Fall 2007 Math 151
Final Exam Practice
(covering Sections 1.1 - 6.5)
courtesy: Amy Austin
NOTE: These problems are to serve merely as practice for
your final exam. The final exam for Math 151 is NOT a common exam. Each instructor makes up his or her own final
exam. In addition to working this problem set, it is advised
that you work the first three exams as well as the night before
drills. Since the final is not a common exam, there will not
be a ’night before drill’ for the final.
1. Given the vectors
compute:
a.)
and
and
,
d.) Find the vector and scalar projection of
onto
2. Find a vector equation and a set of parametric equations for the line passing thru the points
and
.
3. An object on the ground is pulled with two forces as
shown below. Find the magnitude of each force if the
object is moved horizontally with a total force of 50 N.
45
30
4. Using the definition of the derivative, find
! " $# & %
( b.) ! '
for:
a.)
5. Compute the following limits without the aid of a calculator.
(
)-/.*+0!, 132 54 %
+
# (
, 6 8
b.) -/)+*+.7
: 9 0 ;9
c.) -/)+*+. , -/0
4 4
d.) -/)+*+.7, < ! where
# %(
=
! ' < (4 ?
a.)
if
if
8>$
limits:
7
6
5
f(x)
4
3
2
1
−9 −8 −7 −6 −5 −4 −3 −2 −1
1
2
3
4
5
6
7
8
9
−1
−2
−3
−4
−5
b.) Find a unit vector in the direction of
c.) Find the angle between
B (
= B
e.) -/)+. *+,
4A@ # (%( E
% *+, C D4 FG
f.) -/)+.
6. Use the graph of ! below to compute the following
/- )+.I*+, H @ ! *+, J32 ! b) -/)+.
, < ! c) -/)+*+.I
d) -/)+.
*+, 0!1 ! −6
a)
e) Discuss the continuity and differentiability of the
function whose graph is given above.
4G
4 8 < (8
?
$
a.) ! '
if ? > $
if 8K if $
$
b.) ! "
L < ifif $
c.) Find the value of M and N that make ! differentiable everywhere.
! " MO 4 % N ifif 8>$
8. Find the equations of both tangent lines to the parabola
! " 4 % that pass thru the point .
9. Find all horizontal and vertical asymptotes for ! 54 .
10. Using the limit definition, find the slope of the tangent
4 $= %$ at Q .
line to the graph of ! P
What is the equation of this line?
11. Find ?
for ! '$R 4
12. What is the domain of ! ' ) ST R U ?
7. Discuss the continuity of the following functions:
if
RV
13. Compute the inverse of !
I %(R WX
14. Find W for:
< X ( < % X 6 % X
a.)
X '$ X b.) Y[Z
\A X F < (
D4
<
'
] Y[Z
\^ F ] 4 b.) !]
<
c.) _ `acb S E
L
16. If d " ! 4 %( and V$e , compute d .
X R - at 17. Find the tangent line to the graph of ) S f
< X
18. Given ] , g
= ] ] 4 , find the cartesian equation
for the tangent line at ] h . Also, find the equations
15. Compute the derivative:
a.)
vwrx ' ) SF] ] R u ?
Rc
28. Find the derivative of ! ' ) StyY[Z
\5 X X
%( < {z 29. Find for F%
L
30. A cup of coffee ?/| is placed in a = | room. If the
L
temperature of the coffee is | after 2 minutes, when
will the coffee be } ? | ?
27. If
, find parametric equations for the
tangent line to the curve at the point
.
31. A bacteria culture starts with 1500 bacteria, and the population triples every 2 hours. How long will it take for
the population to reach 2520 bacteria?
! ' 6 P < ~=
! 32. If
, Find the intervals where
is
increasing/decreasing, locate all local extrema, intervals
of concavity and points of inflection. Sketch the curve.
! ' ) S& concave up?
< 8 4 over
34. Find the absolute extrema for ! V
the interval € ‚ .
of all vertical and horizontal tangent lines.
35. In part (a) below, the graph of is given. Using the
graph of , determine all critical values of , where is
19. To approximate a root to the equation
increasing and decreasing, local extrema of , where < % 4 %ij$? , we apply Newton’s method with 1 is concave up and concave down, and the x-coordinates
. Find the second iteration, 4 .
of the inflection points of . Assume is continuous.
20. Find the linear and quadratic approximation for !
V
k
) S& at M . What is the usefullness of the linear and
quadratic approximations?
21. A filter in the shape of a cone is e
lnm high and has a
radius of lnm at the top. A solution is poured in at the
</o mprq . Find the rate at which the height of
rate of lnm
the residue is increasing when the height is lnm .
33. Where is
100
x
–10
–8
–6
–4
–2
2
4
6
8
10
0
–100
22. A kite is flying 100 feet above the ground at the end of a
string. The girl flying the kite lets out the string at a rate
of 1 foot per second. If the kite remains 100 feet above
the ground, how many feet per second is the horizontal
distance from t he girl increasing when the string is 125
feet long?
23. The length of a rectangle is decreasing at the rate of 1
foot per second, but the area remains constant. How fast
is the width increasing when the length is 10 feet and
the width is 5 feet?
) Z
s` % ) Z
sD %(
" ) Z
s =
25. Solve for : ) SF ) St %E
"$
< 0!1
26. Solve for ] : u" u
24. Solve for :
y
–200
a.)
–300
&% % L 4
# ) SF
)S 54
36. Compute the following limits using L’Hospital’s rule:
-/)+.*+, C
b.) -/)+.7*+, ƒO2
c.) -/)+.
*+, C
a.)
37. Find the most general antiderivative of
4 ( %(
#
1 R-; W
„
38. Approximate
using 4 equally spaced interƒ‡
vals, and take †… to be the midpoint of each subinterval.
0!1 L 39. Compute Y[Z
\A acb S
0!1 \ˆ* S ‚ ‰ 40. Compute \ˆ* S
X
0!1 ] Š(Dacb S 0!1 # ] 41. Find the derivative of Y[Z
\
42. A rectangular poster is to have an area of 180 square
inches with 1.5 inch margins at the top and bottom, and
a 1 inch margin on the sides. What poster dimensions
will give the largest printed area?
43. A farmer has 20 feet of fencing, and he wishes to enclose
a rectangular pen with the barn as one of it’s sides. What
are the dimensions of the fence with maximum area?
W - ; F 6 W
] ]
44. Find W
4
L 4 W
45.
< E
4
1 R -
‹{0ŒH W
4 4
46. ƒ
0Œ6 W
47. ƒ
# F(
] W
48.
] 4 E ]
1Ž 4 ‰ W
Y[Z
\
49. 1
6 % # 8 0!1 W
50. 1 4
51.
52.
53.
54.
acb St W
4 L %( W
0!1
\ˆ* SIH ] W ]
Y[Z
\ ]
W
# %(