3 grams? 4) ) Krypton 8 gas leaks into the reactor room of an electric power plant. Its half-life is years. How long is it before 99.9% of the krypton decays? Enter your answer as just a real number rounded to one decimal place (no units). ) 6) A fossil was discovered that had about 7% of the 4C level found today in living matter. Given that the decay constant for 4C is., determine the age of the fossil. Enter your answer exactly as just lna b is a real number to decimal places. where a is a real number to one decimal place and b 7) In a chemical reaction, substance A decomposes at a rate proportional to the amount of A present. It is found that 8 g of A will reduce to 4 g in 4. hours. After how long will there be only g left? 6) 7) 8) $ is invested at 6% interest compounded continuously. What is the value of the investment after years? 8) 3

4 9) A bank pays.% interest on deposits. What is the return on a $ deposit after two years if interest is compounded continuously? 9) 3) Eight years ago, $ was deposited in a savings account paying 3% interest compounded continuously. Three years ago, $ was withdrawn from the account. What is the current value of the account? 3) 3) How long would it take $6 to grow to $3, at 7% compounded continuously? Round your answer to the nearest tenth of a year. 3) 3) After years of continuous compounding at.3% the amount in an account is $,7. What was the amount of the initial deposit? 3) 33) An amount is invested at a certain growth rate, k, per year compounded continuously. The doubling time is years. What is the growth rate k? 33) 34) Find the doubling time for an amount invested at a growth rate 7% per year compounded continuously. 34) 3) Find: 4 x - x dx 3) Evaluate. 36) (x 4 + e 4x ) dx 36) Find the value of k that makes the antidifferentiation formula true. 37) (7 - x) - dx = k ln 7 - x + C 37) 38) Find a function f(x) with the following property: f'(x) = x 6 + x4 3 - x3 + 3 x, f() = e. 38) 39) A ball is thrown upward with initial velocity of 44 feet per second. How high will the ball go? (Recall that from physics, it is known that the velocity at time t is 44-3t feet per second.) Enter just an integer (no units). 39) 4) A newspaper is launching a new advertising campaign in order to increase the number of daily subscribers. The newspaper currently (t = ) has 6, daily subscribers and 4) management expects that number, S(t), to grow at the rate of S'(t) = 8t / subscribers per day, where t is the number of days since the campaign began. How long (to the nearest day) should the campaign last if the newspaper wants the number of daily subscribers to grow to 49,? 4) Find a company's total-cost function if its marginal cost function is C'(x) = x - 3 and its fixed cost is $6. 4) 4

5 4) The rate at which an assembly line worker's efficiency E (expressed as a percent) changes with respect to time t is given by E'(t) = 7-6t, where t is the number of hours since the worker's shift began. Assuming that E() = 9, find E(t). 4) 43) An object's acceleration at time t is given by v'(t) = 4t, and its initial velocity v() is. Find the velocity function v(t). 43) Calculate. 44) dx 44) 3 - x 4) e 3x - dx 4) (x + ) - 46) (x - x ) dx 46) -

6 Answer Key Testname: MATH36CA ) C ).69 3) ln 4 4) ln 3 ) - 6) 6.3 days 7) e.7 8) 9) (e. - ) ) - 9 x/3-8 3 x3/ + C ) p = 9-4x ) ln t - 7e t + C 3) x - 3 x3 + x 4-6 x6 + C 4) v(t) = 8t + 3 ) x ) 7 6 7) 3 8) 4 (e - e 8 ) + ln 9) 3 (- + e4 ) ) 6.79 yr ) 39.9 ) 3, years 3) 9,9 years 4).9 yr ) 99.7 ln.7 6) -. 7).6 hours 8) $ ) $.7 3) $99.4 3) 3. years 3) $63. 33) 6.3% 34) 9.9 years 3) 8 3 x3/ - x / + C 6

7 Answer Key Testname: MATH36CA 36) x + e4x 4 + C 37) - 38) f(x) = x7 7 + x - x x3 + e 39) 34 4) 7 days 4) C(x) = x - 3x + 6 4) E(t) = 7t - 3t + 43) v(t) = 7t + 44) ln 3 4) 3 e3-6 46) 73 7