Clearly mark answers to the multiple choice problems on your paper and your scantron.
1. [5 pts] Find all absolute extrema that exist for f ( x)  12  x 
(a) Absolute minimum at
x  3
(b) Absolute minimum at
x 3
(c) Absolute maximum at
x  3
(d) Absolute maximum at
x 3
9
; x 0 .
x
(e) none of these
________________________________________________________________________________________
2. [5 pts] Find the absolute extrema for f (x)  x3  x2  x over the interval [0, 2].
a. Absolute max: 0; Absolute min: -1
b. Absolute max: 2; Absolute min: 0
c. Absolute max:
5
; Absolute min: 0
27
d. Absolute max: 2; Absolute min: -1
e. none of these
3. [5 pts] Find all absolute extrema that exist for f ( x) 
(a) Absolute minimum at
x  1
(b) Absolute minimum at
x 2
1
on [-1, 2].
x2
(c) No absolute extrema exist
(d) Absolute maximum at
x 2
(e) none of these
_____________________________________________________________________________________
4. [5 pts] Suppose that f (x) is continuous on the interval [1, 3] and differentiable on the interval (1, 3). If
3  f  ( x)  5 for all x  (1, 3) then which of the following is correct?.
(a) 3  f (3)  f (1)  5
(b) 6  f (3)  f (1)  10
(c) 1  f (3)  f (1)  3
(d) 5  f (3)  f (1)  7
(e) none of these
5. [5 pts] The recursion equation xt 1  xt2  xt  3 has two fixed points x1*  1 and x2*  3 .
Determine if each of x1*  1 and x2*  3 is approached with or without oscillations.
(a) Both are approached with oscillations.
(b) Both are approached without oscillations.
(c) x1*  1 is approached without oscillations; x2*  3 is approached with oscillations
(d)
x1*  1 is approached with oscillations; x2*  3 is approached without oscillations
(e) none of these
________________________________________________________________________________________
6. [5 pts] The recursion equation xt 1  xt2  xt  3 has two fixed points x1*  1 and x2*  3 . Classify
each fixed point.
(a) Both are locally stable.
(b) Both are unstable.
(c) x1*  1 is locally stable; x2*  3 is unstable
(d)
x1*  1 is unstable; x2*  3 is locally stable
(e) none of these
7. [5 pts] Compute the limit
(a)
lim cosx
1
x2
x 0
.
1
2
1
2
(b) 
(c) e
(d) e
1
2
1
2
(e) none of these
_______________________________________________________________________________________
1  e3x
8. [5 pts] Compute the limit lim
.
x 0
x
(a)  3
(b) 3
(c) 0
(d) DNE
(e) none of these

9. [5 pts.] Compute the limit lim 1 
n 
(a)
n
2
 .
n
e
(b) e1
(c) e 2
(d) e  2
(e) none of these
_______________________________________________________________________________________
10. [5 pts.] Compute the limit
(a) 0
(b)

(c)
1
2
(d) 1
(e) none of these
1
1 
.
lim  
x 0 x
tan x 
11. [5 pts] Compute the limit
 x 
 .
lim x ln
x 
x

1


(a) 0
(b) - 1
(c) 1
(d) DNE
(e) none of these
_______________________________________________________________________________________
12. [8 pts.] Find all fixed points for the recursion xt 1  5 
find lim xt if
t 
x0  1.
1
x , and use the method of cobwebbing to
4 t
13.
Use the 4-Step method for graphing f ( x) 
STEP 1 [7 pts.]: Use f ( x) 
x 2  2x  4
x2
.
x 2  2x  4
x2
.
-----------------------------------------------------------------------------------------------------------------------------STEP 2 [6 pts.]: Use f (x).
STEP 3 [5 pts.]: Use f (x).
---------------------------------------------------------------------------------------------------------------------------------
x 2  2x  4
STEP 4 [4 pts.]: Use information from above to graph f ( x) 
.
x2
14. [8 pts.] A fence is built to enclose a rectangular area of 800 square feet. The fence along three sides is
made of material that costs $6 per foot. The material for the fourth side costs $18 per foot. Find the dimensions
of the rectangle that will allow the most economical fence to be built. (You must use calculus!)
15. [7 pts] A continuous function defined for all x has the following properties:
 f is increasing,
f is concave down,
(a) Sketch a possible graph for f.
f(5) = 2,
f (5) 
1
2
(b) What is the lim f ( x) ?
x 
(c) Is it possible that f (1) 
1
?
4
MATH 147
EXAM III
NAME ___________________________________________
SECTION # _______________
SEAT # ____________
Clearly mark answers to the multiple choice problems on your paper and your scantron.
In order to obtain full credit for partial credit problems, all work must be shown. Credit will not
be given for an answer not supported by work.
"An Aggie does not lie, cheat or steal or tolerate those who do."