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STUDY GUIDE FOR FINAL EXAMINATION
MATH 1113
1.
2.
3.
4.
5.
Let f ( x)  x 2  2 x . Compute the following.
(a) f (3)
(c) f (a)
(b)
(d)
(e)
(f)
f (x + h)
f (0)
f (a + 2)
f ( x  h)  f ( x )
, h0
h
Find the domain of the following functions.
(a)
f ( x)  x 2  2 x
(b)
f ( x) 
(c)
f ( x) 
(d)
f ( x) 
(e)
(g)
(i)
(k)
f ( x)  e
f ( x)  sin x
f ( x)  csc x
x3
x
1
x3
1
x3
f ( x)  ln x
f ( x)  tan x
(f)
(h)
(j)
f ( x)  sin 1 x
(b)
f ( x)  4 x 2  2 x  8; g ( x)  2 x  3
(b)
f ( x)  x 3  3
f ( x)  tan 1 x
Find a simplify ( f ◦ g)(x).
3
2
; g ( x) 
(a) f ( x) 
x2
3x
Find the inverse of the following functions.
1
(a) f ( x)  3
3 x
Sketch the graph of the inverse of the following function.
6.
2
Put the following equations of conic sections in standard graphing form by completing the square.
Then graph the conic sections. Give the following information for each conic section.
Parabola: vertex, axis of symmetry, focus, and directrix.
Ellipse: center, vertices, covertices, major axis, minor axis, foci, and eccentricity.
Hyperbola: center, vertices, transverse axis, conjugate axis, foci, and eccentricity.
(a) 4 x 2  y 2  16 x  2 y  13  0
(b) x  2 y 2  2 y  3
(c) 9 x 2  4 y 2  36 x  8 y  4  0
7.
Find the equation, in standard form, of the parabola with vertex (7, −9) and focus (5, −9).
8.
Find all the asymptotes of the graph of R ( x) 
9.
Consider the function R( x) 
5x  2
.
4x 2  7x
3x 2  5 x  2
. The graph of R will have a vertical asymptote at
x2  4
_______ and will have a missing point at x = __________.
x 2  17 x  72
10. Find the oblique asymptote of the graph of R( x) 
.
x3
11.
Suppose a radioactive substance Barnesvillium has a half-life of 77 years. If 258 grams are present
initially, how much will be left after 212 years?
12.
What is the half-life of a radioactive element that decays to 33% of the original amount present in
77 years?
13.
Find the balance when$19,000 is invested at an annual rate of 6% for 5 years if the interest is
compounded
(a) daily
(b) continuously.
14.
How long will it take money to double at 17% interest compounded monthly?
15.
(a)
(b)
The angle of depression from a tower to a landmark is 19° and the landmark is 400 feet
below the tower. How far away is the landmark?
Suppose the angle of elevation of the sun is 23.4°. Find the length of the shadow cast by
Cindy Newman who is 5.75 feet tall.
3
4

and     , find sec θ and cot θ.
5
2
16.
If cos   
17.
Find where the graphs of the following functions have vertical asymptotes or horizontal
asymptotes or state there are no asymptotes.
(a) y  sin x
(b) y  tan x
(c) y  csc x
(d) y  cos 1 x
(e) y  tan 1 x
18.
Find the exact value of the following.

3

(a) sin 1  

2


19.
(b)
(c)
tan 1 1
(d)
(e)
1

tan  tan 1 


(f)
Solve the following trigonometric equations.
(a) 2 sin x  1  0
1
(c) cos 3 x  
2

2

cos 1  

2


 5 
sin 1  sin

4 

   
cos 1 cos  
  3 
(b)
2 cos 2 x  sin x  1  0
(d)
tan 3 x  tan x  0
20.
Approximate the solutions for each equation on the interval 0 ≤ x ≤ 2π. Round your answer to four
decimal places.
(a) sin x = −0.739
(b) cos x = 0.4201
(c) tan x = −5.17
21.
Establish the following trigonometric identities.
tan x
 sec x
(a)
(b)
sin x
sin x  cos x
 1  cot x
(c)
(d)
sin x
cos x  1
 cot x  csc x
(e)
sin x
22.
Solve the following oblique triangles.
(a) A = 30°, b = 12, c = 16
(c) B = 62.3°, C = 51.9°, c = 11.3
(e) A = 22°, a = 10, c = 15
(b)
(d)
sin 2 t  cos 2 t
 cot t
tan t
1  cos 
cos 2 

cos   1
tan 2 
a = 3.55, b = 4.08, c = 2.83
C = 57.47°, b = 8.841, c = 0.777
4
23.
Find the area of the triangle ABC given a = 8.5, B = 102°, and C = 27°.
24.
Convert the following polar equations into rectangular equations.
(a) r  7
(b) r  4 sin   0
(c) r  2 tan 
25.
Convert the following rectangular equations into polar equations.
(a) x + y = 1
(b) y 2  2 x  x 2
(c) The circle centered at (0, −4) with radius 4.
26.
(a)
(b)
3 

Convert  8, 
 into rectangular coordinates.
4 

Convert 5 3 ,  5 into polar coordinates.


ANSWERS
1.
(a)
(c)
(e)
3
a2 – 2a
x2 + 2xh + h2 – 2x – 2h
(b)
(d)
(f)
0
a2 + 2a
2x + h – 2
2.
(a)
(c)
(e)
(g)
(, )
[3, )
(, )
(, )
(b)
(d)
(f)
(,  3)  (3, )
(3, )
(0, )
(h)
all real numbers except
(2n  1)
where n is an integer; that is, all real numbers except the
2

2
all real numbers except 2nπ where n is an integer; that is, all real numbers except the even
multiples of π
[−1, 1]
(k) (, )
odd multiples of
(i)
(j)
3.
(a)
9x
2  6x
4.
(a)
f 1 ( x) 
(b)
1
3
x3
(b)
16 x 2  44 x  38
f
1
( x)  3 x  3
5
5.
6.
(a)
( y  1) 2
ellipse; ( x  2) 
 1 ; center: (2, −1); vertices: (2, 1), (2, −3); covertices: (1, −1),
4
3
(3, −1); foci: 2,  1  3 , 2,  1  3 ; maj. axis: x = 2; min. axis: y = −1; e 
2
2



2
(b)
(c)
1
1
1
5
1

5
parabola;  y     x   ; vertex:  ,   ; axis of symm: y   ; focus:
2
2
2
2
2

2
19
1
 21
 ,   ; directrix: x 
8
2
 8
( x  2) 2 ( y  1) 2

 1 ; center: (−2, 1); vertices: (0, 1), (−4, 1); foci:
4
9
13
 2  13, 1 ,  2  13, 1 ; trans. axis: y = 1; conj. axis: x = −2; e 
2
hyperbola;



7.
( y  9) 2  8( x  7)
8.
vertical asymptotes: x = 0, x 
9.
vertical asymptote at x = −2, missing point at x = 2
10.
y = x − 20
11.
38.3 grams
12.
48.14 years
13.
(a)
(b)
$25,647.32
14.
4.12 years
15.
(a)
(b)
13.29 feet
16.
5
4
sec    ; cot   
4
3
17.
(a)
no asymptotes
(b)
vertical asymptotes at x 
$25,646.69
1161.68 feet
7
; horizontal asymptote: y = 0
4
(2n  1)
where n is an integer
2
18.
(c)
6
vertical asymptotes at x = 2nπ where n is an integer
(d)
no asymptotes
(a)

(c)
(e)
19.
(a)
(b)
(c)
(d)

4
1


3
horizontal asymptotes: y  
(b)
3
4
(d)

(f)

3

2
, y

4
11
 7

 k 2 ,
 k 2 , where k is an integer 

6
 6

11

 7

 k 2 ,
 k 2 ,  k 2 , where k is an integer 

6
2
 6

2

k
2

4

k
2




,

, where k is an integer 

3
9
3
 9


3
5
7


 k 2 ,
 k 2 ,
 k 2 , where k is an integer 
k ,  k 2 ,
4
4
4
4


20.
(a)
(c)
21.
The answer is in working the problems.
22.
(a)
(c)
(e)
23.
≈ 20.6 square units
24.
(a)
(c)
25.
(a)
26.
(e)
3.9731, 5.4516
1.7619, 4.9036
(b)
1.1372, 5.1459
B ≈ 46.94°, C ≈ 103.06°, a ≈ 8.21
(b) A ≈ 58.54°, B ≈ 78.62°, C ≈ 42.84°
A ≈ 65.8°, a ≈ 13, b ≈ 13
(d) no triangle
B1 ≈ 12.19°, C1 ≈ 145.81°, b1 ≈ 5.636; B2 ≈ 123.81°, C2 ≈ 34.19°, b2 ≈ 22.180
x2  y2  7
x4  x2 y2  4y2  0
(c)
1
cos   sin 
r  8 sin 
(a)
 4
r
2,  4 2

(b)
x 2  ( y  2) 2  4
(b)
r  2 cos
(b)
 
5 

10,  ,   10,
 , etc
6 
6 


2