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Simon Fraser University
Department of Mathematics
Burnaby and Surrey Campus
MATH 151 -3, Spring 2005
Final
April 16th, 2005, 3:30 – 6:30 pm
Last Name (please print): _________________________________________
First Name (please print): _________________________________________
Student Number: _________________________________________
Instructions:
1. DO NOT OPEN THIS BOOKLET UNTIL
TOLD TO DO SO.
- Fill in the above box.
- This exam contains … pages with a total of 9 questions. Once the exam begins please check to make sure your exam is complete.
- SHOW ALL YOUR WORK!
- If you run out of space in a problem, use the space on the back of the previous page and clearly indicate where the solution continues.
- Only scientific calculators are allowed (basic math/stat functions + memory key).
- No book, paper, or device, other than the usual writing instruments, this booklet and a scientific calculator, shall be within reach of a student during the examination.
- During the examination, speaking to, communicating with, or deliberately exposing written papers to the view of other examinees is forbidden.
- Try your best!
Do not write in this table!
Question Marks
Total /
(a) [2 marks] What is meant by saying that L is the limit of f ( x ) as x approaches a?
[ f ( x ) is defined for all x in some neighbourhood of a except possibly at a itself, and f ( x ) can be made as close to L as desired by choosing x sufficiently close to a .]
(b) [2 marks] What is meant by saying that the function f ( x ) is continuous at x = a?
[ lim (^) x → a f ( x ) and f ( a ) are both defined, and lim (^) x → af ( x )= f ( a ) ]
(c) [2 marks] State two properties that a continuous function f ( x ) can have, either of which guarantees the function is not differentiable at x = a. Draw an example of each.
[Any two of: a corner at x = a , a vertical tangent line at x = a , and a cusp at x = a .]
(d) [2 marks] State Newton’s iterative formula that gives a sequence of approximations x 0 (^) , x 1 , x 2 ,…to a solution of f ( x ) = 0, assuming that x 0 is given.
[
1 n
n n n f x
f x x (^) + = x − ]
(e) [2 marks] Draw a labelled diagram showing an example of a function for which Newton’s iterative formula fails to find a solution of
f ( x ) f ( x )= 0. Mark on your diagram x 0 , x 1 and x 2.
- The following questions involve derivatives.
(a) [2 marks] Evaluate Dt cos −^1 (cosh( e −^3 t )),without simplifying your answer.
(b) [5 marks] Use logarithmic differentiation to find (^) y ' ( u )as a function of u alone, where 13
(^2 1 )(^22 )
u u
u u y u , without simplifying your answer.
[
]
(c) [5 marks] Solve the initial value problem (^4) ( 4 7 )
dt t
dx , x (2) = 1.
[ 4
t
x t ]
(d) [5 marks] Let x = 2 sin t + 1 and y^ =^2 t^3 −^3 define a parametric curve. Find 2
2
dx
d y as a
function of t , without simplifying your answer.
[Not too many marks for finding (^2)
2
dx
d y from dx
dy ]
4. The equation e^ y +^ y (^ x −^2 )= x^2 −^8 defines y implicitly as a function of x near a point (3, 0).
(a) [4 marks] Determine the value of y 'at this point.
(b) [4 marks] Use a linear approximation to estimate the value of y when x = 2.98.
- Let 1
x
x f x be defined for all x ≠± 1. You can make use of the following facts:
2
( 1 )
x
x f x
2
( 1 )
x
x x f x.
Showing all your work, determine for the graph of y = f ( x ):
(a) [1 mark] The ( x , y ) co-ordinates of all intercepts.
(b) [4 marks] All asymptotes. For each vertical asymptote, if any, determine the behaviour of f ( x ) as x approaches the vertical asymptote from the left and from the right.
(c) [1 mark] The ( x , y ) co-ordinates of all critical points, if any.
(d) [2 marks] The intervals on which is increasing and the intervals on which is decreasing.
f f
(e) [1 mark] The classification of each critical point, if any, as a minimum or maximum, local or global, or not an extremum.
(f) [2 marks] The intervals on which is concave up and the intervals on which is concave down.
f f
(g) [1 mark] The ( x , y ) co-ordinates of all inflection points, if any.
(h) [3 marks] Sketch the graph using all the above information and label all relevant points and lines. [
-6 -4 -2 2 4
10
20
]
- [8 marks] A helicopter takes off from a point 80 m away from an observer located on the ground, and rises vertically at 2 m/s. At what rate is the elevation angle of the observer’s line of sight to the helicopter changing when the helicopter is 60 m above the ground?
[2/125 rad/s: use related rates or take arctan and differentiate directly.]
9. [8 marks] Sketch the curve whose polar equation is r =− 1 + 2 cos θ, indicating any
symmetries. Mark on your sketch the polar co-ordinates of all points where the curve intersects the polar axis.
[
0.5 1 1.5 2 2.5 3
-1.
-0.
1
Symmetric about polar axis ( x -axis). Intersects polar axis at
(–1, 0) (equivalently (1, π ))
(0, ± π /3): OK to have only one of these since it’s the same point
(–3, π ) (equivalently (3, 0))
]
