Mathematics 2122-002
Calculus for Life Sciences II
Fall 2002
Test #3

Instructor: Dr. Alexandra Shlapentokh
1. The distance between the points with coordinates and is
(a)
(b)
(c)
(d)
(e) None of the above
2. What is the formula of the sphere centered at (-1,3,2) and of radius equal to 3?
(a)
(b)
(c)
(d)
(e) None of the above
3. What is the graph of the equation ?
(a) A circle centered at (0,0) of radius 10.
(b) A sphere centered at (0,0,0) of radius .
(c) A sphere centered at (1,0,0) of radius .
(d) A sphere centered at (-1,0,0) of radius .
(e) None of the above
4. Let . Compute .
(a) -1
(b) -3
(c) -5
(d) -7
5. Find the domain of .
(a)
(b) This function has no domain.
(c)
(d)
(e) None of the above
6. Find the domain of .
(a) This function is not defined for any pair of real numbers.
(b) The domain consists of all pairs of real numbers.
(c)
(d)
(e) None of the above
7. Find the domain of .
(a)
(b)
(c)
(d) The domain consists of all pairs of real numbers.
(e) None of the above
8. Compute for
(a)
(b)
(c) This function does not have this partial derivative.
(d)
(e) None of the above
9. Compute for
(a)
(b)
(c) This function does not have this partial derivative.
(d)
(e) None of the above
10. Compute for
(a)
(b)
(c)
(d) This function does not have this second order partial derivative.
(e) None of the above
11. Compute for
(a)
(b)
(c)
(d) This function does not have this second order partial derivative.
(e) None of the above
12. Consider the differential equation , where is a function of .
(a) This is a first order differential equation.
(b) This is a second order differential equation.
(c) This is a third order differential equation.
(d) This differential equation has no order.
(e) None of the above
13. Consider the differential equation , where is a function of .
(a) This is a non-linear differential equation.
(b) This is a linear differential equation with non-constant coefficients.
(c) This is a linear differential equation with constant coefficients.
(d) This differential equation has no coefficients.
(e) None of the above
14. Consider the differential equation .
(a) This is a non-linear differential equation.
(b) This is a linear differential equation with non-constant coefficients.
(c) This is a linear differential equation with constant coefficients.
(d) This differential equation has no coefficients.
(e) None of the above
15. Which of the functions below is a solution to differential equation ? (Here is a function of .)
(a)
(b)
(c)
(d)
(e) None of the above
16. Let be a solution to the differential equation with . Then what is
(a)
(b) Cannot be determined.
(c)
(d)
(e) None of the above
17. Let be a solution to the differential equation with . Then what is
(a)
(b) Cannot be determined.
(c)
(d)
(e) None of the above
18. Let be a solution to the differential equation with . Then what is
(a)
(b) Cannot be determined.
(c)
(d)
(e) None of the above
19. A certain population of insects, as a function of time measured in days, is growing exponentially with the uniform rate equal to 0.01 of the population size. Suppose that the initial population contains 100 insects. What will the population be at time days (rounded to an integer)?
(a) Approximately 111 insects
(b) Approximately 109 insects
(c) Approximately 107 insects
(d) Approximately 105 insects
(e) None of the above
20. A population of rabbits on an island triples every 5 years. How long does it take this population to quadruple if the growth rate is always proportional to the population size?
(a)
(b)
(c)
(d)
(e) None of the above
21. If a radiation dose of 1 rad kills 2% of cancer cells, how much radiation would kill 90% of the cells? (Assume that the cancer cells die at the rate proportional to the number of cells as a function of the radiation dose.)
(a)
(b)
(c)
(d)
(e) None of the above
22. Suppose a certain radioactive element has a half-life of 10000 years. How long before 99 % of the original amount is lost? Round your answer to the nearest year.
(a) 66441 years
(b) 66439 years
(c) 66443 years
(d) 66437 years
(e) None of the above
23. Suppose a certain population of animals has a uniform birth rate of 10% with 2 young surviving per birth on the average and a uniform death rate of 5% when time is measured in years. What is the population size as a function of time, if the initial population was 100 animals?
(a)
(b)
(c)
(d)
(e) None of the above
24. Let be a solution to , with . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
25. Let be a solution to . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
26. Let be a solution to , with . What is ?
(a)
(b)
(c)
(d)
(e) None of the above
27. Let be a solution to , with . What is ?
(a)
(b)
(c)
(d)
(e) None of the above
28. Suppose a certain population of animals has a uniform birth rate of 10% with 2 young surviving per birth on the average and a uniform death rate of 5% when time is measured in years. Assume also that each year 100 animals are moving into the area. Which of the following differential equations would provide the best model for the population size as a function of time?
(a)
(b)
(c)
(d)
(e)
29. Let be a solution to , with . Then what is rounded to one decimal point?
(a) 4.1
(b) 2.7
(c) 5.2
(d) 6.3
(e) None of the above
30. Let be a solution to , with . Then what is rounded to the nearest integer?
(a) 10
(b) 20
(c) 30
(d) 40
(e) None of the above
31. Let be a solution to . Then which of the following statements are true? (Here is a constant.)
(a)
(b)
(c)
(d)
(e) None of the above
32. Let be a solution to with . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
33. Let be a solution to . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
34. Suppose a population of mice in a house followed a logistic model with the maximum population equal to 300 mice. Initially the house had 10 mice. After 1 year the house had 50 mice. How many mice will live in the house after 5 years? (Round your answer to the nearest integer.)
(a) 295
(b) 296
(c) 297
(d) 299
(e) None of the above
Key
1(b), 2(b), 3(c), 4(c), 5(d), 6(b), 7(e), 8(b), 9(a), 10(a), 11(e), 12(c), 13(c), 14(a), 15(c), 16(c), 17(d), 18(d), 19(d), 20(d), 21(e), 22(b), 23(e), 24(c), 25(c), 26(b), 27(b), 28(b), 29(b), 30(a), 31(a), 32(a), 33(c), 34(d)





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