Mathematics 2122-001
Calculus for Life Sciences II
Fall 2004
Test #3
Instructor: Dr. Alexandra Shlapentokh
Some useful formulas.








1. What is , if and ?
(a) 0
(b) 1
(c) 1/2
(d)
(e) None of the above
2. What is , if and .
(a) 0
(b) 1
(c) 1/2
(d)
(e) None of the above
3. What is if and ?
(a)
(b)
(c)
(d)
(e) None of the above
4. What is if , and ?
(a)
(b)
(c)
(d)
(e) None of the above
5. What is if , and ?
(a)
(b)
(c)
(d)
(e) None of the above
6. What is if and .
(a)
(b)
(c)
(d)
(e) None of the above
7. What is if , , and ?
(a) 1
(b) -1
(c) 0
(d) 1/2
(e) None of the above
8. What is if , , and ?
(a) 1
(b) -1
(c) 0
(d) 1/2
(e) None of the above
9. The distance between the points with coordinates and is
(a)
(b)
(c)
(d)
(e) None of the above
10. What is the formula of the sphere centered at (2,-3,-1) and of radius equal to 2?
(a)
(b)
(c)
(d)
(e) None of the above
11. What is the graph of the equation ?
(a) A circle centered at (0,0) of radius 10.
(b) A sphere centered at (-1,0,-1) of radius .
(c) A sphere centered at (-1,-1,0) of radius .
(d) A sphere centered at (1,0,0) of radius .
(e) None of the above
12. Let . Compute .
(a) 1
(b) 3
(c) 5
(d) 7
13. Find the domain of .
(a)
(b) This function has no domain.
(c)
(d)
(e) None of the above
14. 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
15. Compute for
(a)
(b)
(c) This function does not have this partial derivative.
(d)
(e) None of the above
16. Compute for .
(a)
(b)
(c)
(d) This function does not have this second order partial derivative.
(e) None of the above
17. Compute for

(a)
(b)
(c)
(d) This function does not have this second order partial derivative.
(e) None of the above
18. 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
19. 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
20. 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
21. 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
22. Let be a solution to the differential equation with . Then what is
(a)
(b) Cannot be determined.
(c)
(d)
(e) None of the above
23. Let be a solution to the differential equation with . Then what is
(a)
(b) Cannot be determined.
(c)
(d)
(e) None of the above
24. A certain population of insects, as a function of time measured in days, is growing exponentially with the uniform rate equal to 0.02 of the population size. Suppose that the initial population contains 1000 insects. What will the population be at time days?
(a) insects
(b) insects
(c) Cannot be determined from these data.
(d) 2000
(e) None of the above
25. A population of mice in a house was doubling every year. How long does it take this population to triple if the growth rate is always proportional to the population size?
(a) years
(b) years
(c) years
(d) None of the above
26. If a radiation dose of 2 rad kills 10% of cancer cells, how much radiation would kill 99% of the cells? (Assume that the cancer cells decrease at the rate proportional to their number as a function of the radiation dose.)
(a) Cannot be determined from these data.
(b) rads
(c) rads
(d) rads
(e) None of the above
27. Suppose in a certain population of animals 5% of individuals give birth a year with 2 young surviving per birth on the average, and a uniform death rate is 1% when time is measured in years. What is the population size as a function of time, if the initial population was 5000 animals?
(a)
(b)
(c)
(d)
(e) None of the above
28. Let be a solution to , with . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
29. Let be a solution to . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
30. Suppose in a certain population of animals 5% of the individuals give birth a year with 4 young surviving per birth on the average, and a uniform death rate is 3% when time is measured in years. Assume also that each year 1000 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)
31. Let be a solution to , with . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
32. 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
33. Let be a solution to with . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
34. Suppose a population of roaches in a house followed a logistic model with the maximum population equal to 50000 roaches. Initially the house had 100 roaches. After 1 year the house had 2000 roaches. How many roaches will live in the house after t years?
(a)
(b)
(c)
(d)
(e) None of the above
Key
1(a), 2(a), 3(c), 4(d), 5(d), 6(b), 7(c), 8(c), 9(b), 10(d), 11(c), 12(a), 13(c), 14(b), 15(a), 16(c), 17(e), 18(b), 19(c), 20(a), 21(d), 22(c), 23(d), 24(b), 25(c), 26(d), 27(c), 28(d), 29(d), 30(b), 31(b), 32(b), 33(c), 34(c)





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