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Math 103 Final Exam April 2006 2.5 hours.
- No calculators, books, notes, or electronic devices of any kind are permitted.
- Unless otherwise indicated, show all your work. Answers not supported by calculations or
reasoning may not receive credit. Messy work will not be graded. Read each question
carefully to be sure you are answering the question being asked.
Rules governing formal examinations:
- Each candidate must be prepared to produce, upon request, a Library/AMS card for identification;
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions;
- No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination;
- Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action; (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners; (b) Speaking or communicating with other candidates; (c) Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received;
- Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator; and
- Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
Problem # Value Grade
Total 50
Formulae
∑^ 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
∫ (^) b
a
f (x) dx = lim ∆x→ 0
∑^ N
k=
f (xk)∆x, xk = a + k∆x, ∆x = (b − a)/N
∫ (^) b
a
F ′(x) dx = F (b) − F (a),
∫ (^) b
a
udv = uv
b
a
∫ (^) b
a
vdu
Vshell =
∫ (^) b
a
2 πxf (x) dx, Vdisk =
∫ (^) b
a
πf (x)^2 dx, L =
∫ (^) b
a
1 + f ′(x)^2 dx
sin^2 θ + cos^2 θ = 1, tan^2 θ + 1 = sec^2 θ, cos^2 θ =
1 + cos 2θ 2
, sin^2 θ =
1 − cos 2θ 2 sin(A + B) = sin A cos B + sin B cos A, cos(A + B) = cos A cos B − sin A sin B,
sec θdθ = ln | sec θ + tan θ| + C,
sec^3 θdθ =
(sec θ tan θ + ln | sec θ + tan θ|) + C
∫ 1 1 + u^2
du = arctan u + C,
1 − u^2
du = arcsin u + C
d dx
(tan x) = sec^2 x,
d dx
(cot x) = − csc^2 x,
d dx
(sec x) = tan x sec x,
d dx
(csc x) = − cot x csc x,
x ¯ = Ex =
∑^ n
k=
xkp(xk), Vx =
∑^ n
k=
x^2 kp(xk) − (Ex)^2 ,
x ¯ = Ex =
∫ (^) b
a
xp(x) dx, Vx =
∫ (^) b
a
x^2 p(x) dx − (Ex)^2
F (x) =
∫ (^) x
a
p(x) dx, F (xmedian) =
p(E 1 and E 2 ) = p(E 1 )p(E 2 |E 1 ), p(E 1 or E 2 ) = p(E 1 ) + p(E 2 ) − p(E 1 and E 2 )
C(n, k) = n! (n − k)!k!
, P (n, k) = n! (n − k)!
p(k heads out of n tosses) = C(n, k)pk(1 − p)n−k
Tn(x) =
∑^ n
k=
f (k)(x 0 ) k!
(x − x 0 )k
(d) The concentration of a protein in a long, thin cylindrically shaped bacterium is given by the function c(x) where c is measured in mol/μm along the long axis of the cell and 0 < x < 3 is measured in μm. Suppose that when the cell divides, it splits at x = 3/ 2 μm and that all protein to the left of this point ends up in the left daughter cell and any to the right ends up in the right daughter cell. Determine which of the following statements is necessarily true: i. If the center of mass of the protein density is to the left of the median of the density, than the cell on the right gets more than half of the total protein. ii. If the center of mass of the protein density is at x = 1μm, then the cell on the left gets more than half of the total protein in the mother cell. iii. If the median of the protein density is at x = 1μm, then the cell on the left gets more than half of the total protein in the mother cell. iv. If the mean and the median are at the same point, both cells get the same amount of protein.
(e) 1 − 2 x^2 is the second order Taylor polynomial (around x = 0) for which one of the following functions? i. 1 − sin(4x) ii. e−x 2
iii. cos(2x) iv. e−^4 x v. cos(x)
(f) Which of the following is essentially a form of the Fundamental Theorem of Calculus? i. The area under the graph of f (x) between x = a and x = b is given by
∫ (^) b a f^ (x)^ dx. ii.
∫ (^) b a f^ (x)^ dx^ = limn→∞
∑n k=1 f^ (xk)∆x. iii. The derivative with respect to time of the position of a molecule is v. The net displacement of the molecule from t = 0 to t = T is given by
∫ T
0 v(t)^ dt. iv.
∫ (^) b a f^ (x)^ dx^ =^ f^
′(b) − f ′(a).
- For each of the following short questions, write your answer in the box – it will be graded simply right or wrong.
(a) Calculate
k=1(k^ + 3)
(b) If
1 f^ (x)dx^ = 2 and^
3 f^ (x)dx^ = 5, find^
1 f^ (x)dx.
(c) If the continuous random variable x has probability density p(x) = π 2 sin(πx), 0 ≤ x ≤ 1, find
the mean value of x.
(d) A bag contains 5 red balls, 3 green balls, and 2 yellow balls. If balls are always replaced into the bag after being drawn, what is the probability of drawing the same colour out of the bag on two
successive attempts?
(e) The Taylor series for the function (^) 1+^1 x 2 around x = 0 is
1 + x^2
∑^ ∞
n=
(−1)nx^2 n^ = 1 − x^2 + x^4 − x^6 + x^8 − · · ·.
Use this fact to find the Taylor series for arctan(x) around x = 0 (hint: the formula sheet might
be useful).
- You are driving your car at 30 m/sec (approx. 108 km/hr) to catch your flight to Costa Rica for summer holidays. A pedestrian runs across the road, forcing you to brake hard. Suppose it takes you 1 sec to react to the danger, and that when you apply your brakes, you slow down at the rate a = − 10 m/sec^2. After applying the brakes, how long will it take you to stop? How far will your car move from the instant that the danger is sighted until coming to a complete stop?
- Consider the region below the graph y = 1 − x^3 (and above the x-axis) between x = 0 and x = 1.
(a) Find the area of this region. (b) Find the volume of the solid “dome” obtained by rotating this region about the y-axis. (c) Suppose the density of the solid from part (b) is p(y) = ky. Find its mass.
- In an experiment involving a bacteria population, N (t) denotes the size of the population (measured in thousands of individuals) as a function of time, starting at t = 0. The initial population is N (0) = N 0.
(a) Suppose the population growth is governed by the differential equation
dN dt
N
t + 1
Find N (t) if N (0) = 2. (b) Suppose the population growth is governed by the differential equation
dN dt
N 2
(t + 1)^2
Find N (t) if N (0) = 2. (c) What happens to the solution from part (b) as t → 1? Can you find an initial population N 0 for which this problem doesn’t occur for any time t?
