Download Final Exam Review

New York City College Of Technology
Department Of Mathematics
MAT 1375 – Final Examination Review Sheet
Revised by Professor S. Singh (2009), A. Mukhin (2010), G. Niezgoda (2011)
1. Solve the following inequalities and express your answers in interval notation.
a. x  6 x  5  0
2
u  ln x
2. Let
a)
ln

and
x y2

c. x  9  0
b. x 2  3 x  2
2
d.
x5
0
x 2  2x  3
v  ln y . Write the following expressions in terms of u and v.
b)
ln

3
x5 y 4

3. For each of the following functions,
(a) state the domain of the function, (b) find x- and y-intercepts (if any),
(c) find vertical and horizontal asymptotes (if any) of the graph, (d) sketch the graph.
2x  3
ii. g ( x) 
,
x 1
3x  6
i. f ( x)  2
,
x  6x  8
4. Let
3x 2  x  2
iii. h( x)  2
.
x x2
q( x)  ax 2  bx  c , where a, b and c are non-zero constants.
Evaluate the following expressions:
ii. q ( 2  x )
i. q (  x )
iii.
q( x 2 )
iv.
q ( x  h)  q ( x )
, h ≠ 0.
h
5. Solve the absolute value inequalities
a. | 4  5 x | 4
b. | 5  2 x | 7
6. Find all roots of
f ( x)  x 3  x 2  5 x  5 exactly. Sketch a complete graph of f(x) and label all roots
clearly.
7.
Let
v =  5 3,5  . Find the magnitude and direction angle of v.


2
2
p = 2(cos( )  i sin( )) , and q = 5(cos( )  i sin(
)) . Write the following in standard ( a  bi )
2
2
3
3
p
form: a.
b. pq
q
8. Let
c. Find the product and write the result in a+bi form:
[4(cos 45


+ i sin 45 )][3(cos 15


+ i sin 15 )]
9. For the functions below, find the amplitude, period, and phase shift. Draw the graph over
a one-period interval. Label all maxima, minima and intercepts.
a.
b.
y = 3cos(2x)
y = sin (
2x
)
3
c.

y = -3sin ( 2 x  )
3
10. Use graphical methods to find all real solutions of the equations, approximating to the nearest tenth
if necessary:
a.
x 4  5x 2  x  2  0
b.
x 4  2 x 3  8 x 2  10 x  15  0
11. Find all exact solutions of trigonometric equations. Use radian measure of angles.
a.
2 cos x 1  0
b. tan
2
x 1
c. 4 sin
2
x  4 sin x  1  0
12. The initial population of a colony is 10,000 and is decreasing exponentially
at 1.5% per year.
a. What is the size of the colony in 5 years?
b. How long will it take for the population to be half of its initial amount?
13. Use the Binomial Theorem, write in simplest form:
8
y

a. The first four terms of the expansion of  2 x   .
2

8
y

b. The 7 term in the expansion of  2 x   .
2

th
14. a. Find the sum of the first 65 terms in the arithmetic sequence below:
-6, -2, 2, 6, 10, . . .
b. Find the sum of the infinite geometric sequence:
24, -12, 6, -3, . . .
MAT 1375 Review. Answer Sheet.
1. a. (−∞, 1] ∪ [5, +∞); b. [1, 2]; c. (−∞, -3) ∪ (3,+∞); d. (−5, −1) ∪ (3, +∞);
2. a.
1
u  2v
2
b.
5
u  4v
3
3. i. (a) (−∞, -4) ∪ (-4, -2) ∪ (- 2, +∞); (b) x-intercept: none,
(с) vertical asymptotes:
x  4 and x  2
y-intercept (0,  3 )
4
, horizontal asymptote:
y0
(d) Graph
ii.
(a) (−∞, -1) ∪ (-1, +∞); (b) x- intercept: ( 
(с) vertical asymptote:
3
, 0 ), y-intercept (0, 3)
2
x  1 , horizontal asymptote: y  2
(d) Graph
iii. (a) (−∞, -1) ∪ (-1, 2) ∪ (2, +∞); (b) x- intercepts: ( 
(с) vertical asymptotes:
(d) Graph
x  1 and x  2 ;
2
, 0 ) and (1, 0 ) ; y-intercept (0, 1)
3
horizontal asymptote:
y3
ax 2  bx  c
8
a. 0  x 
5
4. i.
5.
ii.
ax 2  (4a  b) x  (4a  2b  c)
x  1
b.
or
iii.
ax 4  bx 2  c
iv.
x6
x  1, x   5 ; Graph :
6.
7.
|
v | = 10,
θ=
11
.
6
3 1
 i
5 5
8. a.
b.
 5 3  5i
c.
6  6 3i
A  3, T   ,   0
9. a.
Graph
c.
Graph
A  3, T   ,  

Graph:
6
10.
a. x = -2.3, x = 0.6, x = 0.8, x = 2
11.
a.
x
b.
x
c.
x  (1) n 
2
 2  n, and
3

4
12. a. 9272;
13. a.
A  1, T  3 ,   0
b.
   n, and

6
x 
x 
   n, where

4
b. x = -2.2, x = -1, x = 2.2, x = 3
2
 2  n , where
3
   n , where
n  0,1,2,3,...
n  0,1,2,3,...
b. Approximately 45.9 years
256 x 8  512 x 7 y  448x 6 y 2  224 x 5 y 3
14. a. 7930;
n  0,1,2,3,...
b. 16
;
b.
7 2 6
x y
4
2ax  ah  b