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Math 112
Review Exercises for the Final Exam
The following are review exercises for the Math 112 final exam. These exercises
are provided for you to practice or test yourself for readiness for the final exam. There are
many more problems appearing here than would be on the final. These exercises
represent many of the types of problems you would be expected to solve on the final, but
are not meant to represent all possible types of problems that could appear on the final
exam.
Your final exam will be in two parts: the first part does not allow the use of a
calculator, and the second part does allow the use of a graphing calculator. Since the
exercises in this review sheet are mixed together, we have put a symbol next to
exercises or parts of exercises where you WILL be allowed to use the graphing
calculator: otherwise you should be able to solve the problem WITHOUT a calculator.
Such a symbol will not be on the final exam. Please note that for the final, you may use
any graphing calculator except the TI- 89 , TI-Inspire, and any calculator with a QWERTY
keypad.
Show all your work: unsupported results may not receive credit.
Chapters 1, 2 and 3 review.
I. Factor completely:
(a) 2 𝑥^7 /^2 − 5 𝑥^5 /^2 − 3 𝑥^3 /^2 (b) (𝑥^2 + 1 )^1 /^2 − 5 (𝑥^2 + 1 )−^1 /^2
II. Solve the following for real values of x :
(a) 𝑥 4
- 3 𝑥 2 − 4 = 0 (b) | 3 𝑥 − 2 |^ + 4 = 5 (c) | 5 𝑥 + 2 |^ + 1 ≤ 5
III. Find the equation of a circle with center (− 1 , 3 )^ and radius 4.
IV. Given 𝑓(𝑥) = 2 𝑥^2 − 3 𝑥 + 1 and 𝑔(𝑥) =
𝑥 𝑥− 2
, find and simplify the
following:
(a)
𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ
(b)
𝑔(𝑥+ℎ)−𝑔(𝑥) ℎ
V. Sketch the graph of 𝑓(𝑥) = {
𝑥^2 𝑖𝑓 − 1 ≤ 𝑥 < 1
VI. Find the domain of the following functions and express your answer using interval notation.
(a) 𝑓(𝑥) =
𝑥 + 4 𝑥^2 − 9 (b) 𝑓(𝑥) = (^) √ 3 − 𝑥 − (^) √ 2 + 𝑥
(c) 𝑓(𝑥)^ = √𝑥^2 − 7 𝑥 − 18
VII. Given 𝑓(𝑥) =
1 𝑥
and 𝑔(𝑥) =
𝑥 𝑥 + 3
, find and simplify (𝑔𝑜𝑓)(𝑥)
VIII. Find the exact Maximum/Minimum value of the function 𝑓(𝑥) = 2 𝑥^2 + 7 𝑥 + 5
1. Sketch the graph of the following:
Using interval notation, state the domain and the range. State the equation(s) of the asymptote(s). Find the x- and y-intercepts where they exist.
(a) ( ) 3 4 2 = + x − f x
(b) f ( x )= ln( x − 2 )+ 3
(c) ( ) 5 1 3 = + x − f x
(d) f ( x )= log( x − 1 )+ 2
2. Given log (^) a 5 = 2. 3 and log (^) a 3 = 1. 6 , fill in the table below with the appropriate values.
X 15 9 35 5a^32 a
log ax
3. Which is the following is larger: log 328 or log 463? JUSTIFY YOUR ANSWER
FOR CREDIT
4. Find the EXACT solution for the following:
(a)log( x + 2 )−log( x )= 3
(b)log(^ x^ −^3 )=^1 −log( x )
(c)
2 x 1 3 x 2 = 8
(d) 2 3 3 3 27 − + = x x
(b) sin 2 x given: 5
sin x = and x is in Quadrant II.
(c) sin( x + y )given: 3
sin x =− and x is in Quadrant III; and 3
sin y = and y is in
Quadrant II.
13. Verify the identity:
(a) x x
x (^) 2 2
2 sin cot
1 sin
(b) x x
x
x
x 2 sec 1 sin
cos
1 sin
cos
−
14. Find the following exactly in degrees:
(a)
− 2
sin
1
(b)
− 2
cos
1
15. Find the following EXACT:
(a)sin[cos ( 2 / 3 )] − 1
(b)sin[tan ( 6 )]
1 −
−
16. If =arcsin x , express tan in terms of x.
17. (a) Suppose on Jan 1, 1997 Dave invested $2,000 into a bank account at 5% interest compounded continuously. Let y(t) be the value of Dave's investment after t years. Give an exact formula for y(t) (b) Also on Jan 1, 1997 John decides to invest. He put $2,500 into an account at 3% interest compounded monthly. Let g(t) be the value of John's investment after t years. Give an exact formula for g(t). (c) Which account is worth more after 9 years? [ Must show work to receive credit.] (d) To the nearest tenth, at what time t is the value of both accounts the same? 18. Evaluate the following EXACTLY.
(a) 7 log 2 64
(b) 7 log 3 81
19. Evaluate to two decimal places.
(a)log 527
(b)log 315
20. Given the following functions f ( x )below, find ( ) 1 f x
− .
(a) x f ( x )= 3
(b) f ( x )=log 3 x
(c) ( ) 10 3 1 = − x + f x
(d) f ( x )= log( x − 4 )+ 5
(e) ( ) 4 2 1 = −
x + f x e
(f) f ( x )= ln( 2 x + 3 )− 5
21. Sketch a graph of the following. Label the asymptotes. Find the intercepts EXACT.
Find ( ) 1 f x −
. Find the intercepts accurate to two decimal places.
(a) ( ) 3 2 = − x + f x e
(b) f ( x )= ln( x + 4 )+ 1
22. From 1990 to 2000 the student tuition at a University grew from $12,000 to $18,000.
(a) Using the exponential growth model, determine r , the annual rate of increase for the population as a decimal accurate to 3 places (b) Assuming the same growth rate use r found in Part (a) above, find in what year (to the nearest year) the tuition of Rutgers will reach $30,000.
23. Carbon dating is commonly used to determine how old an object is by measuring the
amount of carbon-14 that is left in an object as the object decays over the years. This decay proceeds exponentially with a half-life of approximately 5800 years. How old (to the nearest year) would carbon dating say a piece of bone is when the amount of carbon-14 has decayed from its original amount of 100 grams to a final amount of 22 grams?
27. Find the length of a 25 o arc with radius 12 inches accurate to 2 decimal places. 28. Two trains, Train A and Train B, leave a train station at 10:00 AM traveling along straight tracks at 80 and 90 mi/hr respectively. If the angle between their directions of travel is 118 o, to the nearest mile, how far are the trains from each other at 11:30 AM? 29. Given t = 4 , complete the following:
a. Using t = 4 , sketch on the unit circle the approximate location of P(x,y) , the terminal point
b. Find the reference number for t (to two decimal places)___________________
c. What is the terminal point determined by t?
(Give to two decimal places) P = ( , )
30. The top of a volcano is viewed from a safe distance of 20,000 meters level to the base of the volcano. The angle of inclination is found to be 22 degrees. If the angle of incline from the base of the volcano to its summit is found to be 42 degrees, to the nearest foot, how high is the volcano?
31. A pilot in an airplane flying at 25,000 ft sees two towns directly ahead of her in a straight line. The angles of the depression to the towns are 25 o and 50 o , respectively. To the nearest foot, how far apart are the towns?
ANSWERS: MATH 112 FINAL EXAM REVIEW EXERCISES
I(a) 𝑥^3 /^2 ( 2 𝑥 + 1 )(𝑥 − 3 ) I(b) (𝑥^2 + 1 ) −^12 (𝑥 − 2 )(𝑥 + 2 )
II(a) 𝑥 = − 1 , 1 II(b) 𝑥 =
1 3 , 1 III(c) −
6 5 ≤ 𝑥 ≤
2 5 III (𝑥 + 1 )^2 + (𝑦 − 3 )^2 = 16
IV(a) 4 𝑥 + 2ℎ − 3 IV(b)
− 2 (𝑥− 2 )(𝑥+ℎ− 2 ) (V)
(VI)(a) (−∞, − 3 ) ∪ (− 3 , 3 ) ∪ ( 3 , ∞) (VI)(b) [− 2 , 3 ] (VI)(c) (−∞, − 2 ] ∪ [ 9 , ∞)
(VII)
1 1 + 3 𝑥
(VIII) Minimum value is −
9 8
4. (a) 999
x = (b) x = 5 (c) 7
x = (d) x =− 12 (e) x = 2
5. (a) (i) 3
x = (ii) x n n
= + + , n an integer.
(b) (i) 2
x = (ii) x n
2 2
= + , n an integer.
(c) (i) 4
(^) x = (ii) x n n n n
2 4
(^) = + + + + , n an integer.
6. x 0. 23 , 0. 50 7. (a)
( ) 6 cos 2
f x x (b)
( ) 5 sin 3
f x x
Period = Period =^23
Amplitude = 6 Amplitude = 5
Phase shift = 6 right Phase shift = 6 left
8. (a) 4
(b) 4
9. 45 o^ , 2
10. (a) 7
sin t =− , 4
tan t =− , 33
cot t =− , 4
sec t = , 33
csc t =−
(b) 8
cos t = , 55
tan t =− , 3
cot t =− , 55
sec t = , 3
csc t =−
sin t = , 41
cos t =− , 4
tan t =− , 5
cot =− , 4
sec t =− , 5
csc t =
12. (a) 25
(b) 25
− (c) 9
13. (a) x x
x x
x
x
x
x
x
x
x (^) 2 2
2 2
2
2
2
2
2
2
2 sin cos
sin cos
sin
cos
cos
cot
cos
cot
1 sin = = = =
(b) = − +
( 1 sin )cos
( 1 sin )( 1 sin )
( 1 sin )cos
1 sin
cos
1 sin
cos
x x
x x
x x
x x
x
x
x
x
= − +
− + +
− +
− + +
( 1 sin )( 1 sin )
cos sin cos cos sin cos
( 1 sin )( 1 sin )
( 1 sin )cos ( 1 sin )cos
x x
x x x x x x
x x
x x x x
x x x
x
x
x
x x
x 2 sec cos
2
cos
2 cos
1 sin
2 cos
( 1 sin )( 1 sin )
2 cos 2 = 2 = = −
= − +
14. (a) – 30 o (b) 150 o 15. (a) 3
(b) 37
2 1 x
x
−
17. (a) t y t e
- 05 ( )= 2000 (b)
t g t
12
= + (c) y(9) = $3,136.62 and
g(9) =$3,273.81 Hence John’s account was worth more. (d) t = 11.1 years
18. (a) 6/7 (b) 4/7 19. (a) 2.05 (b) 2.46 20. (a) f x 3 x 1 ( )=log − (b) x f ( x ) 3 1 = −
(c) ( ) log( 3 ) 1 1 = + − − f x x (d) ( ) 10 4 1 5 = + − x − f x (e) 2
ln( 4 ) 1 ( )
− x f x
(f) 2
5 1 − =
−
x e f x
22.(a) r 0. 041 (b) 2012 23. 12,670 years old 24. (a) 75 o^ (b) 49 o
