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Transcript 
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
1. Write the following expression as a logarithm of a single quantity.
ln x  6ln  x 2  1
Ans:


x

ln 
  x 2  16 


2. Find the following limit if it exists: lim ln  x – 5 . Use  when appropriate.
–
x 5
Ans: does not exist
3. Find the following limit if it exists: lim ln  x – 2  . Use  when appropriate.

x 2
Ans: 
4. Find an equation of the tangent line to the graph of y  ln  x11  at the point (1,0).
Ans: y  11( x  1)
5. Find the derivative of the function y  ln(ln x 71 ) .
Ans: dy
71

dx ln( x )
6.
 11x 
Find the derivative of the function f ( x )  ln  2
.
 x  16 
Ans:
11
2x
f ( x )   2
x x  16
7.
Find the derivative of the function y  ln x x 2  2 .

Ans: 1
x
 2
x x 2
8. Find the derivative of the function y  ln x 2  4 .
Page 1

Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
Ans: dy
x
 2
dx x  4
9. Differentiate the function f  x   ln  4 x 2 – 2 x + 13 .
Ans:
8x – 2
4 x – 2 x + 13
2
10.
dy
.
dx
Use implicit differentiation to find
4 x 2 + 5ln xy  12
Ans:
y (8 x 2 + 5)

5x
11.
dy
.
dx
Use implicit differentiation to find
x 6  7ln y  5
Ans:
6 x5 y

7
12.
Find
Ans:
1
 x ln  x  dx .
4
1
1
 x ln  x  dx  4 ln ln  x   C
4
4
 ln x 
16
13.
Find the indefinite integral
Ans:
 ln x 

x
dx .
17
17
C
14. Find the indefinite integral.
x 2  20 x  9
 x3  30 x 2  27 x  9 dx
Page 2
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
Ans: 1
ln x 3  30 x 2  27 x  9  C
3
15. Find the indefinite integral.
x3
 –2 x 4  1 dx
Ans:
1
– ln –2 x 4  1  C
8
16.
x2
Find the indefinite integral  3
dx .
6x  3
Ans: 1
ln 6 x 3  3  C
18
17. Find the indefinite integral.
cos  4 
 sin  4  dx
Ans: 1
ln sin  4   C
4
18. Find the indefinite integral.
 csc(18x)dx
Ans: 1
ln csc(18 x )  cot(18 x )  C
18
19. Find
Ans:
 tan8 d .
1
 tan8 d   8 ln cos8
C
20. Evaluate the definite integral.
Page 3
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
1  cos 16 
d
1 16  sin 16 
Ans: 1
240  sin  240 
ln
16
16  sin 16 

21.
15
Evaluate the definite integral

e6
e
1
dx .
x ln  x 2 
Ans: 1
ln  6 
2
22.
Evaluate

e
 3  ln x 
5
dx .
x
Ans: e  3  ln x 5
3367
1 x dx  6
1
23. Find f 1 ( x ) if f ( x)  x 3  7 .
Ans: f 1 ( x )   x  7 1/ 3
24. Find f 1 ( x ) if f ( x)  x 9 .
1
Ans:
f 1 ( x )  x 9
25. Find f 1 ( x ) if f ( x )  2 x 2 , ( x  0 ).
Ans:
x
f 1 ( x ) 
2
26. Solve the following equation for x .
ln  x  6  2
3
Ans: x  e 3 2  6
27. Solve the following equation for x .
Page 4
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
ln x –4  2
Ans: x  4 e2
28. Solve the following equation for x .
2  8e3 x  10
Ans:
1  3
x  ln  
3 2
29.
Find G( x ) if G( x )  
e6 x
0
ln  t  4  dt .
Ans: G( x )  6e6 x ln  e6 x  4 
30. Find the derivative of the function f  x   6e x cos  x  . Simplify your answer.
Ans: 6e x  cos( x   sin  x) 
31.
ex
Find the derivative of the function f  x  
. Simplify your answer.
1 – 2 x2
Ans: e x  –2 x 2 + 4 x  1
1 – 2 x 
2 2
32.
ex + 4
Find the derivative of the function f  x   x
. Simplify your answer.
e –4
Ans:
–8e x
e
x
– 4
2
33. Find the derivative of the function f  x   5e7 x – 5e –6 x .
Ans: 35e7 x + 30e–6 x
34.
 e6 x  1 
Differentiate the function f  x   ln  5 x
.
 e 1
Page 5
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
Ans:
35.
6e6 x
5e5 x
–
e 6 x  1 e5 x  1
6
dy
if y  x8e x .
dx
6
Ans: dy
 e x 8 x 7  6 x13 
dx
Find
36. Find the derivative of the function f  x   x 5e x .
Ans: x 4e x  x  5
37.
6
dy
if y  e8 x .
dx
6
Ans: dy
 48 x 5e8 x
dx
Find
38. Evaluate the following expression without using a calculator.
1
log 2  
 16 
Ans: 4
39. Solve the following equation for x .
log5 x  log5  x  4  1
Ans: 5
40. Solve each of the following equations for x .
(i) x 2  3x  log5 625
Ans:
(i) –1, 4
,
(ii) 9 x  2  log4 64
,
(ii)
1
9
41. Solve each of the following equations for x .
Page 6
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
log4 256  x , log 4
Ans: 4 , 2
1
x
16
42. Solve the following equation for x .
log2  x  2   3
Ans: x  10
43. Solve the following equation for x .
3  22 x4   105
Ans:
x
log 2 35  4
2
44.
 x 4  10 
Find f ( x ) if f ( x )  log7  4
 .
 x 5 
Ans:
20 x 3
f ( x )  4
 x  10 x 4  5 ln  7
45.
 x2  8 
Find f ( x ) if f ( x )  log3 
.
 x7 
Ans:
x 2  14 x  8

f ( x)  2
 x  8  x  7 ln  3
46. Find f (t ) if f (t )  t 3108t .
Ans: f (t )  3t 2108t  8ln 10 t 3108t
47. Find the following indefinite integral.
x
6
4  dx
Ans:
 x7
7
4  x
C
7 ln  4 
Page 7
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
48. Find the indefinite integral.
6
5x
Ans:
dx
1 5x
6 C
5ln 6
49.

2
Evaluate the expression arcsin  –
 without using a calculator.
 2 
Ans: 5
4
50.
 1
Evaluate the expression arcsin  –  without using a calculator.
 2
Ans:

–
6
51.
7 

Evaluate the expression cos  arcsin  without using a calculator.
25 

Ans: 24
25
52. Write the following expression in algebraic form.
sin  arccos 8x  
Ans:
53.
1  64x 2
6x  1
 x
Find the derivative of the function y  arctan   
.
2
 9  9  x  8
Ans:
dy 1 
1
48  2 x  6 x 2 
 

2
2
dx 9  1   x / 9 2

x

8




Page 8
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
54. Find the derivative of the function h  t   11sin  arccos t  .
Ans:
11t

1  t2
55. Find the derivative of the function f  x   arcsec 12 x  .
Ans:
1
x 144 x 2  1
56. Find the derivative of the function f ( x)  6arcsin  x  10 .
Ans:
6
f ( x ) 
20 x  x 2  99
57. Find the derivative of the function f ( x)  2arcsin  3x 2  5x  7  .
Ans:
2  6 x  5
f ( x ) 
1   3x 2  5 x  7 
2
58. Find the indefinite integral.
x
1
36 x 2  25
dx
Ans: 1
 6x 
arcsec 
C
5
 5 
59. Find the indefinite integral.
1
 49   x  7
2
dx
Ans: 1
 x7
arctan 
C
7
 7 
60.
Find the integral
t
4
t
dt.
 36
Page 9
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
Ans:
61.
Find the integral
Ans:
62.
1
t2
arctan  C
12
6

2
x
dx.
 23

23
arctan 
cos x   C
23
 23

Evaluate the integral
Ans:
sin x
 23  cos

3
3
14
dx.
9  x2
7

18
63. Find the indefinite integral.
x
2
Ans:
2x  3
dx
 8 x  32
ln x 2  8 x  32 
11
 x4
arctan 
C
4
 4 
64. Find the indefinite integral.

dx
 x  20 x
Ans:
 x  10 
arcsin 
C
 10 
2
65. Find the derivative of the function g  x   2sec h 2 8 x.
Ans: 32sec h2 8x tanh8 x
66.
Find the derivative of the function h  x  
13
13x
sinh 2 x 
.
4
2
Ans: 13sinh 2 x
Page 10
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
67.
1
x x
Find the derivative of the function y  sinh  4 x   14coth    .
8
7 8
Ans: dy 1
 x 1
 cosh  4 x   2csch 2   
dx 2
7 8
68. Find the derivative of the function y  ln  cosh5 10 x   .
Ans: dy
 50 tanh 10 x 
dx
69. Find the derivative of the function y  sech 5x  6 .
Ans: dy
 5sech 5 x  6 tanh 5 x  6 
dx
70.
Find the integral of
Ans:
x4
 6 x10  6.
1
arctan  x 5   C
30
71. Find the indefinite integral.
7
6
2 x 
x
csch
 7  dx

 
Ans:
 x7 
 coth    C
 7 
72. Find the indefinite integral.
 sinh 5  10x  dx
Ans: 1
cosh  5  10 x   C
10
73.
Evaluate the integral

3
0
1
64  x 2
dx.
Page 11
Prof. Israel N Nwaguru
MATH 2414
CHAPTER 5 - REVIEW.
PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/PAPERS THEN
CHOSE THE BEST ANSWER. YOU MUST SHOW WORK. AGAIN " NO WORK NO CREDIT ".
Ans:
74.
arcsin
3
8
Evaluate the integral
Ans:

ln 5
0
6 tanh xdx .
 13 
6 ln  
5
75. Find the derivative of the function y  4sinh 1  tan 6 x  .
Ans: y  24 sec6 x
76. Find the derivative of the function y  cosh 1  4 x  .
Ans:
4
y 
16 x 2  1
77. Write the following expression as a logarithm of a single quantity.
6ln x  3ln  x 2  15
Ans:


x6

ln 
  x 2  153 


Page 12
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