MATH 101 Final, Fall 2000
American University of Beirut · Calculus and Analytical Geometry I · 8 pages
More MATH 101 files
Show the text of this final exam
Time : 2 hours MATHEMATICS 101 NAME
Prof. H. Abu-Khuzam Final Examination ID#
February 3, 2001 ( First Semester 2000-01) Section (5 or 6 or 7)
Vers
l. Find the area of the region between the curve y= x°, -1< x $1, and the x-axis,
L 8 points]
2 Let ys [BF +2 4¢3dr Find gg
dx
{ 8 points}
3. Find any asymptotes, any maxima or minima . and sketch the graph for
4
fix)=
xl
{ 12 points]
4, Let g(x) = 2x4+sinx. Finda function f(x) so that f’(v) = g’(x) for allx, and f(r) =1.
[ 8 points]
Circle the correct answer in each of the following questions ( problem 5 to problem
20) 1 4 points each]:
[ 4 points}
does not exist .
[ 4 points]
7. If x+siny=xy . then y" at the point (0,0) is equal to:
0
1/2
i
-|
none of the above
Se Boe
{ 4 points}
8. Thegraphof y= -x*+I has
a. a local minimum at (0,1)
b. a local maximum at (( 1)
c. an absolute minimum at ( 2, -15 )
d. a pointof inflection at (0,1)
e. no extreme values.
[ 4 points]
9. The graph of the function y= eel has an oblique asymptote of equation :
xotxtl
a y= xX
b y= -x-1
Cc. y=x+2
do y=x+l
e& y=x-l.
[ 4 points}
10. The value of the constant k that will make the function
3x-l. if x <2
-kx°, ifx>2
f(x) =
continuous, is
none of the above.
[ 4 points]
|x [(3x* +4)
lt. lim
-2"|
is equal to :
- 3/2
3/
0
a.
b.
c 2
d.
e. docs not exist.
{ 4 points]
0 _ f-l
12. If f(x) = (2x4+1"" . then tim LUTE
is equal to:
ho h .
a. 200
b. 100
e. does not exist.
{ 4 points]
2x+ | if x<-2
Lx | if x>-2
13. Het
then
a. f(x) is continuous at x + -2
lim f(x )=-3
r(-2}
fC -2.) is not detined
lim f(x ) does uot exist
roe?
none of the above.
[ 4 points]
(% sec x tan x dv
if
32
V2
2V2-2
0
none of the above.
[ 4 points]
If f is continuous onan interval [a,b] and ce (a,b). then which one
of the following statements is TRUE ? .
If f’ (c)=0, then f has a local maximum or a local minimum at x =c.
If f has alocal maximumat x=c. then ff’ (c)=0)
fr ff’ (c)=0,and f" (ce) <0, then f has a local minimum at x =c.
If f’(c)=0, then f has a point of inflection at x=c.
e. ff is integrable over fa,b].
[ 4 points].
16. Let
fo = 3x", if x<l
axth, if x>l
be differentiable at x=T are :
a
b:
ron
d
€
17, Let
‘
a=6 and b= -3
a=3 and b=()
a=6 and b=6
a=1 and b= t
a=6 and b=1
t(x) = x°-3x+3, then :
f(x) is always increasing.
f(x) has an inflection point at x=0.
f(x) has a local maximum at x=1.
f(x) a local minimum at x= -1.
F(x) has no extreme values.
. Then, the values of a and b so that f will
{ 4 points]
[ 4 points].
18. Let t(x)=x*+1. [f g(x) is a polynomial such that g(0)=1 and g’(0)=3, then (f © g)’(0) is equal to:
none of the above.
[ 4 points]
19. If x and y are two nonnegative numbers whose sum is [, then the minimum value of the sum
of their squares is :
1/2
none of the above
[ 4 points]
ff ftnar= vet . then f(x) =
e none of the above.
[ 4 points]