# 1.4E: Exercises - Mathematics

## Practice Makes Perfect

Use Negatives and Opposites of Integers

In the following exercises, order each of the following pairs of numbers, using < or >.

Exercise (PageIndex{1})

1. 9___4
2. −3___6
3. −8___−2
4. 1___−10
1. >
2. <
3. <
4. >

Exercise (PageIndex{2})

1. −7___3
2. −10___−5
3. 2___−6
4. 8___9

In the following exercises, find the opposite of each number.

Exercise (PageIndex{3})

1. 2
2. −6
1. −2
2. 6

Exercise (PageIndex{4})

1. 9
2. −4

In the following exercises, simplify.

Exercise (PageIndex{5})

−(−4)

4

Exercise (PageIndex{6})

−(−8)

Exercise (PageIndex{7})

−(−15)

15

Exercise (PageIndex{8})

−(−11)

In the following exercises, evaluate.

Exercise (PageIndex{9})

−c when

1. c=12
2. c=−12
1. −12
2. 12
1. Exercise (PageIndex{10})

−d when

1. d=21
2. d=−21

Simplify Expressions with Absolute Value

In the following exercises, simplify.

Exercise (PageIndex{11})

1. |−32|
2. |0|
3. |16|
1. 32
2. 0
3. 16

Exercise (PageIndex{12})

1. |0|
2. |−40|
3. |22|

In the following exercises, fill in <, >, or = for each of the following pairs of numbers.

Exercise (PageIndex{13})

1. −6___|−6|
2. −|−3|___−3
1. <
2. =

Exercise (PageIndex{14})

1. |−5|___−|−5|
2. 9___−|−9|

In the following exercises, simplify.

Exercise (PageIndex{15})

−(−5) and −|−5|

5,−5

Exercise (PageIndex{16})

−|−9| and −(−9)

Exercise (PageIndex{17})

8|−7|

56

Exercise (PageIndex{18})

5|−5|

Exercise (PageIndex{19})

|15−7|−|14−6|

0

Exercise (PageIndex{20})

|17−8|−|13−4|

Exercise (PageIndex{21})

18−|2(8−3)|

8

Exercise (PageIndex{22})

18−|3(8−5)|

In the following exercises, evaluate.

Exercise (PageIndex{23})

1. −∣p∣ when p=19
2. −∣q∣ when q=−33
1. −19
2. −33

Exercise (PageIndex{24})

1. −|a| when a=60
2. −|b| when b=−12

In the following exercises, simplify each expression.

Exercise (PageIndex{25})

−21+(−59)

-80

Exercise (PageIndex{26})

−35+(−47)

Exercise (PageIndex{27})

48+(−16)

32

Exercise (PageIndex{28})

34+(−19)

Exercise (PageIndex{29})

−14+(−12)+4

-22

Exercise (PageIndex{30})

−17+(−18)+6

Exercise (PageIndex{31})

135+(−110)+83

108

Exercise (PageIndex{32})

−38+27+(−8)+12

Exercise (PageIndex{33})

19+2(−3+8)

29

Exercise (PageIndex{34})

24+3(−5+9)

Subtract Integers

In the following exercises, simplify.

Exercise (PageIndex{35})

8−2

6

Exercise (PageIndex{36})

−6−(−4)

Exercise (PageIndex{37})

−5−4

-9

Exercise (PageIndex{38})

−7−2

Exercise (PageIndex{39})

8−(−4)

12

Exercise (PageIndex{40})

7−(−3)

Exercise (PageIndex{41})

1. 44−28
2. 44+(−28)
1. 16
2. 16

Exercise (PageIndex{42})

1. 35−16
2. 35+(−16)

Exercise (PageIndex{43})

1. 27−(−18)
2. 27+18
1. 45
2. 45

Exercise (PageIndex{44})

1. 46−(−37)
2. 46+37

In the following exercises, simplify each expression.

Exercise (PageIndex{45})

15−(−12)

27

Exercise (PageIndex{46})

14−(−11)

Exercise (PageIndex{47})

48−87

-39

Exercise (PageIndex{48})

45−69

Exercise (PageIndex{49})

−17−42

-59

Exercise (PageIndex{50})

−19−46

Exercise (PageIndex{51})

−103−(−52)

-51

Exercise (PageIndex{52})

−105−(−68)

Exercise (PageIndex{53})

−45−(−54)

9

Exercise (PageIndex{54})

−58−(−67)

Exercise (PageIndex{55})

8−3−7

-2

Exercise (PageIndex{56})

9−6−5

Exercise (PageIndex{57})

−5−4+7

-2

Exercise (PageIndex{58})

−3−8+4

Exercise (PageIndex{59})

−14−(−27)+9

22

Exercise (PageIndex{60})

64+(−17)−9

Exercise (PageIndex{61})

(2−7)−(3−8)

0

Exercise (PageIndex{62})

(1−8)−(2−9)

Exercise (PageIndex{63})

−(6−8)−(2−4)

4

Exercise (PageIndex{64})

−(4−5)−(7−8)

Exercise (PageIndex{65})

25−[10−(3−12)]

6

Exercise (PageIndex{66})

32−[5−(15−20)]

Exercise (PageIndex{67})

6.3−4.3−7.2

-5.2

Exercise (PageIndex{68})

5.7−8.2−4.9

Exercise (PageIndex{69})

(5^{2}−6^{2})

-11

Exercise (PageIndex{70})

(6^{2}−7^{2})

## Everyday Math

Exercise (PageIndex{71})

Elevation The highest elevation in the United States is Mount McKinley, Alaska, at 20,320 feet above sea level. The lowest elevation is Death Valley, California, at 282 feet below sea level.

Use integers to write the elevation of:

1. Mount McKinley.
2. Death Valley.
1. 20,329
2. −282

Exercise (PageIndex{72})

Extreme temperatures The highest recorded temperature on Earth was 58° Celsius, recorded in the Sahara Desert in 1922. The lowest recorded temperature was 90° below 0° Celsius, recorded in Antarctica in 1983.

Use integers to write the:

1. highest recorded temperature.
2. lowest recorded temperature.

Exercise (PageIndex{73})

State budgets In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of $540 million. That same month, Texas estimated it would have a budget deficit of$27 billion.

Use integers to write the budget of:

1. Pennsylvania.
2. Texas.
1. $540 million 2. −$27 billion

Exercise (PageIndex{74})

College enrollments Across the United States, community college enrollment grew by 1,400,000 students from Fall 2007 to Fall 2010. In California, community college enrollment declined by 110,171 students from Fall 2009 to Fall 2010.

Use integers to write the change in enrollment:

1. in the U.S. from Fall 2007 to Fall 2010.
2. in California from Fall 2009 to Fall 2010.

Exercise (PageIndex{75})

Stock Market The week of September 15, 2008 was one of the most volatile weeks ever for the US stock market. The closing numbers of the Dow Jones Industrial Average each day were:

 Monday −504 Tuesday +142 Wednesday −449 Thursday +410 Friday +369

What was the overall change for the week? Was it positive or negative?

-32

Exercise (PageIndex{76})

Stock Market During the week of June 22, 2009, the closing numbers of the Dow Jones Industrial Average each day were:

 Monday −201 Tuesday −16 Wednesday −23 Thursday +172 Friday −34

What was the overall change for the week? Was it positive or negative?

## Writing Exercises

Exercise (PageIndex{77})

Give an example of a negative number from your life experience.

Exercise (PageIndex{78})

What are the three uses of the “−” sign in algebra? Explain how they differ.

Exercise (PageIndex{79})

Explain why the sum of −8 and 2 is negative, but the sum of 8 and −2 is positive.

Exercise (PageIndex{80})

Give an example from your life experience of adding two negative numbers.

## Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

## 1.4E: Exercises - Mathematics

Ordered by date of discovery. Also available in order of appearance in the text.
Items marked [2] have been fixed in the 2nd and later printings. Page 348, Definition 9.17, first line. [2]
Reported 3/5/05 by Kurt L. Van Etten. Page 65, Example 1.53, item 4. [2]
Change (01+)* to 1*(01+)*.
Reported 3/21/05 by Kurt L. Van Etten. Page 92, Problem 1.62. [2]
Change an DFA to a DFA.
Reported 3/21/05 by Kurt L. Van Etten.
Erratum corrected 11/3/05 by Evangelos Georgiadis of MIT.
Page 92, Problem 1.64(d). [2]
Change both occurrences of (b) to (c).
Reported 3/21/05 by Kurt L. Van Etten. Page 125, lines 5 and 6 from bottom. [2]
Change both occurrences of |V| + 2 to |V| + 1.
Reported 4/5/05 by Kurt L. Van Etten. Page 132, Solution 2.7(a), lines 3 and 4. [2]
Change both occurrences of a a to an a.
Reported 4/5/05 by Kurt L. Van Etten. Page 223, second line from bottom. [2]
Change if string to if a string.
Reported 4/18/05 by Kurt L. Van Etten. Page 370, Proof of Lemma 10.5. [2]
A slight adjustment in the proof is needed because &epsilon is an upper bound on the error probability, not the actual value. Additionally there is a variable conflict between the input w and the number of wrong results.
Reported 5/9/05 by Nancy Lynch of MIT. Page 93, Solution to 1.4b, first line. [2]
Change exactly three to exactly two.
Reported 6/7/05 by Gregory Roberts of North Carolina State University. Page 83, Exercise 1.4e, first line. [2]
Part (e) is the same as part (c) so substitute as follows:
Part e. <w| w starts with an a and has at most one b>.
Reported 6/7/05 by Suzanne Balik of North Carolina State University. Page 305, paragraph following Algorithm N. [2]
Change accepts, accept, and accepted (twice) to rejects, reject, and rejected.
Change ALLNFA to overline<>NFA>.
Add sentence at end of paragraph: Note that N accepts improperly formed inputs, too.
Reported 6/8/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 305, second paragraph following Algorithm N. [2]
Change markers, to markers and the repeat loop counter,
Reported 6/8/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 328, paragraph following Algorithm M. [2]
Add sentence at the beginning of the paragraph: Note that M accepts improperly formed inputs, too.
Reported 6/8/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 125, second line of second paragraph. [2]
Change b |V| + 1 to b |V| + 1 > b |V| + 1.
Reported 6/8/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 376, sixth line of second paragraph. [2]
Change another to an.
Reported 6/8/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 131, Problems 2.32, 2.33, and 2.42. [2]
In 2.32, remove the star indicating a difficult problem.
In 2.33, add a star to indicate this problem is difficult. Additionally, change i &ne kj for every to i = kj for some.
In 2.42, add a star to indicate this problem is difficult.
Found 6/13/05. Page 132, Solution 2.7(a), lines 2 and 6. [2]
Change count is 0 to count is positive.
Change if b is on top to if a is on top.
Reported 7/14/05 by Matthew Kane of Indiana University.
Erratum corrected 7/15/05 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 305, Stage 4 of description of N. [2]
Change Stage 4 to read as follows:
Accept if Stages 2 and 3 reveal some string that M rejects, that is, if at some point none of the markers lie on accept states of M. Otherwise, reject.
Reported 7/15/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 172, stage 1 of Turing machine MG.
Add a period at the end of the line.
Reported 9/15/05 by Zhidian Du of Clemson University. Page 228, fifth text line from bottom.
In the line which begins with where every, change bi to bj.
Reported 10/8/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 93, Solution to 1.2.
Change M2 to M1 and M3 to M2.
Reported 10/17/05 by Thomas Watson of the University of Wisconsin-Madison. Page 125, lines 7 and 9 from bottom.
Change both occurrences of chose to choose.
Reported 10/17/05 by Thomas Watson of the University of Wisconsin-Madison. Removal suggested 10/15/08 by Alexis Maciel. SEE errata to solution 2.8 -MS 9/6/10 --> Page 213, Problem 5.27.
Change Q × &sum to Q × (&sum &cup <#>).
Reported 10/17/05 by Evangelos Georgiadis of MIT.
Erratum corrected 10/27/05 by Cem Say of Boğaziçi University, Istanbul, Turkey
and 1/25/06 by David Wittenberg of Brandeis University.
Page 371, fifth and sixth line before Theorem 10.6.
Exchange Zp + and Zp.
Reported 10/27/05 by Cem Say of Boğaziçi University, Istanbul, Turkey
and 11/9/05 by Thomas Watson of the University of Wisconsin-Madison.
Page 281, formula after third paragraph.
Change 1 k to 1 &le i k and add a period following the formula.
Reported 10/20/05 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Erratum corrected 11/3/05 by Evangelos Georgiadis of MIT
and 11/6/05 by Peter Fejer at the University of Massachusetts, Boston
and 11/19/05 by John Sieg at the University of Massachusetts, Lowell
and 10/13/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences.
Page 210, twelfth line from bottom.
In the first line of algorithm G, Change The input is to On input.
Reported 10/25/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 285, second line.
Change variables to variable.
Reported 10/25/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 222, first line of proof of Theorem 6.5.
Change purposes to purpose.
Reported 10/21/05 by Tamir Tassa of the Open University of Israel. Page 329, Problem 8.5.
Change intersection to concatenation.
Reported 10/21/05 by Tamir Tassa of the Open University of Israel. Page 267, last line.
Reported 10/30/05 by Evangelos Georgiadis of MIT. Page 160, second line of Exercise 3.7.
Change The input is a polynomial p to On input &lsaquop&rsaquo, where p is a polynomial.
Reported 11/3/05 by Evangelos Georgiadis of MIT.
Erratum corrected 11/30/05 by Andreas Guelzow of Concordia University
and 4/25/06 by Lewis Collier of the University of Rhode Island.
Page 305, third line.
Change a NFA to an NFA.
Reported 11/3/05 by Evangelos Georgiadis of MIT. Page 180, second line after Figure 4.19.
Change 4.18 to 4.19.
Reported 11/3/05 by Jesse Tjang of the Technical University of Delft, The Netherlands. Page 98, third and fourth line of solution to 1.55 part d.
Replace the sentence beginning If s is generated by 0 * 1 + 0 + 1 * .
with the sentences If s is generated by 0 * 1 + 0 + 1 * and s begins either 0 or 11, write s = xyz where x = &epsilon, y is the first symbol and z is the remainder of s. If s is generated by 0 * 1 + 0 + 1 * and s begins 10, write s = xyz where x = 10, y is the next symbol and z is the remainder of s.
Reported 11/6/05 by Peter Fejer of the University of Massachusetts, Boston. Pages 295 and 298, Problems 7.13, 7.30 and 7.35.
The period should be placed before the right parenthesis, not after.
Reported 11/9/05 by Evangelos Georgiadis of MIT. Page 395, line before Phase 1.
Change either fail to that fails.
Reported 11/9/05 by Thomas Watson of the University of Wisconsin-Madison. Page 98, third line of solution to 1.55.
Change onto to into.
Reported 11/9/05 by Thomas Watson of the University of Wisconsin-Madison. Page 298, Problem 7.32.
Change TM to NTM.
Reported 11/11/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 92, Problem 1.64 part b.
Change Show that, to Show,.
Reported 11/21/05 by Thomas Watson of the University of Wisconsin-Madison. Page 133, Solution 2.8.
The problem asks for leftmost derivations, but the first derivation given is not leftmost. Also, some steps are skipped in the derivations so these should be included. Lastly, in the problem statement, the girl touches the boy, but in the solution, the boy touches the girl.
Reported 11/21/05 by Thomas Watson of the University of Wisconsin-Madison
and 11/30/05 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences.
Erratum corrected 6/1/07 by Kayla Jacobs of MIT.
--> Page 128, third line of Exercise 2.1.
The + should be in the typewriter font.
Reported 11/23/05 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 145, fourth line of Example 3.9.
Change q14 to q8.
Reported 11/18/05 by Promita Chakraborty of Louisiana State University, Baton Rouge. Page 415, Reference 4.
The new URL for this paper is http://www.cse.iitk.ac.in/users/manindra/algebra/primality_original.pdf .
An updated version is at http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf .
Reported 11/19/05 by John Sieg at the University of Massachusetts, Lowell.
Erratum updated 2/21/07 by Santhosh Samarthyam of Anna University, India.
Page 298, second line of Problem 7.31.
Change to .
Reported 11/24/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 284, fourth line from bottom of proof idea.
Reported 11/26/05 by Victor Bandur of McMaster University.
Erratum corrected 11/27/05 by Ryan Lortie of McMaster University.
Page 301, seventh line of Solution 7.22.
Change k > 2m to k > m/2.
Reported 11/26/05 by Victor Bandur of McMaster University.
Erratum corrected 11/27/05 by Ryan Lortie of McMaster University.
Page 318, Figure 8.15.
The topmost x should be x1.
Remove the spurious 1 that is near the node across from x3.
Reported 11/27/05 by Evangelos Georgiadis of MIT. Page 421, eighth and ninth lines in the left column.
Remove the eighth line and add 4, before 328 in the ninth line.
Reported 11/27/05 by Evangelos Georgiadis of MIT. Page 301, first line of Solution 7.15.
Change second A to A * .
Reported 11/29/05 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 65, fifth line before Example 1.53.
Change that that to that.
Reported 11/30/05 by Evangelos Georgiadis of MIT. Page 370, second to last line of the proof of Lemma 10.5.
Change log2 to -log2.
Reported 12/1/05 by Cem Say of Boğaziçi University, Istanbul, Turkey
and 12/8/05 by Paul Beame of the University of Washington at Seattle.
Page 344, fifth line.
Add a period at the end of the line.
Reported 12/3/05 by Evangelos Georgiadis of MIT. Page 412, second line of Problem 10.21.
Add a period at the end of the line.
Reported 12/3/05 by Evangelos Georgiadis of MIT. Page 330, sixth line from bottom.
Remove the comma after G,c,m,h.
Reported 12/5/05 by Evangelos Georgiadis of MIT.
Erratum corrected 12/16/05 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 93, second line of Problem 1.65.
Change a NFA to an NFA.
Reported 12/6/05 by Evangelos Georgiadis of MIT. Page 117, last line.
Add a period at the end of the line.
Reported 12/6/05 by Evangelos Georgiadis of MIT. Page 38, first line of Example 1.11.
Add a period at the end of the line.
Reported 12/6/05 by Evangelos Georgiadis of MIT. Page 307, ninth from last line.
Add a period after 1,2,3. .
Reported 12/6/05 by Evangelos Georgiadis of MIT. Page 307, last line.
The period should be placed before the right parenthesis, not after.
Reported 12/11/05 by Evangelos Georgiadis of MIT. Page 332, second line of Problem 8.26.
Place a line over BIPARTITE so it indicates the complementary language.
Reported 12/8/05 by Paul Beame of the University of Washington at Seattle
and 12/15/05 by Kevin Matulef of MIT.
Page 363, fifth line of Solution to Exercise 9.2.
Change the second and third occurrences of big O to small o.
Reported 12/16/05 by Evangelos Georgiadis of MIT. Page 163, fifth and seventh lines of Solution 3.16(a).
Change both instances of accept to accepts.

Found 12/17/05. Page 383, fifth line.
Change On input &phi: to On input &lsaquo&phi&rsaquo:.
Reported 12/17/05 by Evangelos Georgiadis of MIT. Page 125, third and fourth lines of the third paragraph.
Change so it must contain a path from the root to a leaf of length to so its longest path from the root to a leaf has length.
Reported 12/7/05 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences.
Erratum corrected 8/15/07 by Atsushi Fujioka of NTT Laboratories.
Page 139, the two lines preceding Figure 3.2.
Remove in stages 2 and 3.
Reported 1/4/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 145, first line of the last paragraph.
Change q6 to q7.
Reported 1/9/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 281, eighth line from bottom.
Change the the to the.
Reported 1/20/06 by Yongwook Kim of MIT. Page 416, reference 32.
Change 229 to 299.
Reported 1/20/06 by Yongwook Kim of MIT. Page 162, last sentence of solution 3.3.
Change D to D2.
Reported 1/23/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 181, first line.
Change 4.19 to 4.20.
Reported 1/24/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 83, problem 1.4b.
Remove the first at.
Reported 1/25/06 by Vladimir Lifschitz of the University of Texas at Austin.
Erratum corrected 10/27/05 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 129, first line of exercise 2.17.
Change Problem to Exercise.
Reported 2/1/06 by David Wittenberg of Brandeis University. Page 96, first sentence of solution 1.40.
Add (a) to the beginning of the line. Change an NFA to a DFA.
In 2., change &delta(r,a) to .
Reported 2/15/06 by Barbara Kaiser of Gustavus Adolphus College.
Erratum corrected 3/1/06 and 4/17/06 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 134, fourth line from bottom.
Add , equal numbers of 0s and 1s, and three times as many b's as a's to the end of the sentence beginning Strings in C.
Reported 2/20/06 by Thomas Watson of the University of Wisconsin-Madison. Page 233, last line before section 6.4.
Change Problem to Exercise.
Reported 2/20/06 by Thomas Watson of the University of Wisconsin-Madison. Page 281, last line of second paragraph.
Change Exercise to Problem.
Reported 2/20/06 by Thomas Watson of the University of Wisconsin-Madison. Page 331, last line of problem 8.22.
The period should be placed before the right parenthesis, not after.
Reported 2/20/06 by Thomas Watson of the University of Wisconsin-Madison. Page 413, second line from last line.
Change member to members.
Reported 2/20/06 by Thomas Watson of the University of Wisconsin-Madison. Page 22, proof of 0.24, fourth line.
Change that integer. to the largest such integer..
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 81, footnote.
Remove the footnote, it is redundant.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 95, solution to 1.11, second line.
Change that accepts to that recognizes.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 96, solution to 1.29c, third line.
Change A_1 to A_3.
Reported 2/24/06 by Christos Kapoutsis of MIT
and 4/18/05 by Evangelos Georgiadis of MIT.
Erratum corrected 4/24/05 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 109, third line.
Remove which follows from end of sentence.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 132, solution to 2.3d,e.
Change G to L(G), both occurrences.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 134, solution to 2.30c, ninth line.
Remove only and change nonempty to empty, both occurrences.
Reported 2/24/06 by Christos Kapoutsis of MIT.
Erratum corrected 6/1/07 by Kayla Jacobs of MIT.
Page 184, problem 4.25.
The 1 and 0 should be in the typewriter font.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 196, second line.
Change computation for to computation history for.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 228, eighth line of proof.
Change &Sigmai * to &Sigmai.
Reported 2/24/06 by Christos Kapoutsis of MIT.
Erratum corrected 4/24/05 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 228, second and third lines from bottom.
Change b1 through bi to a1 through ai.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 229, lemma 6.14, second line.
Change language of Th(N,+,×) to language of (N,+,×).
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 244, solution to 6.12, second and third lines.
Change language of Th(N, Reported 2/24/06 by Christos Kapoutsis of MIT. Page 263, stage 8 of algorithm D.
Change comma to period.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 281, fourth line.
Add or a state symbol after the boundary symbol #.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 298, sixth line.
Remove the period after the parenthesis.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 300, last sentence of problem 7.45.
Add Z is in DP and before every language.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 300, first line of problem 7.46.
Change the largest to a largest.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 301, second from last line of solution 7.22.
Change kk to k.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 314, fifth line.
Change player E to Player E.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 315, Theorem 8.11.
Add period at end of line.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 320, sixth and ninth lines.
Change &phi to &psi, both occurrences.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 362, problem 9.16.
Add period at end of line.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 362, problem 9.18.
Change REG&uarr to REX&uarr, both occurrences.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 362, problem 9.21.
Second line: Change k-oracle to k-query oracle.
Fourth line: Change k-oracle A to k-query A-oracle.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 386, first line after Definition 10.27.
Change &Sigmai alternating to &Sigmai-alternating.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 391, seventh line before Claim 10.31.
Change verifier to Verifier.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 391, fifth line before Claim 10.31.
Change wj+1 to mj+1.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 392, fifth line.
Remove stray period.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 396, first line.
Change accepts to recognizes.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 397, second displayed equation in proof idea.
Change the period after otherwise to a comma.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 398, 14th line.
Change If S is to If Si+1 is, both occurrances.
Reported 2/24/06 by Christos Kapoutsis of MIT. Erratum corrected 11/26/06 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 398, last two lines.
Change S to Si.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 399, lines 2 and 4.
Change S to Si.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 400, second paragraph third line.
Change make to makes.
Reported 2/24/06 by Christos Kapoutsis of MIT. Page 403, third line of proof of Theorem 10.40.
Change Ci to Cn.
Reported 2/24/06 by Christos Kapoutsis of MIT.
Erratum corrected 11/6/06 by Rodney Bliss of Brigham Young University.
Page 408, Definition 10.45.
Sixth line and second from last line: Change on input w to on input f(w).
Last line: Change Pr to PrM,w.

Reported 2/24/06 by Christos Kapoutsis of MIT.
Erratum corrected 4/26/06 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 410, Definition 10.47, fourth from last line.
Change Pr to PrE,w.

Reported 2/24/06 by Christos Kapoutsis of MIT.
Erratum corrected 10/4/06 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 412 and 413, Problem 10.16 and solution.
Change Zp to Zp + in problem and twice in last paragraph of solution.

Reported 2/24/06 by Christos Kapoutsis of MIT. Page 189, first line of proof of Theorem 5.1 and second line of proof idea of Theorem 5.2.
Change both purposes to purpose.
Reported 2/28/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 215, last line of solution to 5.28.
Change &lsaquoM,w&rsaquo to &lsaquoMw&rsaquo.
Reported 3/1/06 by Joseph Wilson of the University of Florida. Page 214, solution to 5.8.
We also change the first domino (as shown in Figure 5.16 on page 201 in the standard solution) to [# over ##q0w1w2. wn].
Reported 3/8/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 268, second line of the proof of Theorem 7.24.
Add a period at the end of the sentence.
Reported 3/12/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 186, fifth line from end of solution to 4.21.
Change states to state. Add or if two different accept states of N have red pebbles on them. to the end of that sentence.
Reported 3/29/06 by Thomas Watson of the University of Wisconsin-Madison. Page 84, problem 1.6, first line.
Add a period at the end of the line.
Reported 4/18/06 by Evangelos Georgiadis of MIT. Page 212, problem 5.15, last line.
Change is to Roman font.
Reported 4/18/06 by Evangelos Georgiadis of MIT. Page 263, stage 1 of algorithm D.
Change to For w = &epsilon, if S&rarr&epsilon is a rule, accept, else reject.
Reported 4/18/06 by Marvin Nakayama of the New Jersey Institute of Technology. Page 199, end of second paragraph.
Add a period after the displayed set of dominos.
Reported 5/26/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 347, last paragraph.
Change both occurrences of R to R2.
Reported 8/21/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 184, problem 4.20.
Change <>to <&lsaquoR&rsaquo|.
Reported 10/2/06 by Hjalmtyr Hafsteinsson of the University of Iceland. Page 267, definition 7.21.
Change a O(t(n)) to an O(t(n)).
Reported 10/13/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 360, 18th line from bottom.
In the sentence beginning Similarly, add is before equivalent.
Reported 10/13/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 377-378.
Figure 10.12 (a), put the labels, 0 and 1, of the two output nodes to be in the regular (not typewriter) font, and on page 378, the 7th line from the top and the 18th line from the bottom, put the 1 in the regular font.
Reported 10/13/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page viii, seventh line from bottom.
Capitalize consistently with other chapter entries.
Reported 10/13/06 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 162, seventh line of solution to 3.3.
Change Stage 5 to Stage 3.5.
Reported 11/8/06 by Sara Miner More of McDaniel College. Page 166, sixth line of first paragraph.
Change CFL to CFG.
Reported 11/15/06 by Elazar Birnbaum of the Open University, Israel. Page 395, fourth line.
Change when the ai's to when a1 thru ai.
Reported 11/24/06 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 395, Phases 1, 2, and i.
In the third line of each of these phases, replace the words evaluate . checks with the word check.
Reported 12/9/06 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 398, Phase i.
In the fifth line of this phases, replace the words evaluate . checks with the word check.
Reported 12/9/06 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 328, paragraph following Algorithm M.
Reported 4/13/06 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 354, fourth line of Example 9.29.
Change 2-parity to parity2.
Reported 4/13/06 by Cem Say of Boğaziçi University, Istanbul, Turkey. Pages 341 and 342, proof of Theorem 9.10.
In stage 2 of D, change 3, 4, and 5 to 4 and 5.
In the beginning of the fifth paragraph on page 342, change stages 3 and 4 to stage 4.
Reported 4/14/07 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 213, problem 5.33.
Move this problem to Chapter 6.
Reported 4/25/07 by Paul Beame of the University of Washington at Seattle. Page 129, problem 2.8.
Exchange the words boy and girl for consistency with the solution.
Reported 5/17/07 by David Warren of Brigham Young University. --> Page 358, second line before row of dots.
Change qaccept0 to qaccept&minus, where "&minus" is the blank symbol.
Reported 11/3/05 by Thomas Watson of the University of Wisconsin-Madison,
and 6/14/07 by Goutam Biswas of IIT Kharagpur.
Erratum corrected 10/24/08 by Peter Drake of Lewis & Clark College.
Page 326, second line before Section 8.6.
Change implies to imply.
Reported 8/15/07 by Atsushi Fujioka of NTT Laboratories. Page 132, Solution 2.8 (2.9 in Internation Edition).
Here is one leftmost derivation:
&lsaquoSENTENCE&rsaquo &rArr
&lsaquoNOUN-PHRASE&rsaquo &lsaquoVERB-PHRASE&rsaquo &rArr
&lsaquoCMPLX-NOUN&rsaquo &lsaquoVERB-PHRASE&rsaquo &rArr
&lsaquoARTICLE&rsaquo &lsaquoNOUN&rsaquo &lsaquoVERB-PHRASE&rsaquo &rArr
The &lsaquoNOUN&rsaquo &lsaquoVERB-PHRASE&rsaquo &rArr
The girl &lsaquoVERB-PHRASE&rsaquo &rArr
The girl &lsaquoCMPLX-VERB&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl &lsaquoVERB&rsaquo &lsaquoNOUN-PHRASE&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches &lsaquoNOUN-PHRASE&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches &lsaquoCMPLX-NOUN&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches &lsaquoARTICLE&rsaquo &lsaquoNOUN&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches the &lsaquoNOUN&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches the boy &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches the boy &lsaquoPREP&rsaquo &lsaquoCMPLX-NOUN&rsaquo &rArr
The girl touches the boy with &lsaquoCMPLX-NOUN&rsaquo &rArr
The girl touches the boy with &lsaquoARTICLE&rsaquo &lsaquoNOUN&rsaquo &rArr
The girl touches the boy with the &lsaquoNOUN&rsaquo &rArr
The girl touches the boy with the flower

Here is another leftmost derivation:
&lsaquoSENTENCE&rsaquo &rArr
&lsaquoNOUN-PHRASE&rsaquo &lsaquoVERB-PHRASE&rsaquo &rArr
&lsaquoCMPLX-NOUN&rsaquo &lsaquoVERB-PHRASE&rsaquo &rArr
&lsaquoARTICLE&rsaquo &lsaquoNOUN&rsaquo &lsaquoVERB-PHRASE&rsaquo &rArr
The &lsaquoNOUN&rsaquo &lsaquoVERB-PHRASE&rsaquo &rArr
The girl &lsaquoVERB-PHRASE&rsaquo &rArr
The girl &lsaquoCMPLX-VERB&rsaquo &rArr
The girl &lsaquoVERB&rsaquo &lsaquoNOUN-PHRASE&rsaquo &rArr
The girl touches &lsaquoNOUN-PHRASE&rsaquo &rArr
The girl touches &lsaquoCMPLX-NOUN&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches &lsaquoARTICLE&rsaquo &lsaquoNOUN&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches the &lsaquoNOUN&rsaquo &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches the boy &lsaquoPREP-PHRASE&rsaquo &rArr
The girl touches the boy &lsaquoPREP&rsaquo &lsaquoCMPLX-NOUN&rsaquo &rArr
The girl touches the boy with &lsaquoCMPLX-NOUN&rsaquo &rArr
The girl touches the boy with &lsaquoARTICLE&rsaquo &lsaquoNOUN&rsaquo &rArr
The girl touches the boy with the &lsaquoNOUN&rsaquo &rArr
The girl touches the boy with the flower
Reported 11/21/05 by Thomas Watson of the University of Wisconsin-Madison
and 11/30/05 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences
and 8/15/07 by Atsushi Fujioka of NTT Laboratories.
Erratum corrected 6/1/07 by Kayla Jacobs of MIT
and 8/21/10 by Benjamin Bing-Yi Wang of the Society of Actuaries.
Page 413, second line of second paragraph of Solution 10.16.
Change da mod p to (da mod p) p-1 .
Reported 7/17/07 by Cem Say of Boğaziçi University, Istanbul, Turkey. %%% unnecessary change 1/10/12 % Page 163, second line from end of Solution 3.16(a).
% Change both M1 and M2 reject to % both M1 and M2 fail to accept %
% Reported 9/26/07 by Atsushi Fujioka of NTT Laboratories. % --> Page 228, second to last line.
Add using &epsilon transitions after branching.
Reported 10/20/07 by Flemming Jensen of Aalborg Universitet, Denmark. Page 400, Definition 10.34.
Change C1, C2 to C0, C1.
Reported 10/24/07 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 133, first line of last paragraph.
Change xv 2 wy 2 z to uv 2 xy 2 z.
Reported 10/26/07 by Xiangdong Liang of MIT. Page 97, last line of Solution 1.46.
Change both occurrences of xyz to s'.
Reported 11/4/07 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 117, ninth line from bottom.
The dollar sign should be in the typewriter font.
Reported 1/09/08 by Hiroki Ueda of NTT Laboratories. Page 66, fourth line from end of subsection.
The numbers +7. and -.01 should be in the typewriter font.
Reported 1/23/08 by Rajagopal Nagarajan of the University of Warwick. Page 197, fourth line of proof of Theorem 5.13.
Remove have to after but.
Reported 2/11/08 by Peter Dillinger of Northeastern University. Page 362, Problem 9.21 part b.
Change P to NP.
Reported 2/26/08 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 82, first line of Example 1.76 and next to last line of Example 1.77.
The 1 at the end of both lines should be in the typewriter font.
Reported 3/18/08 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 183, both lines of Exercise 4.10.
Both PDA subscripts should be in the san-serif font.
Reported 3/18/08 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 225, sixth line.
Change Rl to Rk.
Reported 3/18/08 by Hans-Rudolf Metz of the Fachhochschule Gießen-Friedberg, University of Applied Sciences. Page 116, second line of proof of Lemma 2.21.
Change ql to qstart.
Reported 4/17/08 by Kishan Yerubandi of the University of Connecticut. Page 363, Solution 9.15.
In lines 5 and 6, change
second test runs in time poly(|w|) . is in polynomial time. to
second test runs in time poly(|s|), and because |s| &le |w|, the test runs in time poly(|w|).
In the last line, change tests to actions.
Reported 5/5/08 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 125, line 11 from bottom of proof.
Change both v and y are not &epsilon to v and y are not both &epsilon.
Reported 5/30/08 by Chinawat Isradisaikul of the University of Pennsylvania.
Erratum corrected 10/29/08 by Peter Drake of Lewis & Clark College
and 10/15/08 by Alexis Maciel of Clarkson University
and 4/17/09 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 96, solution to 1.29c, third line of second paragraph.
Change |y| > 1 to |y| > 0.
Reported 9/28/08 by Mark Testa of the Stevens Institute of Technology.
Erratum corrected 10/29/08 by Peter Drake of Lewis & Clark College
and 4/17/09 by Cem Say of Boğaziçi University, Istanbul, Turkey.
Page 87, Problem 1.22.
Change the last sentence in the first paragraph to For simplicity, assume that the alphabet for C is .
Reported 1/30/09 by Jerry Grossman of Oakland University. Page 63, last sentence.
Following standard convention, change AWK, GREP and PERL to lowercase, awk, grep and Perl.
Reported 4/18/05 by Jonathan Deber of the University of Toronto,
and 11/24/06 by Matt Diephouse of the University of Michigan,
and 1/15/09 by Rob Bittner of Oakland University.
Erratum corrected 11/28/10 by John Trammell of the University of Minnesota.
Page 184, Problem 4.27.
Add G is a CFG and before L(G).
Reported 3/20/09 by Phanisekhar Botlaguduru Venkata of the University of Florida. Page 185, third line of solution to problem 4.13.
Change decides A to decides the language of this problem.
Reported 4/3/09 by Arthur Hall III of the University of Kentucky. Page 323, fifth line before section 8.5.
Change page 305 to page 306.
Reported 5/15/09 by Cem Say of Boğaziçi University, Istanbul, Turkey. Page 416, Reference 18.
Change a p-1 &equiv p to a p-1 &equiv 1 mod p.
Reported 8/8/09 by Eleazar Leal of the Universidad Simón Bolívar, Caracas, Venzuela. Page 231, Proof of Theorem 6.17, Stage 4 of Algorithm S.
Remove the sentence If it halts and rejects, reject.
That situation can never arise.
Reported 9/8/09 by Mladen Mikša of the University of Zagreb, Croatia. Page 90, Problem 1.49, parts a and b.
The 1 specifying the alphabet should be in the typewriter font, both times.
Found 9/14/09. Page 280, Figure 7.40b.
Change the lower row from q1 a a to q2 a a.
[change is for added clarity, not because the text is incorrect]
Reported 12/26/10 by Cheng-Chung Li of the National Taiwan University, Taiwan. Page 95, solution to 1.11, end of fifth line.
Change &Sigma to &Sigma&epsilon.
Reported 12/31/10 by Simon Dexter of Brooklyn College of the City University of New York. Page 97, solution to 1.44, seventh line.
Change &Sigma to &Sigma&epsilon.
Reported 12/31/10 by Simon Dexter of the City University of New York. Page 85, Exercise 1.15, item d.
Change Q to Q1.
Reported 2/11/11 by Simon Dexter of the City University of New York. Page 172, second from last line.
Hyphenate context free.
Reported 5/9/11 by Edwin Sze Lun Khoo of MIT. Page 125, second line. [2]
Add the sentence If b=1 then reset b so that b=2. before the words In any.
Reported 10/15/08 by Alexis Maciel of Clarkson University, and 6/19/11 by Jeroen Vaelen of Hasselt University, Belguim. Page 101, last word of first paragraph.
The period in "or." should be outside the quotes.
Reported 9/15/11 by Jonas Nyrup of the University of Southern Denmark.

## Extrema - Exercises

egin[100mm] To find the maximum and minimum values, we need to first take a derivative to find the slopes and where the slope switches from positive to negative. egin<>> frac(x^3-x^2-8x+1) &=& 3x^2-2x-8
&=& (3x+4)(x-2)
&Rightarrow& x = <-frac<4><3>,2>
end<>
> Then we test the endpoints and where the slope is 0 to find out which point is the maximum/minimum. egin<>> f(-2) &=& (-2)^3-(-2)^2-8(-2)+1
&=& -8-4+16+1
&=& 5
f(-frac<4><3>) &=& (-frac<4><3>)^3-(-frac<4><3>)^2-8(-frac<4><3>)+1
&=& -frac<64><27>-frac<14><9>+frac<32><3>+1
&+& -frac<64><27>-frac<42><27>-frac<288><27>+frac<27><27>
&=& frac<209> <27>Rightarrow Maximum
f(2) &+& (2)^3-(2)^2-8(2)+1
&=& 8-4-16+1
&=& -11 Rightarrow Minimum
end<>
> To confirm the solution, we can simply plug in a value between $-frac<4><3>geq x leq 2$ into $f’(x)$. I’ll use $f’(0)$ for simplicity.
egin<>> f’(0) &=& (3(0)+4)((0)-2)
&=& (4)(-2)
&=& -8 Rightarrow Negative
end<>
> Seeing that the slope was negative hence decreasing from $-frac<4><3>$ to $2$, the calculated values from before are correct. end

question[$*$] Find the minimum and maximum values of the function on the interval $[-1,1]$. [f(x) = x^5 + x + 1]

egin[80mm] To find the maximum and minimum values, we need to first take a derivative to find the slopes and where the slope switches from positive to negative. egin<>> frac(x^5+x+1) &=& 5x^4+1
end<>
> We should notice that $5x^4+1$ is always positive as it’s a quartic function and a positive constant.
From this, we can conclude that the minimum and maximum on the given interval is at $x = -1$ and $x = 1$ respectively.
To confirm the solution, plug in $x = -1$ and $x = 1$
egin<>> f(-1) &=& (-1)^5+(-1)+1
&=& -1 Rightarrow Minimum
f(1) &=& (1)^5+(1)+1
&=& 3 Rightarrow Maximum
end<>
> end

question[$**$] Find the minimum and maximum values of the function on the interval $[-frac<1><2>,1]$. [f(x) = frac<1>]

egin[110mm] To find the minimum and maximum values, we take the derivative of $f(x)$ using the quotient rule to find the critical values where the slope may switch from positive to negative.
egin<>> frac(f(x)) &=& frac(frac<1>)
&=& frac(1)cdot(x^5+x+1)-1cdotfrac(x^5+x+1)><(x^5+x+1)^2>
&=& -frac<5x^4+1><(x^5+x+1)^2>
end<>
> Analyzing the derivative, we can deduce that the derived function is always negative because the numerator is a second degree function translated upwards and the denominator is a quintic function which is squared.
However, we must check the denominator of the original function to ensure it isn’t 0 at any point in the domain mentioned. We can check for $x=-frac<1><2>$ to see if it’s below 0.
egin<>> f(-frac<1><2>) &=& frac<1><(-frac<1><2>)^5+(-frac<1><2>)+1>
&=& frac<1><-frac<1><32>-frac<16><32>+frac<32><32>>
&=& frac<32><15>
end<>
> We can see that were are no asymptotes or holes within the domain of $[-frac<1><2>,1]$
Thus, the maximum is at $x=-frac<1><2>$ and minimum at $x=1$. egin<>> f(-frac<1><2>) &=& frac<32><15>
f(1) &=& frac<1><(1)^5+(1)+1>
&=& frac<1><3>
end<>
> end

question[$*$] Find the critical numbers of $r( heta) = 3 heta - arcsin heta$.

egin[80mm] Take a derivative and set it to 0 to find the critical numbers. egin<>> frac(r( heta)) &=& frac(3 heta-arcsin heta)
&=& 3-frac<1>>
heta &=& pm 1 enspace for enspace singular enspace values
&Rightarrow& Equate enspace to enspace 0 enspace for enspace critical enspace values
0 &=& 3-frac<1>>
frac<1>> &=& 3
frac<1> <3>&=& sqrt<1- heta^2>
frac<1> <9>&=& 1- heta^2
heta^2 &=& frac<8><9>
heta &=& pmfrac<2sqrt<2>> <3>enspace for enspace critical enspace values
end<>
> end

question[$**$] Find the global maximum value of $f(x)$ using any method you can think of, you do not need to be rigorous. [f(x) = frac<1> <1+|x|>+ frac<1><1+|x-a|>]

egin[100mm] First, we notice that the constituent parts that sum to the function are the smallest when the denominator is the smallest. We can use the fact that the function behaves similarly to 1/x in terms of growth with some offset from the asymptote. Since a is just a constant, the denominator is minimized when the absolute values are 0, hence at $x=0 enspace and enspace x=a$. At both x values, for a given constant a, the output would be equal and are global maximums. end

question[$**$] For the function $f(x) = e^<-2x^2+4x+3>$, find (if they exist) egin[(i)] item Critical points item Maximum points item Minimum points item Intervals of increase item Intervals of decrease item Points of inflection item Intervals of concave upward item Intervals of concave downward item Sketch the function end

egin First we calculate the derivatives of $f$: [f’(x) = 4(1-x)e^<-2x^2+4x+3>] [f’‘(x) = 4(4x^2-8x+3)e^<-2x^2+4x+3>] egin[(i)] item Critical points: Setting $f’(x) = 0$, we find that $x = 1$ is a critical point of $f$. There are no points where the derivative does not exist, so $x = 1$ is the extit critical point. item Maximum points: Plugging $x = 1$ into $f’’$ we obtain [f’‘(1) = -4e^5 < 0] this means that $x = 1$ is a maximum of $f$. item Minimum points: There are no critical points of $x$ that yield a positive second derivative, so there are no minimum points. item Intervals of increase: We solve the inequality $f’(x) > 0$, to find that $x < 1$ is the interval of increase. item Intervals of decrease: We solve the inequality $f’(x) < 0$, to find that $x > 1$ is the interval of decrease. item Points of inflection: We solve the equation $f’‘(x) = 0$: This yields $4x^2-8x+3 = 0$, so $x = 1/2, 3/2$. These are the two points of inflection. item Intervals of concave upward: We solve the inequality $f’‘(x) > 0$. This yields $x in (-infty,1/2)$, and $xin(3/2,infty)$. item Intervals of concave downward: We solve the inequality $f’‘(x) < 0$. This yields $xin(1/2,3/2)$. item Sketch the function:
includegraphics[scale = 0.4] end

## 1.4E: Exercises - Mathematics

Instructor: Sam Buss
Email: [email protected]
Office: APM 6210
Phone: 534-6455
Office hours: Monday 9:00-9:50 and Wednesday 9:00-9:50 and 1:30-2:30. (valid thru March 10).

Teaching Assistant: Tom Langley, [email protected]
Office: APM 2202
Office hours: Wednesday 12:15-1:15, and Thursday 2:00-4:00. (thru March 11)

## ANNOUNCEMENTS FOR FINAL EXAM WEEK

Review session: Tuesday, March 16, 3:00-4:30pm. APM 2301.

Exam week office hours for Sam Buss, APM 6210:
Monday and Thursday: 11:00-12:00 and 1:30-2:30.
Tuesday: 1:30-2:45.

Exam week office hours for Tom Langley, APM 2202:
Wednesday: 10:00-12:00 and 2:00-4:00.
Thursday: 3:00-6:00.

Soft reserves: By Wednesday, March 10, will have all midterm, homework and quiz answers (except for last quiz and homework).
Some past years' final exams and other sample final exam problems were handed out in class March 10. If you missed class, please pick them up from Tom or Sam.

## Course announcement & Grading Policy

This course will cover Automata and Regular and Context-free languages (Part I of the textbook), plus selected topics from Abstract Computability (Part II, Turing machines, etc.) and from Feasible Computability (Part III). These are basic topics in the foundations of the theory of computation, plus form the mathematical foundations for compiler design theory and for analysis of algorithmic efficiency.

The course will have a midterm exam on February 12 and a final exam. There will be quizes in the Thursday section meeting on most weeks. Popquizes have been given in class, but not many more (if any) are planned. Homework assignments will typically be due in class on Friday. Course grading will be 50% final, 30% midterm, 10% homework, 10% quizes, 0% popquizes. You may drop your lowest two quiz scores. The percentages may change slightly, if so I will announce the change.

## Homework Assignments

Homework #2, due Friday, January 22.
Pages 83-90: Problems 1.4e,f,g,i,k,l,m,n, 1.5a-d.
And: Prove that the language over the alphabet <0,1. 9>containing the integers (in base 10 representation) which are multiples of 7 is a regular language.

Homework #3, due Friday, January 29.
Pages 83-88: Exercises: 1.5f,g, 1.6a, 1.7a, 1.8a, 1.9, 1.12
Pages 88-90: Problems: 1.24, 1.44* (* = challenge problem).
Hint for 1.24: given an finite automaton M, construct an NFA recognizing the reversal of L(M).

Homework #4, due Friday, February 5.
Pages 83-88: Exercises: 1.10b, 1.14a, 1.15, 1.16, 1.17.
Pages 88-90: Problems: 1.23, 1.32a.
Revised due date: 1.17 and 1.23 are due on Monday, February 8.

Homework #5, due Friday, February 19.
Pages 119-121: Exercises: 2.1, 2.3, 2.4, 2.6a,b, 2.9.
Pages 121-122: Problems: *2.21.

Homework #6, due Friday, February 26.
Pages 119-121: Exercises: 2.5, 2.10, 2.11.
Pages 121-122: Problems: 2.15, 2.17a.
Hint: For 2.17a, use the equivalence of CFG's and PDA's (so you may work with PDA's). This will make problem 2.17a conceptually quite simple. Problem 2.15 may be done with either CFG's or with PDA's.

Homework #7, due Friday, March 5.
Pages 121-122: Problems: 2.17b, 2.18, *2.26, *2.27.

Homework #8, due Friday, March 12.
Page 149: Problems: 3.14, 3.15.
Also: Construct a Turing machine which decides the language containing all odd length length strings which have a "0" as their center character. (Alphabet = <0,1>).

## Quizes

Pop Quiz #1 . Set notation, if .. then . set formation.
Also in pdf format and gif format.

Pop Quiz #2 . Set notation, if .. then . set formation. Again!
Also in pdf format and gif format.

Quiz #1 , January 14.
Set notation, if .. then . set formation. Again.
Also in pdf format and gif format.

Quiz #2 , January 21.
DFA's. State diagrams. The 5-tuple definition of DFA's.
Also in pdf format and gif format.

Quiz #3 , January 28.
NFA's. Construct NFA's recognizing given language. Convert an NFA into an equivalent DFA.
Also in pdf format and gif format.

Quiz #4 , February 4.
Regular expressions and the languages they represent. Convert a 2-state DFA into a regular expression.
Also in pdf format and gif format.

Quiz #5 , February 10 - Note different time: Wednesday in class
Proving languages are non-regular. The Pumping Lemma.
Answer key available immediately after class.
Also in pdf format and gif format.

Quiz #6 , February 25.
Construct a context-free grammar. And construct a pushdown automaton.
Also in pdf format and gif format.

Quiz #7 , March 4.
Prove that a language is not context-free with the aid of the pumping lemma.
Also in pdf format and gif format.

Quiz #8 , March 11.
Construct a Turing machine to recognize a specified language. (By drawing its state diagram).
Also in pdf format and gif format.

## Course Handouts

Soft reserves has homework, midterm and quiz answers.

Practice problems and old midterm handed out in class, February 8.
Old final exams and more practice problems handed out in class, March 10.

The value of a exceeds, or is lesser than the smallest number that a float can accept. Check it out here.

MIN_VALUE - A constant holding the smallest positive nonzero value of type float, 2^(-149).

Floats cannot hold the value of all real numbers. If you need that kind of precision use BigDecimal.

The thing is that you are beyond the range of float.

float: 4 bytes, IEEE 754. Covers a range from 1.40129846432481707e-45 to 3.40282346638528860e+38 (positive or negative).

## Jul 6 Nikon 28mm f/1.4E - How is the 35mm f/1.4 still a thing?

How is a fast 35mm lens still a thing? For the longest time, the only fast wide angle lens available was the venerable 35mm f/1.4. It was such an essential tool that all professional photographers found it mandatory to have one in their bag. Having said that, I never really liked the 35mm focal length. I've always felt it wasn't wide enough to document at closer focusing range. As such, my favorite lens for the longest time was the Nikon 28mm f/1.4D. I adored it. It was my hands-down go-to lens on my Nikon F4 and D700.

Besides, it's not as if Nikon had an autofocus version of a fast 35 until the release of the 35mm f/1.4G.

Despite how I felt about the fast 35, even I had to eventually succumb to getting one. With improvement in digital sensors and autofocus technology, my trusty 28mm f/1.4D could no longer keep up with the times. But the 35mm f/1.4G was just never wide enough. For that reason, my go-to lens on the Nikon D800E was the much wider 24mm f/1.4G. Unfortunately, 24mm is much too wide for me. Whenever I shot with it, I always ended up cropping the resulting images to the 28mm equivalent - albeit unintentionally. Still, cropping was better than not being wide enough.

So when Nikon announced the release of the new 28mm f/1.4E, I was over the moon. I couldn't wait to get one. Unfortunately, I was away from Hong Kong when they started to ship. Like clockwork, I was stuck in New York. As such, I had to wait an extra week before I could finally lay my hands on one.

Fortunately, that week passed by quickly.

Long story short. I'm in Hong Kong. I have my 28mm f/1.4E. And it's everything I dreamed it would be. It is a fantastic lens.

Nikon Df + 28mm f/1.4E - Raw unedited image

Nikon Df + 28mm f/1.4E - Raw unedited image

I have always been of the opinion that the purpose of such a lens is to shoot up close, in low light, for maximum bokeh. So naturally, the first thing I did right after picking it up from my vendor was fire off a couple of shots to see how the 28mm f/1.4E performed at its closest minimum focusing distance. As a Leica photographer, you forget how close you can focus with a DSLR. I kept going closer and closer until finally I stopped at around 0.92 feet (roughly 11 inches or 0.28 meters).

Admittedly, the 28mm f/1.4E does not focus as close as the 24mm f/1.4G, which can go in an inch closer. But in my opinion, the impression of close focusing is stronger with the 28mm f/1.4E given the relatively tighter focal length. And when compared to the previous 28mm f/1.4D, this updated version can focus almost 3 inches closer.

Of course, what this all means is the background blur of the 28mm f/1.4E is monumental. Though more importantly, it also means that the 28mm f/1.4E might possibly be the most perfect all around lens to ever grace a full frame autofocus camera system. Hear me out. I know I am putting the cart before the horse in making such an outlandish claim. I mean, I've only fired off two sample shots. But OMG, they're amazing!

What I saw in the 28mm f/1.4E was the idiomatic "triple threat" with regards to its functionality. It shoots fast, shoots close up, and shoots just wide enough. In all intent and purpose, I saw no equal to this lens. I was in love with it. Not even my favorite Leica 28mm f/1.4 Summilux-M could match that - with a minimum focusing distance of 2.3 feet (or 0.7 meters). It's no wonder why I never take food pictures with my Leica. But with this Nikon, I'm sure I will.

Nikon Df + 28mm f/1.4E - as you can see, shooting Leica has made me out of practice with food!

These are exciting times to be a Nikon photographer. For so many years, we were forced to endure the indignity of uninspiring new product releases. How disappointed I was with the 58mm f/1.4G. Nikon couldn't even give us an extra half a stop in evoking the 58mm f/1.2 Noct-Nikkor as its spiritual descendent. And don't get me started on the Nikon Df. I only kept mine for the sake of reviewing Nikon lenses.

But with the release of the 105mm f/1.4E and now, the 28mm f/1.4E, Nikon is doing something it hasn't done in a very long time. Nikon is making niche lenses that no other manufacturers are making. The 105mm f/1.4E is the ultimate portrait lens. As for the 28mm f/1.4E, it is the ultimate everyday lens.

It is just unfortunate that Nikon doesn't have a decent full frame camera to do it justice - other than the D5. For this test, I used the Nikon Df and the D800E. Both cameras felt like dinosaurs compared to the Sony A9 and the Canon 5D Mark IV. Having said that, I was enormously jet lagged during the lens review, so I shot everything in live view, face detection, set to aperture priority. So as you could imagine, the slow shutter lag wore at my jet lagged diminished patience.

In conducting a review for the 28mm f/1.4E, I decided to wait until nightfall. It only made sense, given the lens's maximum aperture. Mind you, diminished capacity from jet lag prevented me from doing it during the day. Plus, Mother Nature made it that way - given the calm from the unending downpour unleashed throughout the week on only the night we tested. Besides, Anna wasn't available earlier. Really, it was the perfect storm of reasons that pushed the scheduling beyond the hours of available light.

Of course, we could have conducted all our testing indoors, and avoided the drama unfolding from the tempest all week. But, that wouldn't make sense. The 28mm f/1.4E begs to be tested outside the confines of a controlled environment. At that focal length, it enabled one to environmentally document a subject close enough without the threat of photobombing becoming a nuisance.

Naturally then, Anna and I descended onto the street to put the 28mm f/1.4 through its paces. We shot at varying close focusing distances to avoid run-ins from the crowds. But more importantly, we shot closer-up to see how proficiently it rendered the characteristic background lights of Hong Kong into balls of lights. After all, this is the main reason why anyone would want a lens like this. It's for the bokeh.

In handling the 28mm f/1.4E, the autofocus is smooth and silent as one would expect from any contemporary Nikon lens. Focusing manually, the lens also feels good in hand. In terms of size, the 28mm f/1.4 is noticeably longer than its predecessor or the 35mm f/1.4G. But in my opinion, the increase in size is both expected and within reason. I also didn't find the increase in size impacted its weight adversely.

On normal use, the 28mm f/1.4E performs exceptionally well. Unlike the 28mm f/1.4 Summilux-M, I didn't notice any chromatic aberration in areas of extreme contrast of light and darkness. There wasn't any purple fringing at wide aperture. Also, I am of the opinion that the lens handles perspective distortion rather well at normal focusing distances. However, it does become an issue when you begin to edge in closer than arms length (roughly three feet or a meter).

Like the 105mm f/1.4E, the 28mm f/1.4E also appears slightly sharper than Nikon's G-Series lenses. However, to determine whether this is the case, I went back to my office to compare it with the following lenses on the Nikon D800E:

1. Nikon 35mm f/1.4G
2. Nikon 24mm f/1.4G
3. Nikon 28mm f/1.4D

I included the 28mm f/1.4D because I thought it would be interesting to see how it compared to its predecessor.

Nikon D800E + 28mm f/1.4E - 200% Magnification

Nikon D800E + 28mm f/1.4D - 200% Magnification

Nikon D800E + 35mm f/1.4G - 200% Magnification

Nikon D800E + 24mm f/1.4G - 200% Magnification

Admittedly, my jet lagged reliance on face detection might have adversely impacted the accuracy of the autofocus when comparing these lenses. For some reason, the autofocus kept locking onto the recessive eye. The only time the autofocus locked onto the lead eye was with the 28mm f/1.4E. Coincidence - I don't know. Having said that, I thought I should disclose this, given how consistent this anomaly was in focus selection.

Insofar as rendering details, the Nikon 28mm f/1.4E is significantly sharper and more contrasty than the 28mm f/1.4D. This is especially evident under high magnification at 36 megapixels. From that perspective, I don't see how anyone would still covet the 28mm f/1.4D - or at least at its current market value.

As for how it compared to the G-series lenses, I am of the opinion that the 28mm f/1.4E retains more details at higher resolution. Making this evaluation was difficult, given the differences in focal length with focusing distance more or less the same. However, the deciding factor for me were the details of the eyelashes reflected on Anna's pupils. It was phenomenal.

In retrospect, I should have conducted this test stopped down to increase the depth of field for ensuring focus. Call it an oversight on account of jet lag. Having said that, I am pretty confident that the 28mm f/1.4E is sharper at higher resolution. Even wide open, the focused regions of the 28mm f/1.4E at high magnification still looks sharper than the G-Series lenses.

Leica M10 + 28mm f/1.4 Summilux-M ASPH

It may appear that I am overly biased in my assessment of the 28mm f/1.4E. However, there is reason for my opinion. To prove my point, I decided to do a second day of informal shooting with Anna during the daytime. No, I didn't go outside. I wasn't in any condition. Besides, after my last incident with the 50mm f/1.4 Summilux-SL, I knew better than to push my luck.

My final test was simple. I photographed Anna at the focusing distance of a typical restaurant table (of three feet or one meter) without backing up or tilting on my chair. In my opinion, 35mm wasn't wide enough since it made the subject appear too prominent, while 24mm was too wide in capturing too much unnecessary foreground and background. To me, 28mm was the Goldilocks of focal length. It was the perfect balance between subject isolation and background documentation.

Please note I also included the Leica 28mm f/1.4 Summilux-M in this comparison - just for fun. Also included for fun is the image set below photographed stopped down at f/5.6. I realize I didn't test corner sharpness at smaller apertures. But I figured there would be loads of other reviewers doing that. Besides, I'm pretty sure that corner and edge sharpness of this lens will be sufficiently detailed.

Nikon Df + 28mm f/1.4E - stopped down at f/5.6

Nikon Df + 28mm f/1.4D - stopped down at f/5.6

Nikon Df + 35mm f/1.4G - stopped down at f/5.6

Nikon Df + 24mm f/1.4G - stopped down at f/5.6

Leica M10 + 28mm f/1.4 Summilux-M ASPH - stopped down at f/5.6

Overall, I love the Nikon 28mm f/1.4E. Not only does it make me wonder how a fast 35mm lens is still a thing, but it also makes me ask how Canon is still a thing! Admittedly, I know why. Nikon makes horrible DSLRs when compared to the Canon 5D Mark IV. Though in defense of Nikon, they're probably up for an update soon. I can only hope they won't disappoint.

What people need to understand is why the 35mm f/1.4 has been so dominant for so long. It's largely because it offers the best compromise of focal length and speed for reportage or candid documentation. With that said, 35mm isn't wide enough. But since it was the best option for a fast wide angle lens, the 35mm f/1.4 became the standard. Besides, it's not as if there were a fast 28mm alternative.

Professional photographers always knew that the 28mm focal length was better for documentation than 35mm - hence the popularity of the Leica 28mm f/2 Summicron - albeit a stop slower. Unfortunately, the only fast 28mm option for the longest time was Nikon's outdated 28mm f/1.4D, which was expensive due to its uniqueness in construction. Then last year came the Leica 28mm f/1.4 Summilux-M, which was noticeably exclusive in pricing. If only there were a reasonably priced 28mm f/1.4 in existence.

Well, now there is. If only Nikon had a better full frame DSLR. But then again, it would only be a matter of time until Canon releases their own version. And when they do, the fast 35mm would no longer be the "proverbial thing".

All images have been optimized in Lightroom. Only images under high magnification have been cropped. All images photographed wide open, unless stated otherwise.

## Provide all the questions of 10.3 please it is urgent kindly request to meritnation to provide answer of these exercise

( i )   1 x + 1 y + 1 z = – 2 ,   1 x – 2 y + 1 z = 3 ,   2 x – 1 y + 3 z = – 1 ( i i )   2 x – 1 – 1 y + 1 + 2 z = 1 ,   3 x – 1 + 2 y + 1 – 1 z = 1 , 1 x – 1 – 3 y + 1 – 3 z = 2

(5) Solve the following equations for x,y,z if sin x + cosy + tan z =3

2 sin x + cos y+ tanz=4, where 0 ≤ x,y,z ​ ≤ 90 °
3 sin x + 4 cos y&ndash2 tan z=5.

(6) Find x,y,z if 5e x + 4log10 y &ndash 3 z = 1 ,
4e x + 3log10 y &ndash 2 z = 2 and 3e x &ndash 2 log10 y+ z = 3 .

(7) The sum of three numbers is 6. Thrice the third number when added to the first number gives 7. On adding the sum of second and third number to three times the first number, we get 12. Find the three numbers using determinants.

(8) The sum of three numbers is 2. If twice the second number is added in the sum of first and third, we get
1. On adding the sum of second andthird number to five times the first number, we get 6. Find the three numbers using Cramer'sRule.

(9) The cost of 4 kg potato, 3 kg wheat and 2 kg rice is Rs. 150. The cost of 1 kg potato, 2 kg wheat and 3 kg Rice is Rs. 125. The cost of 6 kg potato, 2 kg wheat and 3 kg rice is Rs. 175. Find the cost of each item per kg, by using Cramer's Rule .

(10) An amount of Rs. 5000 is invested in three investmnets at rule 6%, 7% and 8% per annum respectively. The total annual income from these investments is Rs. 350. If the total annual income from the third, find the amount of each investment by using determinant methods.

So here is the abstract approach:

$langle psi | H | psi angle = frac<1><5>igg( langle phi_1 | H | phi_1 angle + 2langle phi_1 | H | phi_2 angle + 2langle phi_2 | H | phi_1 angle + 4 langle phi_2 | H | phi_2 angle igg) ,.$

Now you know that $H|phi_1 angle = E_1 |phi_1 angle$ and $H|phi_2 angle = E_2 |phi_2 angle$ --- or rather, you can easily check that the functions you've given are indeed eigenstates of the Hamiltonian:

The functions $phi_n$ are also normalised, as you can check, and are orthogonal to one another --- this must be the case, because they are the eigenfunctions of a Hermitian operator (with different eigenvalues). Hence the expression above becomes:

$langle psi | H | psi angle = frac<1><5>igg( E_1langle phi_1 | phi_1 angle + 2E_2 langle phi_1 | phi_2 angle + 2E_2langle phi_2 | phi_1 angle + 4E_2 langle phi_2 | phi_2 angle igg) ,.$ $langle psi | H | psi angle = frac<1><5>igg( E_1 + 4E_2igg) ,.$

Substituting in the form of the energies gives:

This is, to me, easier than computing the integral you've given, although the integral you've given is correct (or almost --- the Hamiltonian should have factor of $hbar$ squared in front of the second derivative). If you tried to compute the integral, you would find a good deal of cancellation due to the orthogonality of the functions involved. If you're good at quickly spotting when an integral vanishes, e.g.:

then the above prescription might not seem any simpler. But as far as cleanness of approach goes, it's nicer to invoke the orthogonality of the eigenfunctions --- which is a result of central importance in this kind of problem, and one which you will prove in any introductory QM course --- before delving into explicit computations.

NB: I anticipate that you may not have met the above notation yet. There isn't anything to it. For our purposes, just take $langle psi | H |psi angle$ to mean

from which you should be able to see how the first line follows. The statement that two functions are orthogonal amounts to $langle phi_1 | phi_2 angle = 0$, whilst the statement that a function is normalised amounts to $langle phi_1 | phi_1 angle = 1$.

## Aug 25 Nikon 105mm f/1.4E

This is going to be a quick writeup. It's just going to be a bunch of sample images from the Nikon 105mm f/1.4E.

Overall, it is amazing. It's sharp - really sharp. I mean crazy CRAZY sharp SHARP. And the bokeh is out of this world. The closer you are to the subject, the more amazing it is. And trust me, I've seen a lot of bokeh over the last couple of months. Click here if you haven't seen it already.

But enough of what I have to say. I think that the images speak for themselves. The only goof that happened on this first look was user error - that is to say, I haven't shot this much with a Nikon DSLR in over four years. So I had some problem with the focusing points - no surprise. Honestly, I just don't understand why the auto focusing points doesn't go all the way to the edge of the frame. spoken like someone who's been spoiled manually focusing on the Leica SL!

Plus I accidentally put the wrong Nikon rear cap onto the rear of the lens. so the rear cap was stuck! I couldn't get it off. It was very embarrassing. Turns out you have to unscrew the rear cap slowly, and with much pressure. I had to make an emergency call to my vendor for that bit of wisdom. So now you know, should the same thing happen to you.

But enough about my pitfalls. Onto the images. All my observations are in the captions under the posted images. All images shot at ISO 400, at f/1.4, on the Nikon D800E.

Image 1. 1/200th of a second. A perfect combination of sharpness and bokeh.

Image 2. 1/125th of a second. And it renders colors well too!

Image 3. 1/250th of a second. Doing my best to follow in the example of Stanley Kubrick in getting tack focus on the subject's lead eye. #BarryLyndon

Image 4. 1/250th of a second.

Image 5. 1/160th of a second. Extreme close up. Look at the eye lashes and the eyes. Crazy sharp, isn't it!

Image 6. 1/125th of a second. But at this distance, there isn't much background left for bokeh!

Image 7. 1/250th of a second. No amount of Lightroom could save this interior's awful AWFUL white balance.

Image 8. 1/250th of a second. Oh yes. and bokeh at this distance from the subject isn't all that nice either.

Image 9. 1/160th of a second. Depth of field on this lens is paper thin. I couldn't get tack focus on the lead eye of the subject on the left.

Image 10. 1/160th of a second. This time, I got tack focus. But I had to get both the subject's lead eye on the same focal plane.

Image 11. 1/80th of a second. Surprised there wasn't any camera shake! Going up the escalator.

Image 12. 1/80th of a second. Again, surprised that there wasn't any camera shake.

Image 13. 1/160th of a second. Almost reaching the top. Surprisingly at this distance, bokeh is still respectable.

All images in this writeup are full crop. Images have been optimize in Lightroom to some extent.

Special thanks to Anna and her friend for being wonderful subjects! Sorry, I forgot to catch her name.

## Jul 12 Awesome Comparison: Bokeh Shootout between Leica, Canon, and Nikon

In retrospect, I really should have conducted this comparison on another day, given that I had been awake that day since 3:00AM Hong Kong Standard Time, and was conducting this shootout after work from 7:00PM-9:00PM. Something was going to give, and in this case, it was a Nikon 50mm f/1.2 that I had I accidentally replaced with a Nikon 58mm f/1.2 Noct-Nikkor - essentially repeating the result of the Noct-Nikkor lens. Hey, you can't blame me. Those AIs Nikon lenses all look the same, especially when you're sleep deprived. Needless to say, I will do a second day of shooting.

The lenses used in this comparison are as follows:

1. Canon 50mm f/1.0L shot with the Canon 1Dx MKII
2. Canon 50mm f/1.2 shot with the Canon 1Dx
3. Leica 50mm f/0.95 Noctilux shot with the Leica MP240
4. Leica 50mm f/1.2 Noctilux shot with the Leica MP240
5. Nikon 50mm f/1.2 Ais shot with the Nikon D4
6. Nikon 58mm f/1.2 Noct-Nikkor shot with the Nikon D4
7. Nikon 58mm f/1.4G shot with the Nikon D4

In addition to that, all these lenses were also shot with an adapter on the Sony A7r MKII. Never before has there been a comparison on the internet in which all these lenses were compared on the same sensor, in order to level the playing field. Also, it would be interesting to see how each lens resolves at higher resolution. Of course, that also depends on whether I actually hit tack focus. I mean, I did try, and I think I did okay. But somehow, I feel I just missed it by a fraction on some of those shots. I relied on Sony's focus peaking, but that only helps so much. To get tack focus wide open, you really have to try.

You may be wondering why I also included the two Nikon 58mm lenses. Well, let's put it this way. Nikon doesn't really have a halo 50mm lens. The closest they have are the 58mm f/1.4G and the vintage 58mm f/1.2 Noct-Nikkor. Without those two lenses included, I would be stuck with only the 50mm f1.2 Ais.

It should also be noted that I am only comparing images shot wide open between the selected lenses. There is no point to stop down, because they all perform more or less just as well as each brand's slower versions. Besides, the only reason to get these lenses is to shoot wide open, for the sharpness in the middle and the amazing bokeh in the background. To use them only to stop down would be so wrong.

Given that, I didn't bother with placing any props at the corner of the frame, since assessing corner sharpness is somewhat irrelevant with regards to a bokeh comparison. I can save you all a lot of time by saying that corner sharpness wide open is not very good - there's vignetting - and honestly, you're shooting wide open, so it's not going to be good. Either you stop down or use a lens with a maximum aperture in the neighborhood of f/2.