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Şule Yazıcı
Spring 2007, Math 101 FINAL EXAM
Show all your work to get full credit
Closed book & notes, calculator allowed, 2 hours and 15 minutes
ABSOLUTELY NO QUESTIONS WILL BE ANSWERED ABOUT
THE EXAM, BY ANYONE, DURING THE EXAM.
Instructions: There are six questions in this exam. Please inspect the exam and make sure you
have all 7 pages of questions. Do all your work on these pages. If you use the back of a page,
make sure to indicate that.
Remember: You must show your work to get proper credit.
Academic Honesty Code: Koç University Academic Honesty Code stipulates that “copying from
others or providing answers or information, written or oral, to others is cheating.” By taking this
exam, you are assuming full responsibility for observing the Academic Honesty Code.
NAME: ________________________________
NUMBER: _______________________________
1
18
2
23
3
16
4
16
5
15
6
12
Total:
100
A list of formulas: I = P rt ; A = P(1 + rt )
A = P (1 + i ) n ; APY = (1 + r / m) m − 1 ; APY: effective rate
FV = PMT
r
(1 + i ) n − 1
1 − (1 + i ) − n
; PV = PMT
; i = ; n = mt
i
i
m
1.
a) (8 points) You want to purchase an automobile for 28,500YTL. The dealer offers you %0
financing for 60 months or a 6000YTL rebate (indirim). You can obtain 6.2% financing for
60 months at the local bank. Which option should you choose? Explain.
b) (10 points) An ordinary annuity pays 6.48% compounded monthly.
A person wants to make equal monthly deposits into the account for 15 years in order to
then make equal monthly withdrawals of $1,500 for the next 20 years. How much should be
deposited each month for the first 15 years? What is the total interest earned during the 35-year
process?
2.
i-) (12 points) Find the values of the following expressions. Show all work to get full credit.
a) sin −1 (
b) cos −1 (
c) sin (
−1
2
)=
3
)=
2
5π
)=
12
d) tan (
11π
)=
6
ii) (3 points) Solve the following equation for x.
log b ( x + 2) + log b x = log b 24
iii) (8 points) Sketch the graph of the function f ( x) = 1 + 2 cos( x +
shifts and transformations.
π
2
) . Show each step of
3. a) (8 points) Maximize
z = 15 x1 + 10 x 2
subject to 2 x1 + x 2 ≤ 10
x1 + 3 x 2 ≤ 10
x1 , x 2 ≥ 0
by sketching the graph of the feasible region. Indicate the optimal value of the objective
function and the optimal solution to this linear programming problem explicitly.
b) (8 points) Using slack variables, write the initial system for the linear programming
problem given in part(a). Solve the system using the simplex method. And compare the result
you find with the solution found at part (a)
4. a) (10 points) Solve the following system of linear equations. Write the solution set and
determine if the system is dependent, independent, consistent or inconsistent where
applicable.
2 x1 + 3 x2 + 5 x3 = 21
x1 − x2 − 5 x3 = −2
2 x1 + x2 − x3 = 11
b) (6 points) Determine if the following systems are in reduced form. If not then put them
into reduced form. Write the solution set and determine if the system is dependent,
independent, consistent or inconsistent where applicable.
⎡1 3 1⎤
⎡1 3 1⎤
i) ⎢
ii) ⎢
⎥
⎥
⎣0 2 − 4 ⎦
⎣2 6 3 ⎦
5. (15 points) Evaluate the limits in a) through e). Specify infinite limits and if the limit does
not exist give the reason.
a)
x2 − x + sin x
lim
=
x→ 0
2x
b)
lim
x→4
4x − x 2
2− x
=
c)
5x 3 + 8 y 2
lim 5
=
x→ 0 3x − 16x 4
d)
5x3 + 8y 2
=
lim
x→ ∞ 3x5 −16x 4
e)
5x7 + 8y 2
lim
=
x→ −∞ 3x5 −16x6
.
6. (12 points) Sketch a function f (x) that satisfy all of the following properties
lim f ( x) = 2 ; lim f ( x) = 1 ; lim f (x) = +∞; lim− f ( x) = +∞ ;
x→+∞
x → −∞
lim f ( x) = −∞ ; f (4) = 3 ;
x→1+
x→−1
f (2) = 0
x →1