final review calculus i, fall 2006. given a percent growth rate of 10%, what is the value of a in...
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Final ReviewFinal Review
Calculus I, Fall 2006Calculus I, Fall 2006
Given a percent growth rate of 10%, what is Given a percent growth rate of 10%, what is the value of the value of aa in the equation in the equation f(x)=Paf(x)=Paxx
A. 0.01
B. 0.001
C. 1.10
D. 1.01
E. None of the Above
Find the indefinite integral of the function Find the indefinite integral of the function graphed below:graphed below:
A. ln|sec(s)tan(x)|+C
B. sec2(x)+C
C. ln|sec(x)|+C
D. tan(x)+C
E. None of the Above
Given logGiven logbbx=A, what is the equivalent x=A, what is the equivalent
exponential expression?exponential expression?
A. x=Ab
B. lnA=x
C. bA=x
D. b=Ax
E. None of the Above
Given the graph of f(x)=x(xGiven the graph of f(x)=x(x22-3) below, what is -3) below, what is the area between the curve and the x-axis from the area between the curve and the x-axis from 0 to 1?0 to 1?
A. 1/4 - 3/2
B. 3/2 – 1/4
C. -2
D. 2
E. None of the Above
Exponential functions have a constant rate of Exponential functions have a constant rate of changechange
A. TRUE
B. FALSE
What is the equation describing the following graph?What is the equation describing the following graph?
above theof None.
32
sin2.
32
sin5.
3)(4
1cos5.
34
sin2.
E
xD
xC
xB
xA
2 4 6-2-4-6
1
3
5
Given the graph of f(x)=x(xGiven the graph of f(x)=x(x22-3) below, what is the total -3) below, what is the total change in the anti-derivative function F(t) from -1 to 1?change in the anti-derivative function F(t) from -1 to 1?
above theof None.
0.
4.
4.
)3(.1
1
2
E
D
C
B
dxxxA
Given Given y=16y=16sinsin(2x)+5(2x)+5, what is the period?, what is the period?
A. 2
B. 4
C. 2
D.
E. None of the Above
What is the equation describing the following graph?What is the equation describing the following graph?
above theof None.
above theof All.
)cos(.
2sin.
2sin.
E
D
xC
xB
xA
-
1
-1
-/2-3/2 /2 3/2
loglogbb(C+D)=log(C+D)=logbbCC•log•logbbDD
A. TRUE
B. FALSE
Which of the following is the graph of Which of the following is the graph of x3log
AB
CD
E None of these
Given the following graph of g(x), order:Given the following graph of g(x), order:a) g’(2) a) g’(2)
b) average value of g(x) over the interval 0<x<8b) average value of g(x) over the interval 0<x<8c) Integral of g(x) from 1 to 3c) Integral of g(x) from 1 to 3
from least to greatest. from least to greatest.
A. a<c<b
B. a<b<c
C.c<a<b
D.c<b<a
E. None of the Above
2 4 6 8
Given F(x)=f’(x), what does the shaded region Given F(x)=f’(x), what does the shaded region represent?represent?
above theof None.
&.
)4(.
)(.
)(.
4
1
4
1
E
CBD
fC
dxxFB
dxxfA
F(x)
Given the graph of f(x), which of the following is the Given the graph of f(x), which of the following is the graph of f(x+k)?graph of f(x+k)?
A
B C
D None of these
The following graph shows the velocity of a The following graph shows the velocity of a rocket taking off from earth versus time. rocket taking off from earth versus time. When did the rocket hit the ground?When did the rocket hit the ground?
A. t=2
B. t=4
C. t=6
D. t=8
E. None of the Above
2 4 6 8
Given the following graph of g(x), determine which of Given the following graph of g(x), determine which of the following statements are true about the anti- the following statements are true about the anti- derivative.derivative.
i. G(t) is positive for all 0<t<2
ii. Over the interval 1<t<2, G(t) is increasing
iii. G(t) is concave up at t = 1/2
2 4 6 8
g(x)
t
dxxgtG1
)()(
A. i only
B. ii only
C. iii only
D. i and ii only
E. ii and iii only
F. None of the above
Given sinGiven sin22(()-1=x, x is equal to)-1=x, x is equal to
above theof None.
csc.
cos.
cos.
csc.
2
2
E
D
C
B
A
Given the following table of values, find a formula Given the following table of values, find a formula for for f f ((xx))
above theof None.
)5.1(16.
1643.3.
162
16.
2
D
C
xxB
xA
x
xx f(x)f(x)
00 1616
11 2424
22 3636
33 5454
44 8181
Evaluate the following integral:Evaluate the following integral:
dxx)7cos(7
above theof None.
)7sin(.
)sin(7.
)7sin(.
)sin(7.
E
CxD
CxC
CxB
CxA
A person runs around a circular track. Their A person runs around a circular track. Their horizontal distance from the starting line is a horizontal distance from the starting line is a periodic function. If the track is 400 meters long, periodic function. If the track is 400 meters long, and the runner takes 1 minute to run completely and the runner takes 1 minute to run completely around the track once, which of the following around the track once, which of the following equations could model the distance from the starting equations could model the distance from the starting line as a function of time given the runner starts line as a function of time given the runner starts running at time t=0?running at time t=0?
above theof None.
)cos(200.
)2sin(200.
)2cos(400.
)sin(400.
E
xD
xC
xB
xA
Evaluate the following integral:Evaluate the following integral:
dx
x
x
42
above theof None.
)4ln(.
)4(2.
42
1.
)4(2
1.
2
2
2
2
E
CxD
CxC
CxB
CxA
The total number of undergraduate students at the The total number of undergraduate students at the Willamette University is given in the following Willamette University is given in the following table. What was the average rate of change in the table. What was the average rate of change in the number of students from 1990 to 2005?number of students from 1990 to 2005?
above theof None.
5.1674.
55.15.
33.22.
D
C
B
Ayearyear enrollmentenrollment
19851985 15311531
19901990 15071507
19951995 15251525
20002000 16591659
20052005 18421842
Given a runner’s horizontal distance from the Given a runner’s horizontal distance from the starting line on a circular track is given by the starting line on a circular track is given by the equation equation
d = 200sin(2d = 200sin(2x)x)
Which of the following formulas gives the total Which of the following formulas gives the total horizontal distance run in 4 minutes?horizontal distance run in 4 minutes?
above theof None.
)42sin(200.
)2cos(400.
)2cos(200.
)42sin(200.
4
0
4
0
E
D
dxxC
dxxB
A
xxeedx
d 2
above theof None.
.
.
)21(.
.
2
2
2
22
2
2
E
eD
xeeC
eexB
xeeA
x
x
x
xe
xx
xex
xxe
Find the critical points of Find the critical points of
52489)( 23 xxxxf
above theof None.
and .
2.
2.
8.
E
BAD
C
B
A
)4ln(3 xedx
d
above theof None.
3.4
12.
4
3.
)4ln(12.
E
De
C
eB
eA
x
x
x
)tan(x
dx
d
above theof None.
)(cos/1.
)(csc.
)cot(.
)(sec.
2
2
2
E
xD
xC
xB
xA
)12sin(12 x
dx
d x
above theof None.
)12sin(
)12)(12)(12cos())12sin()(12)(12ln(.
)12sin(
)12)(6)(12cos())12sin()(12)(12ln(.
)12sin(2
)12)(12cos())12sin()(12)(12ln(.
)12ln(.
E
x
xxD
x
xxC
x
xxB
A
xx
xx
xx
Given Given F’(x)=f(x)F’(x)=f(x), the shaded area on the graph of , the shaded area on the graph of ff((xx) ) representsrepresents
above theof None.
)()(.
)(.
)('.
D
aFbFC
dxxFB
bFAb
a
1
-1
ab
What is the derivative of What is the derivative of )ln()( tttg
above theof None.
1.
1)ln(.
)ln(2.
)ln(1
. 2
E
D
tC
ttB
tt
A
axxFE
baxFDab
aFbFC
aFbFB
aFA
at )( of change of rate ousinstantane The.
and between )( of change of rate average The.
)()(.
)()(.
)('.
Given Given F’(x)=f(x)F’(x)=f(x), the marked quantity on the graph of , the marked quantity on the graph of f(x)f(x) represents: represents:
A1=4
a b
Given Given F’(x)=f(x)F’(x)=f(x), which of the following represents , which of the following represents
on the graph of on the graph of F(x)F(x)??
above theof None.D-
1
-1
-/2-3/2 /2 3/2
a ba
A
-
-1
-3/2 /2
C
-
-1
-/2-3/2 /2a
B
ba b
a b
b
adxxf
ab)(
1
D. None of the above
Given Given F’(x)=f(x)F’(x)=f(x), which of the following represents , which of the following represents f(a) f(a) on the graph of on the graph of FF((xx))
-
1
-1
-/2-3/2 /2 3/2
a
a
a
D. None of the above
A
C
B
)cos(
)sin(
x
x
dx
d
above theof None.
)(sec.
)sec()tan(.
)tan(.
2
D
xC
xxB
xA
)ln(xedx
d
above theof None.
1.
1.
1.
)ln(
De
C
Bx
A
x
xxdx
d24 3
above theof None .
)24(3
2.
212.
)212(2
1.
)24)(16(.
2/33
2
2/12
2/132
E
xxD
xC
xB
xxxA
xexdx
d 43
above theof None.
34.
1612.
43.
4243
3
434
D
exexC
exxeB
exxeA
xx
xx
xx
)ln(x
dx
d
above theof None .
above theof All.
1.
1.
.
)ln(
1
E
De
C
xB
xA
x
22x
dx
d
above theof None.
2)2ln(4.
2.
2.1
2
D
C
B
A
x
x
x
x
dx
d6log
above theof None.
log6.
1
)6ln(
1.
6
1.
D
xC
xB
xA
)()( Find2)(6)( 234 xgxf
dx
dxxxgxxf
above theof None.3
1
2
16
5
1.
264.
2664.
345
23
23
D
xxxxC
xxxB
xxxxA
dx
dy
x
xy Find
4
2
above theof None.
)4(
8.
)4(
2.
2.
2
2
2
2
D
x
xxC
x
xxB
xA
dx
dyey xx Find53
above theof None.5
1
3
1.
)ln(53.
53)3ln(.
De
C
eB
eA
xx
x
xx
dtt
t3
2
1
above theof None.
1ln3.
1ln3
1.
1ln.
3
3
3
D
CtC
CtB
CtA
dtt 2)1(
1
above theof None
)1(
1.
)1(
2.
)1(ln.
1ln2.
3
2
E.
Ct
D
Ct
C
CtB
CtA
What is the amplitude of the function What is the amplitude of the function
33
tan12)(
xxf
above theof None.3
2.
6.
3.
12.
2
E
D
C
B
A
Fill in the blanksFill in the blanksF(x)F(x) F’(x)F’(x)
1.1. IncreasingIncreasing positivepositive
2.2. negativenegative
3.3. 00
4.4. Concave upConcave up
5.5. decreasingdecreasing
6.6. Max/MinMax/Min
a. Concave down b. decreasingc. zero d. inflection pointe. Concave up f. increasingg. Max of Min h. critical point
decreasing
critical point
increasing
Concave down
Inflection point
Which is the best method to use to integrate Which is the best method to use to integrate
dxex )3(
above theof None.
done bet Can'.
Algebra.
.
D
C
B
onSubstitutiA
)tan()sec( xx
dx
d
above theof None.
)sec()(tan)(sec.
)tan()(sec.
)sec(.
23
2
D
xxxC
xxB
xA
is the derivative of is the derivative of )cos(3)( ttg
above theof None.
)3sin(.
)sin(3.
)sin(3.
)sin(3.
E
CtD
CttC
CttB
CtA
Which of the following equations describes the graph Which of the following equations describes the graph of the sine function shifted down by C units, left by D of the sine function shifted down by C units, left by D units, stretched horizontally by A units, and vertically units, stretched horizontally by A units, and vertically by B units?by B units?
above theof None.
)(1
sin.
)(2
sin.
)(1
sin.
)(2
sin.
E
CDxA
ByD
CDxA
ByC
CDxA
ByB
CDxA
ByA
)csc(
)csc(
xdxd
x
above theof None.
)csc(ln.
)tan(.
1.
D
xC
xB
A
What is the equation describing the following graph?What is the equation describing the following graph?
above theof None.
2
5
2
1sin5.
2
5
2
1cos5.
2
5
2sin
2
5.
2
5
2cos
2
5.
E
xyD
xyC
xyB
xyA
2 4 6-2-4-6
5
Given Given f(t)f(t) gives the velocity of a ball in feet per gives the velocity of a ball in feet per second as a function of the time in seconds since the second as a function of the time in seconds since the ball was thrown into the air, what are the units ofball was thrown into the air, what are the units of
A. seconds
B. ft/sec
C. feet
D. ft/sec2
E. None of the above
3
1)( dttf
Given Given F(5)=6F(5)=6, use the graph of , use the graph of F’(x)F’(x) below to find below to find F(-4).F(-4).
A. F’(-4)=-2
B. F’(-4)=14
C. F’(-4)=16
D. None of the above
A1=4
A3=5
A2=1
Which of the following is a possible formula for the Which of the following is a possible formula for the function graphed below?function graphed below?
above theof None.
above theof All.
24
2cos.
22
2sin.
22
2sin.
E
D
xyC
xyB
xyA
Given 3log(x)=12, which of the following Given 3log(x)=12, which of the following must be true?must be true?
A. x=1036
B. 3x=1012
C. x3=1012
D. x=104
E. None of the Above
above theof None.
25log
19log.
519log
25log.
519
25.
519log
25log.
E
tD
tC
tB
tA
Given 19Given 19t+5t+5=25, what is t?=25, what is t?
above theof None.
)ln()ln()ln(.
)ln()ln(ln.
)ln(
)ln()ln(.
)ln(
)ln(ln.
E
BABAD
BAB
AC
B
ABAB
B
A
B
AA
Which of the following formulas is/are correct?Which of the following formulas is/are correct?
Given Given f(x)=af(x)=axx+b+b, when , when aa is greater than one, is greater than one, f(x)f(x) is everywhere ______________. is everywhere ______________.
A. positive
B. negative
C. increasing
D. decreasing
E. None of the Above
Given the following graph of F(x), which are Given the following graph of F(x), which are inflection points of F(x)?inflection points of F(x)?
A. A, C, F, K, M
B. E, G, M
C. E, G
D. B, D, H, I, J, L, N
E. None of the Above
A B C
D
EF G
H
I
J
K
L
MN
Suppose Suppose pp is a critical point of the function is a critical point of the function ff((xx). ). p p is a local minimum if _____________. is a local minimum if _____________.
A. f’(p) is positive
B. f’(p) is negative
C. f’(x) switches from positive to negative at p
D. f’(x) switches from negative to positive at p
E. None of the Above
above theof None .
above theof All.
only and .
.
.
.
)ln(
1
ln
ln
F
E
CBD
CxC
exB
e
exA
A
A
C
A
C
Given AGiven Aln(x)ln(x)=C, what is x?=C, what is x?
above theof None.
65)(''.
260)(''.
260)(''.
65)(''.
3
3
4
4
E
xxfD
xxfC
xxfB
xxfA
Find the second derivative of Find the second derivative of f(x)=-13xf(x)=-13x55
Given the following graph of the derivative Given the following graph of the derivative F’(x), which are inflection points of F(x)?F’(x), which are inflection points of F(x)?
A. A, C, F, K, M
B. E, G, M
C. E, G
D. B, D, H, I, J, L, N
E. None of the Above
A B C
D
EF G
H
I
J
K
L
MN
Given the following graph of the derivative Given the following graph of the derivative F’(x), which are local minima or maxima of F’(x), which are local minima or maxima of F(x)?F(x)?
A. A, C, F, K, M
B. E, G, M
C. E, G
D. B, D, H, I, J, L, N
E. None of the Above
A B C
D
EF G
H
I
J
K
L
MN
How many critical points does f(x)=2xHow many critical points does f(x)=2x3 3 have?have?
A. 0
B. 1
C. 2
D. 3
E. None of the Above
What is the global maximum of f(x)=-(x-2)What is the global maximum of f(x)=-(x-2)22+4+4
A. (-2,4)
B. (4,-2)
C. (4,2)
D. (2,4)
E. None of the Above
Where is f(x)=xWhere is f(x)=x33 concave up? concave up?
A. (-, )
B. (- ,0]
C. [0, )
D. (0, )
E. None of the Above
axxFE
baxFDab
aFbFC
aFbFB
aFA
at )( of change of rate ousinstantane The.
and between )( of change of rate average The.
)()(.
)()(.
)('.
Given Given F’(x)=f(x)F’(x)=f(x), the marked quantity on the graph of , the marked quantity on the graph of f(x)f(x) represents: represents:
A1=4
a b
The following is a graph of the velocity of two cars The following is a graph of the velocity of two cars AA and and B. B. Given this graph, which of the following Given this graph, which of the following statements are true?statements are true?
A. Car A travels further
B. Car A went faster
C. Car A stops first
D. Car B travels further
E. Car B went faster
F. A, B, and C
G. B, C, and D
H. A, C, and E
I. None of these
Car B
Car A
For which graph is the statement “has a local and For which graph is the statement “has a local and global minimum at x=3, but no local or global global minimum at x=3, but no local or global maximum” true?maximum” true?
A
B
C
D
E. None of these
Which of the following is the graph of F(x), which is Which of the following is the graph of F(x), which is the graph of F’(x), and which is the graph of F’’(x)?the graph of F’(x), and which is the graph of F’’(x)?
F’(x)
F’’(x)
F(x)
Between x=-1 and x=2, the graph of the derivative Between x=-1 and x=2, the graph of the derivative f’(x)f’(x) is __________. is __________.
A. increasing
B. decreasing
C. both
D. neither
f(x)
Between x=-1 and x=2, the graph of the derivative Between x=-1 and x=2, the graph of the derivative f’(x)f’(x) is __________. is __________.
A. positive
B. negative
C. both
D. neither
f(x)
Which of the following statements can be true of a Which of the following statements can be true of a continuous function F(x) for -continuous function F(x) for - ≤ x ≤ ≤ x ≤
A. 1
B. 2
C. 3
D. 1 and 2
E. 1, 2, and 3
F. None of these
1. F(x)>0, F’(x)<0, F’’(x)>0
2. F(x)<0, F’(x)<0, F’’(x)<0
3. F(x)<0, F’(x)>0, F’’(x)>0