Download mat170 – final exam – review

FORMULAE THAT WILL BE ON THE FINAL EXAM
FINANCE & TRIGONOMETRY FORMULAS
2
2
cos
(x
)
sin
(x
)
1
2
2
1

tan
(x
) sec
(x
)
2
2
cot
(x
)
1

csc
(x
)
cos(
x

y
)

cos(
x
)
cos(
y
)

sin(
x
)
sin(
y
)
tan(
x
)

tan(
y
)
tan(
x

y
)

cos(
x

y
)

cos(
x
)
cos(
y
)

sin(
x
)
sin(
y
)
1

tan(
x
)
tan(
y
)
sin(
x

y
)

sin(
x
)
cos(
y
)

cos(
x
)
sin(
y
)
tan(
x
)

tan(
y
)
tan(
x

y
)

1

tan(
x
)
tan(
y
)
sin(
x

y
)

sin(
x
)
cos(
y
)

cos(
x
)
sin(
y
)
sin(2x)  2sin(x) cos(x)
1  cos(x)
 x
cos   
2
2
cos2 (x)  sin2 (x)

cos(2x)  2cos2 (x) 1
1 2sin2 (x)

tan(2x) 
1  cos(x)
 x
sin   
2
2
1  cos(x)
 x
tan   
1  cos(x)
2
2tan(x)
1 tan2 (x)
c2 a2b22abcos(C
)
sin(A
) sin(B
) sin(C
)


a
b
c
nt
 r
A P1 
 n
A Pert
u∙v = ||u|| ||v||cos(θ)
MAT170 – FINAL EXAM – REVIEW
PART I – ALGEBRA
A. Difference quotient
Find and simplify the difference quotient,
f(
x

h
)
f(
x
)
,h

0
, of each of the
h
following functions:
1
3x
2
)

x

5
x
9
2) f(x
1)
f (x) 
2
x
3
x

4
x
8
3) f()
B. Function composition
2
1. Find ( f  g)(x) and (g f )(x) , where f(x)x x4and g(x)2x3.
2x
( )ln(x1
).
2. Find (g f )(x) , where f(x)e 1, and gx
C. Inverse functions
Find the inverse of each of the following functions:
4x
1) f (x) 3x
x
ln
(
3
x

2
)
4
2) f()
5x2
3) f (x) e
3
4) f(x)x 10
D. Zeros of polynomials
Find the zeros of each of the following functions:
3
2
)

x

x

2
x
1) f(x
3
2
)
x
x
9
x

9
2) f(x
E. Domains of functions
Find the domain of each of the following functions:
1x
1) f(x)x2 9
x1
2) f(x)3x2
2
x
1
)
3) f(x)log(
4) f(x) 82x
F. Exponential equations
Solve each of the following exponential equations. Give the answer, first in
terms of natural logarithms, and then a decimal approximation to two decimal
places:
2x
x
2
x
0
ex
18
0
1) 3 3 42
3) e 7
x
x 1
2) 5  3
G. Logarithmic equations
Solve each of the following logarithmic equations.
x
)
ln(
x
2
)
1
o
g
()
x
l
o
g
(
x

7
)3
.
1) l
3) ln(
2
2
3x1
)5
2(
2) log
H. Application of rational functions
1) Suppose that an insect population in millions is modeled by
10
x1
f(x)
x  0 is in months. What happens to the insect
0
.2x1, where
population after a long time?
2) A company that manufactures calculators has determined that the
15000

20
x
dollars. In
x
(x
)
average cost for producing x calculators is C
the long run, what value does the average cost approach?
I. Application of quadratic functions
1) An astronaut on the Moon throws a baseball upward. The height of the
(
t
)

2
.
7
t2
30
t
6
.
5
ball is approximated by the function h
feet, t is the
time in seconds after the ball was thrown. When does the baseball
reach its maximum height? What is the maximum height of the
baseball?
2) The percent increase for in-state tuition at Arizona public universities
during the years 1990 – 2002 can be modeled by
2
fx
()0

.
1
5
6
x

2
.
0
51
x

0
.
2
, where x=0 represents 1990. What was the
minimum percentage increase in tuition?
J. Application of exponential functions
1) How long will it take for $5000 to grow to $8400 at an interest rate of
6% compounded a) quarterly? b) monthly? c) continuously?
2) Find the interest needed for $3000 to accumulate to $10000 in an
account where the interest is compounded monthly for 7 years.
K. Transformations
1) Find the function g(x) after all of the following transformations are
applied to x 2 .
a. Reflect about the x-axis
b. Shift left 5 units
c. Shift up 3 units
PART II – TRIGONOMETRY
A. Domain, ranges, and graph of trig functions.
1) What is the domain and range of cos(x) ?
2) What is the domain and range of csc(x) ?
3) What is the domain and range of tan(x) ?
3
c
o
s
(
2
x

); Find the amplitude, period and phase shift.
4) Suppose y
B. Pythagorean identity, summation, difference, double angle, half-angle
formula problems.
3
3
3

)  ,   and cos(
)  , 0    find the following
Given sin(
8
2
5
2
1) cos( )
2) sec( )
3) tan( )
4) cot( )
6) cos(2 )
7) sin(2 )
8) tan(2 )
 
9) cos 2 
 )
11) sin(
)
12) cos(

5) csc( )

 
10) sin 2 
C. Inverse trigonometric functions
1
(3x)), where 3x is positive and
1) Find an algebraic expression for sin(cos
in the domain of the given inverse function.
1
(x)), where x is positive and
2) Find an algebraic expression for cos(tan
in the domain of the given inverse function.
D. Trig identities
Verify the trigonometric identities.
x
)
cot(
x
)

sin(
x
)

csc(
x
)
1) cos(
3) sin(2x)sin(x) + 2cos3(x) = 2cos(x)
3
cos(
x
)

sin(
x
)

sin
(
x
)
2

cot(
x
)

cos
(
x
)
2)
sin(
x
)
E. Trig equations
in
(
2
x
)2
c
o
sx

0
1) Solve s
in the interval [0,2 ] .
2
s
i
n
x

5
s
i
n
x

20
2) Solve 2
in the interval [0,2 ] .
F. Arc length
1. Algiers, Algeria and Barcelona, Spain lie on the same line of longitude.
Algiers is located at 36.8°N latitude and Barcelona is located at 41.4° N
latitude. Find the distance between Algiers and Barcelona. Note the radius of
the earth is approximately 3960 miles.
G. Law of sines
1) Solve the following triangle:
B
c
A = 15º
a=7
b = 15
C
2) An aircraft is spotted by two observers who are 5000 meters apart. As the
airplane passes over the line joining the observers, each observer takes
a sighting of the angle of elevation of the airplane. The first observer
sights the plane at 40° and the second observer sights the plane at 35°.
How far away is the airplane from the first observer?
H. Law of cosines
1)A pilot is flying from Jackson, Michigan, to Chicago, Illinois, a distance of
approximately 200 miles. As he leaves Jackson, he flies 20 degrees off course
for 50 miles. How far is he then from Chicago?
2) A tourist stands 100 ft from the base of the Leaning Tower of Pisa.

Looking up at an angle of 52 he finds that the distance from the top of the
tower to where he is standing is 228 ft. Find the angle the Leaning Tower
makes with the ground?
I. Vectors and Dot Product
1) v = 5i – 4j, w = 2i – j
a) 2v + 3w =
b) v – 5w =
2) v = 6i + j, w = 2i –3j
a) v·w =
b) ||v||=
c) || 2w ||=
d) What is the angle between v and w?
ANSWERS
(Some of these answers have the possibility of not being correct)
PART I – ALGEBRA ANSWERS
1
A. 1) 3x(xh)
2) 2x5h
3) 6x43h
2
2
x)2x
f
g
)
(
x
)

4
xx

1
4

1
6
,
(
g
f
)
(
x
)

2
x

2
x

5 2) (gf)(
B. 1) (
3x
ln(
x
)
2
ex42

1

1
f
(
x
)

f1(x
)
f (x)
f1(x
)3 x
10
C. 1)
2)
3)
4)
4x
5
3
D. 1) zeros are 2, 0, –1
2) zeros are 1, 3i, –3i


,
3
)

(3

,3
)

(
3
,
)
E. 1) (


3.)   2 , 
1
2
2

, )
( ,
)
2) (
3 3
4) (,4]
ln(
7
)
ln(
3
)

log
(
7
)
 
1
.
772) x



2
.
15
9
)2
.197
3
F. 1) x
3) xln(
ln(
3
)
ln(
3
)

ln(
5
)
G. 1) x  8
2) x  11
2
e
3
.16
3) x1e
H. 1) The insect population approaches the size of 50 million.
2) In the long run the average cost approaches 20 dollars.
5
5
I. 1) Maximum time is 5 9 seconds. Maximum height is 89 6 feet.
2) The minimum percentage increase in tuition was approximately 6.57%.
J. 1) a) 8.71 years b) 8.67 years c) 8.65 years
2) 17.3%
()
x

(x
5
)2
3
K. 1) g
PART II – TRIGONOMETRY ANSWERS
A. 1) domain (, ) ; range [ –1, 1]
,
1
]
[1
,
)
2) domain {x|x≠nπ} where n is an integer. ; range (

3) domain {x|x≠ 2 +nπ} where n is an integer. ; range (, )

4) Aplitute = 3, Period =  , Phase shift = – 2
55
8
B. 1) 
2) 
23
6) 32
11)
C. 1)
7)
94 55
40
8 55
55
3 55
32
12)
3)
3 55
55
4)
8)
3 55
23
9) 
8
55
3
5)  3
8 55
10)
4
8 55
4
3 5512
40
1
1 9x2
2) 1  x 2
D. methods vary
 3 5 7
E. 1) 2, 2, 4, 4 2)
 5
6
,
6
F. 1) 318 miles
G. 1) B1 ≈ 33.68° C1 ≈ 131.32°
B2 ≈ 146.32° C2 ≈ 18.68°
2) 2969.05 meter
c1 ≈ 20.31
c2 ≈ 8.66
H. 1) distance ≈ 153.968 mile
2) 77.33°
I. 1) a) 16i – 11j
2) a) 9
b) –5i + j
b) 37
c) 52
 9

cos



65
.
8
d) 


37
13

1


