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This examination has 10 pages of questions excluding this cover
The University of British Columbia Final examination - April 18, 2011
Mathematics 103: Integral Calculus with Applications to Life Sciences
201 (Holmes), 203 (Hauert), 206 (Rolfsen), 207 (Christou), 208 (Lindstrom), 209 (Rolfsen)
Closed book examination Time: 150 minutes (2.5 hours)
Last Name: First Name:
Student Number: Section: circle above
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Question: 1 2 3 4 5 6 7 8 Total
Points: 12 20 10 14 12 10 12 10 100
Score:
Show all your work and explain your reasonings clearly!
- (12 points) Short answer problems (only answers are marked)
a. (5) For each of the following series, indicate whether or not they converge:
i.
∑^ ∞
n=
n
ANSWER: converges; does not converge.
ii.
∑^ ∞
n=
2 n^
ANSWER: converges; does not converge.
iii.
∑^ ∞
n=
2 n^ ANSWER: converges; does not converge.
iv.
∑^ ∞
n=
n^2
ANSWER: converges; does not converge.
v.
∑^ ∞
n=
n 1 − n^3
ANSWER: converges; does not converge.
b. (5) Consider the following four probability density functions (pdf):
a b x
a b x
a b x
a b x
i. Which pdf has the smallest mean? ANSWER: ii. Which pdf has the largest variance? ANSWER: iii. Which pdf has the smallest standard deviation? ANSWER: iv. Which pdf has median larger than the mean? ANSWER: v. Find the maximal probability density in (4). (Note: the peaks at a and b are of equal heights.) ANSWER:
c. (2) A dice is manipulated such that the chance of throwing a 6 is twice as likely as throwing any one of the other numbers (1-5). What is the expected (average) number of throws required to get a 6? Circle the correct answer. i. 3 ii. 3.5 iii. 4 iv. 6 v. 7
c. (6) Consider the differential equation dy dx
= y − x.
Use a Taylor series expansion to find the solution y(x) for the initial value y(0) = 1.
ANSWER: y(x) =
d. (4) A student takes a multiple choice test with 6 questions each of which has 4 possible answers and exactly one is correct. To pass the test at least 5 correct answers are required. (Note: simplify your answers as much as possible but leave fractions and powers.) i. What is the probability that a student who did not study and randomly checks his/her answers still passes the test?
ANSWER:
ii. With what probability does the student have to get every answer correct in order to get a perfect score with a probability of at least 80%?
ANSWER:
- (10 points) Consider the function f (x) = x(x − 2)(x + 1).
a. (3) Sketch the function.
-3 -2 -1 0 1 2 3
0
2
4
b. (7) Find the total, finite area A bounded by f (x) and the x-axis.
ANSWER: A =
- (12 points) In a room two different lightbulbs A and B are installed. According to the packaging, lightbulb A fails before time t with probability FA(t) = 1 − e−t^ (t in months). Lightbulb B has a probability density function for the failure time given by pB (t) = Ce−^2 t.
a. (2) What is the probability density function for the failure time of lightbulb A?
ANSWER: pA(t) = b. (2) Determine the constant C of the probability density function for the failure time of lightbulb B?
ANSWER: C =
c. (4) What is the probability that both lightbulbs are still working after t months?
ANSWER:
d. (4) Which light bulb has the longer expected lifetime? (i.e. longer average time to failure.) Show your reasoning.
ANSWER:
- (10 points) Consider the differential equation
dy dx
= (y^2 − 1)x.
a. (3) Find the steady state solution(s).
ANSWER:
b. (7) Solve for y(t) given the initial value y(0) = 0.
ANSWER: y(t) =
- (10 points) In a lab a bacterial colony is grown in a petri dish. The colony is circular and the area A covered by the colony increases at a rate proportional to its circumference C (because of the high nutrient concentration in the surrounding area): dA dt
= k · C.
One day (t = 0) the colony was observed to cover an area of 1 [mm^2 ]. The next day the colony had grown to cover 2 [mm^2 ]. What area does the colony cover after T days? (Assume that the petri dish is large enough such that the colony never reaches the boundary.)
ANSWER: A(T ) =
Useful Formulæ
Summation
∑^ N
k=
k =
N (N + 1)
∑^ N
k=
k^2 =
N (N + 1)(2N + 1)
∑^ N
k=
k^3 =
N (N + 1)
∑^ N
k=
rk^ =
1 − rN^ + 1 − r
∑^ ∞
k=
krk−^1 =
(1 − r)^2
Trigonometric identities
sin(α + β) = sin α cos β + cos α sin β; for α = β: sin(2α) = 2 sin α cos α
cos(α + β) = cos α cos β − sin α sin β; for α = β: cos(2α) = 2 cos^2 α − 1 sin^2 α + cos^2 α = 1
tan^2 α + 1 = sec^2 α =
cos^2 α
Some trigonometric values
sin(0) = 0, sin(
π 6
, sin(
π 4
, sin(
π 3
, sin(
π 2
) = 1, sin(π) = 0
cos(0) = 1, cos(
π 6
, cos(
π 4
, cos(
π 3
, cos(
π 2
) = 0, cos(π) = − 1
Derivatives
d dx
arcsin x =
1 − x^2
d dx
arccos x = −
1 − x^2
d dx
arctan x =
1 + x^2
Moments of a probability density function
Mk =
∫ (^) b
a
p(x)xkdx
