Mathematics 2122-001
Calculus for Life Sciences II
Fall 2003
Final Examination
Instructor: Dr. Alexandra Shlapentokh
Some useful formulas.





The arguments of all trigonometric functions occurring in the test are in radians.
1. What is if and ?
(a)
(b)
(c)
(d)
(e) None of the above
2. What is if , and ?
(a)
(b)
(c)
(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) 1
(b) -1
(c) 0
(d) 1/2
(e) None of the above
6. What is if , , and ?
(a) 1
(b) -1
(c) 0
(d) 1/2
(e) None of the above
7. Determine which of the following equalities are true for all .
(a)
(b)
(c)
(d)
(e) None of the above
8. Determine which of the following equalities are true for all .
(a)
(b)
(c)
(d)
(e) None of the above
9.
(a) =1.
(b) =0.
(c) does not exist.
(d) =1/2.
(e) None of the above
10.
(a) =1.
(b) =2.
(c) does not exist.
(d) =3.
(e) None of the above
In Problems 11 – 16 compute the derivative of the given function.
11.
(a)
(b)
(c)
(d)
(e) None of the above
12.
(a)
(b)
(c)
(d)
(e) None of the above
13.
(a)
(b)
(c)
(d)
(e) None of the above
14.
(a)
(b)
(c)
(d)
(e) None of the above
15.
(a)
(b)
(c)
(d)
(e) None of the above
16.
(a)
(b)
(c)
(d)
(e) None of the above
In Problems 17 – 21 determine the smallest period of the given function.
17.
(a) This function is not periodic.
(b)
(c)
(d)
(e) None of the above
18.
(a) This function is not periodic.
(b)
(c)
(d)
(e) None of the above
19.
(a) This function is not periodic.
(b)
(c)
(d)
(e) None of the above
20.
(a) 1
(b) 2
(c)
(d)
(e) None of the above
21. _
(a)
(b)
(c)
(d)
(e) None of the above
In Problems 22–24 determine the amplitude of the given function.
22. _
(a) 1
(b) 2
(c) 3
(d) 4
(e) None of the above
23.
(a) 1
(b) 2
(c) 3
(d) 4
(e) None of the above
24.
(a)
(b) 2
(c)
(d) 10
(e) None of the above
25. A certain variable varies sinusoidally between -1 and 3 with a period of 1 day. The variable reaches its highest value at 1 pm each day. Find a formula for .
(a) , where is measured in hours and corresponds to 12 am.
(b) , where is measured in hours and corresponds to 12 am.
(c) , where is measured in hours and corresponds to 12 am.
(d) , where is measured in hours and corresponds to 12 am.
(e) None of the above
26. Suppose an antiderivative of is . Then
(a) cannot be determined.
(b) .
(c) .
(d) .
(e) None of the above
27. Suppose has an antiderivative . Then
(a) has no other antiderivatives.
(b) has infinitely many antiderivatives.
(c) has one more antiderivative.
(d) has two more antiderivatives.
(e) None of the above
28. Let and be antiderivatives of the same function. Then
(a) is a constant.
(b) is equal to .
(c) is equal to .
(d) is equal to .
(e) None of the above
29. Suppose , where is a constant. Then any antiderivative of is of the form
(a) , where is a constant.
(b) .
(c) , where is a constant.
(d) , where is a constant.
(e) None of the above
30. Suppose , where is a constant. Then any antiderivative of is of the form
(a) , where is a constant.
(b) , where is a constant.
(c) .
(d) , where is a constant.
(e) None of the above
31.
(a) 5
(b) -5
(c) 0
(d) 1
(e) None of the above
32.
(a)
(b)
(c)
(d)
(e) None of the above
33.
(a)
(b)
(c)
(d)
(e) None of the above
34.
(a)
(b)
(c)
(d)
(e) None of the above
35.
(a) -4
(b) 5
(c) -6
(d) -7
(e) None of the above
36.
(a)
(b)
(c)
(d)
(e) None of the above
37.
(a)
(b)
(c)
(d)
(e) None of the above
38.
(a)
(b)
(c)
(d)
(e) None of the above
39.
(a)
(b)
(c)
(d)
(e) None of the above
In Problems 40–42 assume that and is a constant. Determine which statements below are always true under the given assumptions.
40. _
(a)
(b)
(c)
(d) does not exist
(e) None of the above
41.
(a)
(b)
(c)
(d) does not exist
(e) None of the above
42.
(a)
(b)
(c)
(d) does not exist
(e) None of the above
43. .
(a) 0
(b) 1
(c) 2
(d) 3
(e) None of the above
44. Suppose . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
45. Suppose and . Then is equal
(a) 1
(b) -1
(c) 0
(d) 6
(e) None of the above
46. Find the area under and above between and .
(a)
(b)
(c)
(d)
(e) None of the above
47. Find the area bounded by .
(a) 1
(b) 2
(c) 3
(d) 4
(e) None of the above
48. Find the area bounded by lines .
(a) 25
(b) 35
(c) 20
(d) 15
(e) None of the above
49. Compute for
(a)
(b)
(c) This function does not have this partial derivative.
(d)
(e) None of the above
50. Compute for
(a)
(b)
(c)
(d) This function does not have this second order partial derivative.
(e) None of the above
51. 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
52. 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
53. 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
54. 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
55. Let be a solution to the differential equation with . Then what is
(a)
(b) Cannot be determined.
(c)
(d)
(e) None of the above
56. Suppose a certain population of animals has a uniform birth rate of 2% with 2 young surviving per birth on the average and a uniform death rate of 1% when time is measured in years. What is the population size as a function of time, if the initial population was 1000 animals?
(a)
(b)
(c)
(d)
(e) None of the above
57. Let be a solution to , with . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
58. Let be a solution to . Then what is ?
(a)
(b)
(c)
(d)
(e) None of the above
Key
1(b), 2(c), 3(d), 4(d), 5(c), 6(c), 7(c), 8(a), 9(a), 10(e), 11(b), 12(d), 13(a), 14(b), 15(b), 16(a), 17(c), 18(c), 19(c), 20(a), 21(b), 22(a), 23(c), 24(a), 25(d), 26(c), 27(b), 28(a), 29(c), 30(a), 31(b), 32(d), 33(a), 34(b), 35(c), 36(a), 37(c), 38(b), 39(d), 40(b), 41(b), 42(a), 43(a), 44(b), 45(c), 46(a), 47(a), 48(a), 49(a), 50(a), 51(c), 52(b), 53(a), 54(e), 55(c), 56(b), 57(b), 58(c)