MATHEMATICS 184 (Section 922) - FINAL EXAM
NAME:
STUDENT ID NUMBER:
SIGNATURE:
INSTRUCTIONS: No notes or books are to be used. Calculators
are allowed. No credit will be given for the correct answer without
the (correct) accompanying work. Use the back of the pages if you
need extra space.
Rules Governing Formal Examinations
The following are the rules governing formal examinations:
1. Each candidate must be prepared to produce, upon request, a
Library/AMS card for identification.
2. Candidates are not permitted to ask questions of the invigilators,
except in cases of supposed errors or ambiguities in examination
questions.
3. 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.
4. 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.
*Having at the place of writing any books, papers or memoranda,
calculators, computers, audio or video cassette players or other
memory aid devices, other than those authorized by the examiners.
*Speaking or communicating with other candidates.
*Purposely exposing written papers to the view of other candidates.
The plea of accident or forgetfulness shall not be received.
5. 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.
1.(a) Find the Taylor Series about x = 0 for the function
f (x) = sin2 (x)
Write down at least the first 4 non-zero terms in the series.
Hint: Use a trigonometric identity first.
[4]
(b) Find the Taylor Series about x = 0 for the function
f (x) =
1
p
1 + x3
Write down at least the first 4 non-zero terms in the series.
[4]
2. Use Newton’s Method with an initial approximation of x0 = −π
2
to find, correct to 5 decimal places, an intersection point of the
curves
f (x) = 2cos(x)
and
g(x) = 1 − x
[10]
3. Find the points on the curve
x2 y 2 = (y + 1)2 (4 − y 2 )
at which the tangent line is horizontal.
[10]
4. For the function
f (x) = cos2 (x) + 1
(a) Identify the domain and the range.
[2]
(b) Find the coordinates of the x and y-intercepts.
[2]
(c) Find all the critical points and corresponding critical values.
[4]
(d) What are the coordinates of the local maxima and minima?
(Indicate which ones are local maxima and which are local minima). Show your work.
[3]
(e) Find the coordinates of any inflection points.
[4]
(f) Using the information from parts (a)-(e), sketch a graph of this
function.
Warning: Your sketch must match the information you determined above. No marks will be awarded for a correct sketch
that does not match your results from (a)-(e).
[4]
5. Differentiate the following functions:
(a) f (x) = x2 sin(x)
(b) y =
[2]
2x
[2]
x3 +7
(c) g(x) = arccos(x2 − 1)
(Simplify your answer).
[3]
(d) z =
(x5 −6x2 )3
(3x3 +8x−6)4
2
(Simplify your answer).
[3]
(e) f (p) = p3 ep + pln(p)
[3]
(f) y = xx
[3]
6. Utopia’s national bank offers an annual percentage interest rate
of 6%, compounded monthly (12 times a year). The local credit
union in competition with the national bank advertises that it will
pay an annual percentage interest rate of 5.8% compounded continuously. Determine which (the bank or the credit union) would
give you the best return on an investment.
[10]
7. The amount of time T , in minutes, it takes me to get to work
in the morning varies according to the equation
πl
T (l) = 45 + l + 15sin( )
−5≤l≤5
4
where l is the number of minutes past 8:30 a.m. that I leave my
house. (I think of l as the number of minutes I am running late).
(a) According to this equation, how many minutes should I leave
after 8:30 a.m. in order to minimize my travel time? (Round your
answer to the nearest minute).
(b) If I leave the number of minutes after 8:30 a.m. you found in
part (a), how long will it take me to get to work?
[6]
[2]
8. A certain ball has the property that each time it falls from a
height h onto a hard, level surface, it rebounds to a height of 0.9h.
Suppose the ball is dropped from an initial height of h = 10 meters.
(a) Assuming that the ball continues to bounce indefinitely, find
the total distance (in meters) it travels vertically. Be sure to count
both the distance travelled up and down.
[4]
(b) If the ball is stopped bouncing after 10 bounces (it is caught
at the top of the bounce) what is the total vertical distance the
ball has travelled?
[3]
(c) What would the height of the 11th bounce have been?
[3]
9. A reasonably realistic model of a firm’s costs is given by the
short-run Cobb-Douglas cost curve
1
C(q) = Kq a + F
where a is a positive constant, F is the fixed cost, and K measures
the technology available to the firm.
(a) Show that C is concave down if a > 1.
[4]
(b) Assuming that a < 1 and that average cost , C(q)
, is minimized
q
when average cost equals marginal cost, find what value of q minimizes the average cost.
[5]