SCORE Problem Points Score 1 6

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1 MA Spring 2015 Exam 1 Name: Section: Do not remove this scoring page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing calculator during the exam, but NO calculator with a Computer Algebra System (CAS) or a QWERTY keyboard is permitted. Absolutely no cell phone use during the exam is allowed. SCORE Problem Points Score Score Spell name correctly

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3 1. Rewrite and simplify the expressions (a) x3 (2x 2 ) 5 x 7 (b) x2n+3 x 3n 3 x n 2 2. Let f (x) = 3x 2 + 3x + 6. Let P be the point (1, f (1)) and Q be the point (3, f (3)). Find the slope and equation of the line between P and Q. 3. As dry air moves upward, it expands and cools. If the ground temperature is 20 C and the temperature at a height of 1 km is 10 C, express the temperature T (in C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. At what height is the temperature 10 C? Page 3 of 10

4 4. One of the main quantities that epidemiologists try to measure for infectious diseases is the so-called basic reproduction number, R 0. Biologically, this is the expected number of new infections that an infected individual will produce when introduced into a completely susceptible population. We can try to modify this by introducing vaccination to control the probability of an outbreak of the disease. We want to know the fraction of the population that we have to vaccinate to achieve a target outbreak probability. If v is the vaccination fraction, then the outbreak probability as a function of v is 1 P = 1 R 0 (1 v). Find the inverse of this function to obtain v, the vaccination coverage needed, as a function of P, the given target outbreak probability. 5. Let f (x) = 2x 5 and g(x) = x 2 + 3x + 4. (a) Find ( f g)(x). (b) Find (g f )(x). (c) Find ( f f )(x). (d) Find (g g)(x). Page 4 of 10

5 6. The length of an insect (L, in millimeters) as a function of the temperature T during development (measured in degrees Celsius) is given by L(T) = 10 + T 12. The volume V of the insect (measured in cubic millimeters) is a function of the length given by V(L) = 5 4 L3. The mass, M, (measured in milligrams) is a function of the volume according to M(V) = 1.47V. (a) Find the mass, M, of the insect as a function of temperature, T, during development. (b) How much would an insect weigh that developed at 24 C? 7. Suppose that the graph of y = log 2 (x) is drawn on a coordinate grid where the unit of measurement is one centimeter. How many kilometers to the right of the origin do we have to move before the height of the curve reaches 25 cm? Page 5 of 10

6 8. Consider the data in the following table (adapted from Parasitoids by H. C. F. Godfray), describing the number of wasps that can develop inside caterpillars of different masses. Mass of Caterpillar (g) Number of Wasps (a) What is the slope of the line connecting the first data point and the last data point? (b) Find the equation of this line. (c) How many wasps does the function predict would develop in a caterpillar with a mass of 0.8 g? (d) How many wasps does the function predict would develop in a caterpillar with a mass of 1.25 g? Page 6 of 10

7 9. A researcher is trying to determine the doubling time of a population of the bacterium Giardia lamblia. He starts a culture in a nutrient rich solution and initiates the process with 30 CFU/ml (colony forming units per milliliter) of the bacteria. He samples the solution in 24 hours and estimates that there are 175 CFU/ml. Assume that you can model the population growth of the bacteria as N(t) = ae rt CFU/ml where t measures the number of hours after the beginning of the culture. (a) Find a and r. (b) Use the model to estimate the doubling time for the culture. 10. A fish farmer has 5000 shrimp fry that he releases into his pond. The number of shrimp increases by 8% per month. (a) Write an expression for S(t), the population s size at t months after the initial release, assuming S(0) = (b) What is the population of the shrimp in the pond after 6 months? (c) The farmer will not start harvesting shrimp until there are 15,000 shrimp in the pond. How many months must he wait? Page 7 of 10

8 11. The table, supplied by Andrew Read, shows the results of an experiment involving malarial parasites. The time t is measured in days and N is the number of parasites per microliter of blood. t (days) N 228 2,357 12,750 26, ,331 2,217,441 6,748,400 We plot the data on a semilog plot and find that the line of best fit is log(n) = t (a) Find the functional relationship for N(t). (b) Estimate the population on day 2.5. Page 8 of 10

9 12. It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function. Cave Area (m 2 ) Number of Species La Escondida 18 1 El Escorpion 19 1 El Tigre 58 1 Misión Imposible 60 2 San Martin El Arenal La Ciudad Virgen Taking data of the number of species S of bats living in caves in central Mexico and the surface area A of the caves in which they were found, our scientists plotted the species-area data in a log-log graph. The line of best fit is given by log S = log A (a) Find the function that relates the number of species, S, to the surface area, A of the cave. (b) If you discover that four species of bats live in a cave, estimate the area of the cave. (c) The cave called El Sapo near Puebla, Mexico, has a surface area of A = 205 m 2. Use the model to estimate the number of bat species you would expect to find in that cave. Page 9 of 10

10 13. The sequence {a n } has first term is a 1 = 2 and is defined recursively by a n+1 = 2n a n + 2. Find a 1, a 2, a 3,a 4, a 5, a 6, a 7, a An invasive species is a plant or animal that is not native to a specific location and has a tendency to spread and is believed to cause damage to the environment, human economy and/or human health. The silver carp is such species here in Kentucky. It competes with other fish, such as shad, for food. It has a much greater birth rate than the native fish. Assume that KFWS found that there were 200 silver carp introduced into Grayson Lake and this population increases at the rate of 11% per year. There is a indigenous population of shad in the lake which numbers about 10,000 but increases at the rate of 1.5% per year. When will the two populations be equal? END OF TEST Page 10 of 10