# 12.5E: Exercises for Section 12.5 - Mathematics

## Finding Components of Acceleration & Kepler's Laws

1) Find the tangential and normal components of acceleration for (vecs r(t)=t^2,hat{mathbf{i}}+2t ,hat{mathbf{j}}) when (t=1).

In questions 2 - 8, find the tangential and normal components of acceleration.

2) (vecs r(t)=⟨cos(2t),sin(2t),1⟩)

3) (vecs r(t)=⟨e^t cos t,e^tsin t,e^t⟩). The graph is shown here:

4) (vecs r(t)=⟨frac{2}{3}(1+t)^{3/2}, frac{2}{3}(1-t)^{3/2},sqrt{2}t⟩)

5) (vecs r(t)=⟨2t,t^2,frac{t^3}{3}⟩)

6) (vecs r(t)=t^2,hat{mathbf{i}}+t^2,hat{mathbf{j}}+t^3,hat{mathbf{k}})

7) (vecs r(t)=⟨6t,3t^2,2t^3⟩)

8) (vecs r(t)=3cos(2πt),hat{mathbf{i}}+3sin(2πt),hat{mathbf{j}})

9) Find the tangential and normal components of acceleration for (vecs r(t)=acos(ωt),hat{mathbf{i}}+bsin(ωt),hat{mathbf{j}}) at (t=0).

10) Suppose that the position function for an object in three dimensions is given by the equation (vecs r(t)=tcos(t),hat{mathbf{i}}+tsin(t),hat{mathbf{j}}+3t,hat{mathbf{k}}).

a. Show that the particle moves on a circular cone.

b. Find the angle between the velocity and acceleration vectors when (t=1.5).

c. Find the tangential and normal components of acceleration when (t=1.5).

c. (a_vecs{T}=0.43, ext{m/sec}^2, quad a_vecs{N}=2.46, ext{m/sec}^2)

11) The force on a particle is given by (vecs f(t)=(cost),hat{mathbf{i}}+(sint),hat{mathbf{j}}). The particle is located at point ((c,0)) at (t=0). The initial velocity of the particle is given by (vecs v(0)=v_0,hat{mathbf{j}}). Find the path of the particle of mass (m). (Recall, (vecs F=mvecs a).)

(vecs r(t)=left(frac{-1}{m}cos t+c+frac{1}{m} ight),hat{mathbf{i}}+left(frac{−sin t}{m}+left(v_0+frac{1}{m} ight)t ight),hat{mathbf{j}})

12) An automobile that weighs 2700 lb makes a turn on a flat road while traveling at 56 ft/sec. If the radius of the turn is 70 ft, what is the required frictional force to keep the car from skidding?

13) Using Kepler’s laws, it can be shown that (v_0=sqrt{frac{2GM}{r_0}}) is the minimum speed needed when ( heta=0) so that an object will escape from the pull of a central force resulting from mass (M). Use this result to find the minimum speed when ( heta=0) for a space capsule to escape from the gravitational pull of Earth if the probe is at an altitude of 300 km above Earth’s surface.

10.94 km/sec

14) Find the time in years it takes the dwarf planet Pluto to make one orbit about the Sun given that a=39.5 A.U.

## MATH 2940: Linear Algebra for Engineers

This is the webpage for MATH 2940-002 ONLY. Sections 001 and 002 are entirely independent: different instructors, syllabi, assignments, and exams. This summer, MATH 2940-001 is using Blackboard.

### Basic information:

Meeting time: MTWThF 10-11:15 am
Location: Malott Hall 207
Instructor: Daniel Jerison
Office: Malott Hall 581
Office hours: MW 11:15 am - 12:15 pm or by appointment Office hours Wednesday, August 10, 10 am - 12 pm
Email: jerison at math.cornell.edu

TA: Sergio Da Silva
Office hours: Th 7:45-9:45 am, Malott Hall 218, or by appointment
Email: smd322 at cornell.edu

This is a very fast-paced class. We will cover all the material of a full-semester linear algebra course in only six weeks. For this reason, keeping up with the course material is of paramount importance. Attendance will not be taken, but I strongly encourage you to come to class every day.

The best way to learn is by solving problems. Classes will include lectures and periods of time for you to work on exercises with your classmates. Outside of class, I may assign parts of the textbook for you to read before the following day's meeting. These readings will be in addition to the assigned homework problems. Overall, you should expect to spend 2 to 3 times as many hours on this class as you would on a similar course during the academic year.

In case you are having trouble understanding a topic, do not wait! Come to office hours and ask questions during class or via email. Office hours are an excellent opportunity to go into greater depth on whichever concepts you need help understanding. The course material builds on itself, so a solid grasp of the earlier material is essential for you to learn and appreciate the later topics.

### Course description:

Linear algebra and its applications. Topics include matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces. Applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations.

Due to an overlap in content, students will receive credit for only one course in the following group: MATH 2210, MATH 2230, MATH 2310, MATH 2940. Taking MATH 2930 and 2940 simultaneously is not recommended.

Note: The official course description says: "May include computer use in solving problems." That will not be a part of the course this summer. In addition, MATH 1920 (multivariable calculus) is listed as a prerequisite. In fact there is very little overlap between the two courses, so you can take MATH 2940 without having previously taken MATH 1920.

### Textbook:

Linear Algebra and its Applications by Lay, Lay, and McDonald, 5th edition. Can be purchased online for around \$30-50 see these links.

15%: Homework
22.5%: Prelim 1
22.5%: Prelim 2
40%: Final exam

### Exams:

All exams are closed-book and closed-notes no calculators or similar aids are allowed. You must take the exams at the scheduled times.

Prelim 1: Tuesday, July 12, in class.
Prelim 2: Tuesday, July 26, in class.
Final exam: Tuesday, August 9, 8:30-11 am in Malott Hall 207.

Prelim 1 covers all course material through Friday, July 8. This includes sections 1.1-1.5, 1.7-1.9, 2.1-2.3, 2.5 (up to the end of p.127), 3.1-3.2, and 3.3 (from Theorem 9 to the end, not including Cramer's Rule). Practice problems and solutions. Actual exam and solutions.

Prelim 2 covers the course material from after Prelim 1 through Friday, July 22. This includes sections 4.1-4.6, 4.8-4.9, 5.1-5.4, and 5.6-5.7. Not included: the end of section 4.8 (starting from "Nonhomogeneous Equations"), the start of section 5.4 ("The Matrix of a Linear Transformation" section). Practice problems and solutions. Actual exam and solutions.

The final exam covers all the course material, with a greater emphasis on Chapters 6 and 7. This includes everything from the two prelims along with sections 6.1-6.5, the "Least-Squares Lines" part of section 6.6, 6.7, 7.1-7.2, and 7.4. Practice problems and solutions.

You are encouraged to work with each other on the homework. Everything you write should be in your own individual words direct copying is forbidden! You are not allowed to get help from any other person or source on an exam, including the textbook, unless that exam's instructions specifically permit it.

### Office of Student Disability Services:

It is Cornell policy to provide reasonable accommodations to students who have a documented disability (e.g., physical, learning, psychiatric, vision, hearing, or systemic) that may affect their ability to participate in course activities or to meet course requirements. Students with disabilities are encouraged to contact Student Disability Services and their instructors for a confidential discussion of their individual need for academic accommodations. Student Disability Services is located in 420 CCC. Staff can be reached by calling 607-254-4545.

### Syllabus:

June 27, 28, 29: Sections 1.1-1.5. HW 1 due Friday, July 1.

June 30, July 1, 5: Sections 1.7-1.9, 2.1, start of 2.2. HW 2 due Thursday, July 7.

July 6, 7, 8: Finish section 2.2, sections 2.3, 2.5, 3.1-3.3. Handout: Invertible matrices. Handout: Determinants and row operations. HW 3 due Monday, July 11.

July 13, 14, 15: Sections 4.1-4.5. HW 4 due Monday, July 18.

July 18, 19: Sections 4.6, 4.8-4.9. HW 5 due Thursday, July 21.

July 20, 21, 22: Sections 5.1-5.4, 5.6-5.7. HW 6 due Monday, July 25.

July 27, 28, 29: Sections 6.1-6.4. HW 7 due Monday, August 1.

August 1, 2, 3: Sections 6.5, start of 6.6, 6.7, 7.1-7.2. HW 8 due Friday, August 5. Handout: Symmetric matrices have real eigenvalues.

August 4: Section 7.4. No required HW from this section, but see optional exercises below. The Wikipedia page for singular value decomposition has a useful animation that goes through the steps of a SVD for a 2 × 2 matrix.

August 5: Final exam review.

### Homework:

Homework is due at the beginning of class on the due date. Solutions will be posted later the same day. Late homework will not be accepted. However, your lowest homework score will be dropped.

HW 1: Due Friday, July 1. Exercises 1.1.25 (hint: start doing row operations and see whether things are zero or not), 1.2.12, 1.2.19, 1.2.28, 1.3.8, 1.3.25, 1.4.1, 1.4.16, 1.5.7, 1.5.31. Solutions.

HW 2: Due Thursday, July 7. Exercises 1.7.30, 1.7.36, 1.8.15, 1.8.17, 1.9.9 (hint: follow what happens to the standard basis vectors e1, e2), 1.9.35, Required additional problems, 2.1.1, 2.2.7, 2.2.24, 2.2.35. Solutions to textbook exercises and additional problems.

HW 3: Due Monday, July 11. Exercises 2.3.15, 2.3.22, 2.3.27, 2.5.4, 3.1.10, 3.2.7, 3.2.24 (hint: see the solution to Practice Problem 2), 3.2.31, 3.3.20, 3.3.27. Solutions.

HW 4: Due Monday, July 18. Exercises 4.1.13, 4.1.19, 4.2.8, 4.2.16, 4.3.11, 4.3.14, Required additional problems, 4.4.3, 4.4.11, 4.4.13 (hint: see the solution to Practice Problem 2), 4.5.13. Solutions to textbook exercises and additional problems.

HW 5: Due Thursday, July 21. Exercises 4.6.2, 4.6.21, 4.8.6, 4.8.21, 4.9.3, 4.9.13. Solutions.

HW 6: Due Monday, July 25. Exercises 5.1.12, 5.1.25, 5.1.35, 5.2.12, 5.3.6, 5.3.13, 5.4.8, 5.4.17, 5.4.23, 5.6.5, 5.6.9, 5.7.5, Required additional problem. For exercise 5.7.5, draw typical trajectories even if the origin is not a saddle point. Solutions to textbook exercises and additional problem.

HW 7: Due Monday, August 1. Exercises 6.1.14, 6.1.28, 6.1.30, 6.2.10, 6.2.28, 6.3.10, 6.3.16, 6.3.17, 6.4.2, Required additional problems. For 6.2.28, an orthogonal matrix is a square matrix whose columns are orthonormal. This is standard terminology, unfortunately. Solutions to textbook exercises and additional problems.

HW 8: Due Friday, August 5. Exercises 6.5.4, 6.5.8, 6.5.10, 6.6.2, 6.7.23, 6.7.25, 7.1.19, 7.1.23, 7.2.6, 7.2.13. Solutions.

Exercises from section 7.4 (not to turn in): 7.4.6, 7.4.13 (hint: see the solution to Practice Problem 1), 7.4.18. Solutions.

## Explore the Volume Worksheets in Detail

Work on the skill of finding volume with this batch of counting cubes worksheets. Count unit cubes to determine the volume of rectangular prisms and solid blocks, draw prisms on isometric dot paper and much more.

Augment practice with this unit of pdf worksheets on finding the volume of a cube comprising problems presented as shapes and in the word format with side length measures involving integers, decimals and fractions.

This batch of volume worksheets provides a great way to learn and perfect skills in finding the volume of rectangular prisms with dimensions expressed in varied forms, find the volume of L-blocks, missing measure and more.

Encourage students to work out the entire collection of printable worksheets on computing the volume of triangular prism using the area of the cross-section or the base and leg measures and practice unit conversions too.

Navigate through this collection of volume of mixed prism worksheets featuring triangular, rectangular, trapezoidal and polygonal prisms. Bolster practice with easy and moderate levels classified based on the number range used.

Motivate learners to use the volume of a cone formula efficiently in the easy level, find the radius in the moderate level and convert units in the difficult level, solve for volume using slant height, and find the volume of a conical frustum too.

Access our volume of a cylinder worksheets to practice finding the radius from diameter, finding the volume of cylinders with parameters in integers and decimals, find the missing parameters, solve word problems and more!

Take the hassle out of finding the volume of spheres and hemispheres with this compilation of pdf worksheets. Gain immense practice with a wide range of exercises involving integers and decimals.

This exercise is bound to help learners work on the skill of finding the volume of rectangular pyramids with dimensions expressed as integers, decimals and fractions in easy and moderate levels.

Help children further their practice with this bundle of pdf worksheets on determining the volume of triangular pyramids using the measures of the base area or height and base. The problems are offered as 3D shapes and in word format in varied levels of difficulty.

Gain ample practice in finding the volume of pyramids with triangular, rectangular and polygonal base faces presented in two levels of difficulty. Apply relevant formulas to find the volume using the base area or the other dimensions provided.

Upscale practice with an enormous collection of printable worksheets on finding the volume of solid shapes like prisms, cylinders, cones, pyramids and revision exercises to revisit concepts with ease.

Learn to find the volume of composite shapes that are a combination of two or more solid 3D shapes. Begin with counting squares, find the volume of L -blocks, and compound shapes by adding or subtracting volumes of decomposed shapes.

## Mathematical Statistics: Old School

Mathematical Statistics: Old School covers three main areas: The mathematics needed as a basis for work in statistics the mathematical methods for carrying out statistical inference and the theoretical approaches for analyzing the efficacy of various procedures. The author, John Marden, developed this material over the last thirty years teaching various configurations of mathematical statistics and decision theory courses in the Department of Statistics, University of Illinois at Urbana-Champaign. It is intended as a graduate-level text.

• Distribution theory: distribution functions, densities, moment generating functions, transformations, the multivariate normal distribution, Bayes theorem, order statistics, convergence in probability and distribution, and the &Delta-method.
• Statistical Inference: estimation and hypothesis testing, as well as confidence intervals and model selection, from both a frequentist and Bayesian perspective exponential family and linear regression models, likelihood methods, and bootstrap and randomization techniques.
• Statistical decision theory: uniformly minimum variance estimators, the Cramér-Rao lower bound, uniformly most powerful tests, invariance, admissibility, and minimaxity.

## 12.5E: Exercises for Section 12.5 - Mathematics

For problems 1 – 6 the given functions perform the indicated function evaluations.

1. (fleft( x ight) = 10x - 3)
1. (fleft( < - 5> ight))
2. (fleft( 0 ight))
3. (fleft( 7 ight))
1. (fleft( <+ 2> ight))
2. (fleft( <12 - x> ight))
3. (fleft( ight))
1. (hleft( 0 ight))
2. (hleft( < - 3> ight))
3. (hleft( 5 ight))
1. (hleft( <6z> ight))
2. (hleft( <1 - 3y> ight))
3. (hleft( ight))
1. (gleft( 0 ight))
2. (gleft( 4 ight))
3. (gleft( < - 7> ight))
1. (gleft( <- 5> ight))
2. (gleft( ight))
3. (gleft( <4sqrt t + 9> ight))
1. (fleft( 0 ight))
2. (fleft( < - 1> ight))
3. (fleft( < - 2> ight))
1. (hleft( <5 - 12y> ight))
2. (fleft( <2+ 8> ight))
3. (fleft( ight))
1. (zleft( 4 ight))
2. (zleft( < - 4> ight))
3. (zleft( 1 ight))
1. (zleft( <2 - 7x> ight))
2. (zleft( > ight))
3. (zleft( ight))
1. (Yleft( 0 ight))
2. (Yleft( 7 ight))
3. (Yleft( < - 4> ight))
1. (Yleft( <5 - t> ight))
2. (Yleft( <- 10> ight))
3. (Yleft( <6t - > ight))

The difference quotient of a function (fleft( x ight)) is defined to be,

For problems 7 – 13 compute the difference quotient of the given function.

1. (Qleft( t ight) = 4 - 7t)
2. (gleft( t ight) = 42)
3. (Hleft( x ight) = 2 + 9)
4. (zleft( y ight) = 3 - 8y - )
5. (gleft( z ight) = sqrt <4 + 3z>)
6. (displaystyle yleft( x ight) = frac<< - 4>><<1 - 2x>>)
7. (displaystyle fleft( t ight) = frac<<>><>)

For problems 14 – 21 determine all the roots of the given function.

1. (yleft( t ight) = 40 + 3t - )
2. (fleft( x ight) = 6 - 5 - 4)
3. (Zleft( p ight) = 6 - 11p - )
4. (hleft( y ight) = 4 + 10 + )
5. (gleft( z ight) = + 6 - 16z)
6. (fleft( t ight) = <2>>> - 8<4>>> + 15)
7. (displaystyle hleft( w ight) = frac<<4w + 5>> + frac<<3w>><>)
8. (displaystyle gleft( w ight) = frac<> - frac<><<4w - 1>>)

For problems 22 – 30 find the domain and range of the given function.

1. (fleft( x ight) = - 8x + 3)
2. (zleft( w ight) = 4 - 7w - )
3. (gleft( t ight) = 3 + 2t - 3)
4. (gleft( x ight) = 5 - sqrt <2x>)
5. (Bleft( z ight) = 10 + sqrt <9 + 7> )
6. (hleft( y ight) = 1 + sqrt <6 - 7y>)
7. (fleft( x ight) = 12 - 5sqrt <2x + 9>)
8. (Vleft( t ight) = - 6left| <5 - t> ight|)
9. (yleft( x ight) = 12 + 9left| <- 1> ight|)

For problems 31 – 51 find the domain of the given function.

1. (displaystyle fleft( t ight) = frac<<4 - 12t + 8>><<16t + 9>>)
2. (displaystyle vleft( y ight) = frac <<- 27>><<4 - 17y>>)
3. (displaystyle gleft( x ight) = frac<<3x + 1>><<5- 3x - 2>>)
4. (displaystyle hleft( t ight) = frac <<- + 1 - 1>><<35+ 2 - >>)
5. (displaystyle fleft( z ight) = frac <<+ z>><<- 9 + 2z>>)
6. (displaystyle Vleft( p ight) = frac<<3 - >><<4+ 10p + 2>>)
7. (gleft( z ight) = sqrt <- 15> )
8. (fleft( t ight) = sqrt <36 - 9> )
9. (Aleft( x ight) = sqrt <15x - 2- > )
10. (Qleft( y ight) = sqrt <4- 4 + y> )
11. (displaystyle Pleft( t ight) = frac <<+ 7>><> >>)
12. (displaystyle hleft( t ight) = frac<<>><> >>)
13. (displaystyle hleft( x ight) = frac<6><- 7x + 3> >>)
14. (displaystyle fleft( z ight) = frac<><- 6 + 9> >>)
15. (Sleft( t ight) = sqrt <8 - t>+ sqrt <2t>)
16. (gleft( x ight) = sqrt <5x - 8>- 2sqrt )
17. (hleft( y ight) = sqrt <49 - > - frac<>>)
18. (displaystyle Aleft( x ight) = frac<><> + 4sqrt <+ 10x + 9> )
19. (displaystyle fleft( t ight) = frac<8><<- 3t - 4>> + frac<3><> >>)
20. (displaystyle Rleft( x ight) = frac<3><<+ >> + sqrt[5] <<- x - 6>>)
21. (Cleft( z ight) = - sqrt[4] <<+ >>)

For problems 52 – 55 compute (left( ight)left( x ight)) and (left( ight)left( x ight)) for each of the given pairs of functions.

## 12.5E: Exercises for Section 12.5 - Mathematics

• Click on the icon below for 32 Youtube videos
of lectures based on the Decision Making book

This text provides an introduction to the topic of rational decision making as well as a brief overview of the most common biases in judgment and decision making. "Decision Making" is relatively short (300 pages) and richly illustrated with approximately 100 figures. It is suitable for both self-study and as the basis for an upper-division undergraduate course in judgment and decision making. The book is written to be accessible to anybody with minimum knowledge of mathematics (high-school level algebra and some elementary notions of set theory and probability, which are reviewed in the book). At the end of each chapter there is a collection of exercises that are grouped according to that chapter&rsquos sections. Complete and detailed answers for each exercise are given in the last section of each chapter. The book contains a total of 121 fully solved exercises.

Part I: Decision Problems

2 Outcomes and Preferences . 15

2.1 Preference relations
2.2 Rational choice under certainty
2.3 Why completeness and transitivity?
2.4 Exercises
2.4.1 Exercises for Section 2.1: Preference relations
2.4.2 Exercises for Section 2.2: Rational choice under certainty
2.5 Solutions to Exercises

3 States and Acts . 27

3.1 Uncertainty, states and acts
3.2 Dominance
3.3 MaxiMin and LexiMin 34
3.3.1 MaxiMin
3.3.2 LexiMin
3.4 Regret: a first attempt
3.5 Exercises
3.5.1 Exercises for Section 3.1: Uncertainty, states and acts
3.5.2 Exercises for Section 3.2: Dominance
3.5.3 Exercises for Section 3.3: MaxiMin and LexiMin
3.5.4 Exercises for Section 3.4: Regret: a first attempt
3.6 Solutions to Exercises

4 Decision Trees . 43

4.1 Decision trees
4.2 Money lotteries and risk neutrality
4.3 Backward induction
4.4 Beyond money lotteries and risk neutrality
4.5 Exercises
4.5.1 Exercises for Section 4.1: Decision Trees
4.5.2 Exercises for Section 4.2: Money lotteries and risk neutrality
4.5.3 Exercises for Section 4.3: Backward induction
4.6 Solutions to Exercises

Part II: Uncertainty and Decision Making

5 Expected Utility Theory . 71
5.1 Money lotteries and attitudes to risk
5.2 Expected utility: theorems
5.3 Expected utility: the axioms
5.4 Exercises
5.4.1 Exercises for Section 5.1: Money lotteries and attitudes to risk
5.4.2 Exercises for Section 5.2: Expected utility theory
5.4.3 Exercises for Section 5.3: Expected utility axioms
5.5 Solutions to Exercises

6 Applications of Expected Utility . 95
6.1 States and acts revisited
6.2 Decision trees revisited
6.3 Regret
6.4 The Hurwicz index of pessimism
6.5 Exercises
6.5.1 Exercises for Section 6.1: States and acts revisited
6.5.2 Exercises for Section 6.2: Decision trees revisited
6.5.3 Exercises for Section 6.3: Regret
6.5.4 Exercises for Section 6.4: The Hurwicz index of pessimism
6.6 Solutions to Exercises

7 Conditional Reasoning . 123
7.1 Sets and probability: brief review
7.1.1 Sets
7.1.2 Probability
7.2 Conditional thinking
7.2.1 The natural frequencies approach
7.2.2 Conditional probability
7.3 Exercises
7.4.1 Exercises for Section 7.1: Sets and probability
7.4.2 Exercises for Section 7.2: Conditional thinking
7.5 Solutions to Exercises

8 Information and Beliefs . 151
8.1 Uncertainty and information
8.2 Updating beliefs
8.3 Belief revision
8.4 Information and truth
8.5 Exercises
8.5.1 Exercises for Section 8.1: Uncertainty and information
8.5.2 Exercises for Section 8.2: Updating beliefs
8.5.3 Exercises for Section 8.3: Belief revision
8.6 Solutions to Exercises

9 The Value of Information . 169
9.1 When is information potentially valuable?
9.2 The value of information when outcomes are sums of money
9.2.1 Perfect information and risk neutrality
9.2.2 Perfect information and risk aversion
9.2.3 Imperfect information
9.3 The general case
9.4 Different sources of information
9.5 Exercises
9.5.1 Exercises for Section 9.1: When is information potentially valuable?
9.5.2 Exercises for Section 9.2: The value of information when outcomes are sums of money
9.5.3 Exercises for Section 9.3: The general case
9.5.4 Exercises for Section 9.4: Different sources of information
9.6 Solutions to Exercises

Part III: Thinking about Future Selves

10 Intertemporal Choice . 215
10.1 Introduction
10.2 Present value and discounting
10.3 Exponential discounting
10.3.1 Time consistency
10.4 Hyperbolic discounting
10.4.1 Interpretation of the parameter b
10.5 Dealing with time inconsistency
10.6 Exercises
10.6.1 Exercises for Section 10.2: Present value and discounting
10.6.2 Exercises for Section 10.3: Exponential discounting
10.6.3 Exercises for Section 10.4: Hyperbolic discounting
10.6.4 Exercises for Section 10.5: Dealing with time inconsistency
10.7 Solutions to Exercises

Part IV: Group Decision Making

11 Aggregation of Preferences . 245
11.1 Social preference functions
11.2 Arrow&rsquos Impossibility Theorem
11.3 Illustration of the proof of Arrow&rsquos theorem
11.4 Application of Arrow&rsquos theorem to individual choice
11.5 Exercises 1
11.5.1 Exercises for Section 11.1: Social preference functions
11.5.2 Exercises for Section 11.2: Arrow&rsquos impossibility theorem
11.5.3 Exercises for Section 11.3: Illustration of the proof of Arrow&rsquos theorem
11.5.4 Exercises for Section 11.4: Application of Arrow&rsquos theorem to individual choice
11.6 Solutions to Exercises

12 Misrepresentation of Preferences . 275
12.1 Social choice functions
12.2 Strategic voting
12.3 The Gibbard -Satterthwaite theorem
12.4 Illustration of the proof of the Gibbard -Satterthwaite theorem
12.5 Exercises
12.5.1 Exercises for Section 12.1: Social choice functions
12.5.2 Exercises for Section 12.2: Strategic voting
12.5.3 Exercises for Section 12.3: The Gibbard -Satterthwaite theorem
12.6 Solutions to Exercises .

Part V: Biases in Decision Making

13 Biases in Decision Making . 303
13.1 Introduction
13.2 Incomplete preferences and manipulation of choice
13.3 Gains versus losses
13.4 Framing
13.5 The confirmation bias
13.6 The psychology of decision making

## How to Make Them Yourself

First, put your data into a table (like above), then add up all the values to get a total:

Next, divide each value by the total and multiply by 100 to get a percent:

Comedy Action Romance Drama SciFi TOTAL
4 5 6 1 4 20
4/20
= 20%
5/20
= 25%
6/20
= 30%
1/20
= 5%
4/20
= 20%
100%

Now to figure out how many degrees for each "pie slice" (correctly called a sector).

A Full Circle has 360 degrees, so we do this calculation:

Comedy Action Romance Drama SciFi TOTAL
4 5 6 1 4 20
20% 25% 30% 5% 20% 100%
4/20 × 360°
= 72°
5/20 × 360°
= 90°
6/20 × 360°
= 108°
1/20 × 360°
= 18°
4/20 × 360°
= 72°
360°

Now you are ready to start drawing!

Then use your protractor to measure the degrees of each sector.

Here I show the first sector .

Finish up by coloring each sector and giving it a label like "Comedy: 4 (20%)", etc.

## 12.5E: Exercises for Section 12.5 - Mathematics

3. Determine all the number(s) (c) which satisfy the conclusion of Mean Value Theorem for (hleft( z ight) = 4 - 8 + 7z - 2) on (left[ <2,5> ight]).

The first thing we should do is actually verify that the Mean Value Theorem can be used here.

The function is a polynomial which is continuous and differentiable everywhere and so will be continuous on (left[ <2,5> ight]) and differentiable on (left( <2,5> ight)).

Therefore, the conditions for the Mean Value Theorem are met and so we can actually do the problem.

Note that this may seem to be a little silly to check the conditions but it is a really good idea to get into the habit of doing this stuff. Since we are in this section it is pretty clear that the conditions will be met or we wouldn’t be asking the problem. However, once we get out of this section and you want to use the Theorem the conditions may not be met. If you are in the habit of not checking you could inadvertently use the Theorem on a problem that can’t be used and then get an incorrect answer.

Now that we know that the Mean Value Theorem can be used there really isn’t much to do. All we need to do is do some function evaluations and take the derivative.

[hleft( 2 ight) = 12hspace<0.5in>hleft( 5 ight) = 333hspace<1.5in>h'left( z ight) = 12 - 16z + 7]

The final step is to then plug into the formula from the Mean Value Theorem and solve for (c).

So, we found two values and, in this case, only the second is in the interval and so the value we want is,

## 12.5E: Exercises for Section 12.5 - Mathematics

#### Week-at-a-glance

• Week 01: 17 January
• Week 02: 24 January
• Week 03: 31 January
• Week 04: 07 February
• Week 05: 14 February
• Week 06: 21 February
• Week 07: 28 February
• Week 08: 07 March
• Week 09: 14 March [Midterm Recess]
• Week 10: 21 March
• Week 11: 28 March
• Week 12: 04 April
• Week 13: 11 April
• Week 14: 18 April
• Week 15: 25 April

#### Special Chapter Assignments [Required]

[Stewart, p. 699, Problems Plus: 4 and 5(a-f)]
[Stewart, p. 822, Discovery Project]
[Stewart, p. 885, Problems Plus: 5 and 7]
[Stewart, p. 978, Problems Plus: 1 and 3]
[Stewart, p. 1032, Discovery Project: 1 - 3]

#### Section Exercises [Optional]

• 10.1: 3 - 24 (x 3s)*, 27, 31, 39
• 10.2: 1, 3, 6, 12, 15, 25, 29, 39, 43, 48, 53, 60
• 10.3: 5, 14, 27, 28, 33 - 48 (x 3s), 61, 64, 65, 67
• 10.4: 3 - 12 (x 3s), 18, 21, 24 - 33 (x 3s), 47, 55
• 10.5: 33 - 48 (x 3s), 51
• 10.6: 3 - 15 (x 3s), 21, 23, 27
• 12.1: 5, 6, 7, 11, 18, 24 - 33 (x 3s), 39
• 12.2: 3 - 6, 13, 18, 21, 25, 27, 29, 30, 32, 37, 39
• 12.3: 3, 6, 9, 12, 14, 18, 24, 34, 39, 40, 41, 47, 49, 53
• 12.4: 5, 10, 13, 14, 15, 21, 30, 31, 34, 36, 39, 40
• 12.5: 5 - 65 (x 5s), 69
• 12.6: 3 - 36 (x 3s)
• 12.7: 5 - 65 (x 5s)
• 13.1: 3 - 24 (x 3s), 26, 34, 40
• 13.2: 1, 9 - 24 (x 3s), 31, 33, 36, 39, 45, 49
• 13.3: 3, 6, 15, 20, 21, 39, 41
• 13.4: 2, 5, 7, 9, 12, 21, 22, 28
• 14.1: 8 - 10, 12, 15, 18, 30, 31, 34, 54, 57
• 14.2: 6 - 18 (x 3s), 27 - 36 (x 3s)
• 14.3: 3, 6, 9, 15 - 54 (x 3s), 74, 80
• 14.4: 3, 6, 17, 19, 24, 27, 32, 38
• 14.5: 3 - 36 (x 3s), 41, 45
• 14.6: 6 - 27 (x 3s), 39, 47, 52, 54
• 14.7: 2, 3, 6, 9, 13, 14, 18, 27, 30, 32, 45, 51
• 15.1: 3, 5
• 15.2: 3 - 30 (x 3s)
• 15.3: 9 - 27 x 3s), 39, 41, 50, 52
• 15.4: 1 - 6, 9, 12, 15, 22, 25, 28
• 15.6: 4, 5, 7, 9, 12, 21, 22, 24
• 15.7: 5, 10, 15, 20, 32, 37, 40(a), 47
• 15.8: 5 - 7, 10, 13, 20, 21, 24, 29, 31

 Week 01 Date Session topic Submissions / Exams Monday 17 January Half-day session Wednesday 19 January Section 10.1 Thursday* 20 January Sections 10.1 & 10.2 Friday 21 January Section 10.2

*Sections refer to Stewart, Calculus: Early Transcendental Multivarible , 5th edition

**Unless otherwise noted as above, Thursdays will be reserved for discussion and questions

## 12.5E: Exercises for Section 12.5 - Mathematics

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