Math 253: HWK 3(DUE 9/29 Friday)

Hi all,

Please start reviewing chapter 2 for upcoming test!

Problems to hand in:

10.1: 11

10.2: 7,8,17

12.1: 6,11(a),31

12.2: 15,19,21,38

12.3: 6,10,39,43,61,62

12.4: 9-12,20

12.5:20

12.6:1-6

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Summary of vectors

Hi all,

This is a brief summary of vectors. We will be using these properties in the rest of chapter 12.

Vector is a quantity with both magnitude (length, norm, etc.) and direction. To completely determine a vector, you will need both magnitude and direction only. In particular, position is irrelevant for describing a vector.  Vector is a concept in any dimensions, not only 2 or 3.

We talk about vectors both with and without coordinate systems, and both ways can be useful.

If we do not introduce coordinate systems: a vector is represented by drawing a line segment with an arrow at the end. We study the addition by using the triangle law or parallelogram law. Subtraction of vector is simply adding the vector of same magnitude but opposite direction. Scalar multiplication of a vector yields either the zero vector, or a parallel vector of same or opposite direction.

Now with coordinate systems: a vector can be represented by its initial and terminal point: say vector \vec{ u} has initial point (x_1,y_1) and terminal point (x_2,y_2). Then the vector \vec{ u} can be simply represented by \vec{u}=<x_2-x_1,y_2-y_1>. Notice this is enough information to determine a vector: direction from slope of the line segment, and magnitude from distance of the two points. As the position of the vector doesn’t impact the vector, we often move vectors so that their initial point is at the origin. Studying the addition (or subtraction) of two vectors \vec{u}=<u_1,u_2> and \vec{v}=<v_1,v_2> now simply becomes the addition or subtraction of the corresponding components. \vec{u} \pm \vec{v}=<u_1\pm v_1, u_2\pm v_2>.

Finally, the standard  basis vectors are useful tools for studying ”skeletons” of all vectors on the plane or in space. We often like to write vectors as combined quantities of standard basis vectors.

 

How to study math?

According to Paul Halmos (image from here):

study_math

Don’t just read it; fight it! Ask your own questions,
look for your own examples, discover your own proofs.
Is the hypothesis necessary? Is the converse true?
What happens in the classical special case? What
about the degenerate cases? Where does the proof
use the hypothesis?

— Paul R. Halmos

Review of Pre-calculus

Hi all,

I found these two resources online for review:

This and this.

I can’t choose between the two, but they definitely have overlaps. Focus on one and see how you feel about the content. Check out review for chapter 1 on page 68 if you are done and let me know if you have any questions.

Welcome to Math 253!

Hi all,

Welcome to Math 253! This is the last course in the Calculus sequence here at PLU. Please read the following note (modified from here) to student, and we will get started!

What makes this course interesting?

The use of calculus and its consequences cuts across many disciplines, ranging from biology to business to engineering to the social sciences. At the risk of oversimplifying, calculus provides powerful tools to study “the rate of change.” For example, we might want to study how fast a disease is spreading through a population, by studying the “number of diagnosed cases per day”. We hope that seeing how calculus can be used to solve real world problems will be interesting. Many practical applications of calculus involve functions that depend on more than one variable. You will begin to learn about the geometry of curves and surfaces and get an introduction to differentiation and integration of functions two or more variables. This course also expands upon the idea of linear approximations learned in math 151. You will learn how to make better approximations and to estimate how good these approximations are.

What makes this course difficult?

The hardest thing about calculus is precalculus. The hardest thing about precalculus is algebra.

You all know from previous math classes how one course will build upon the next, and calculus is no exception. Math 253 will not only use material from precalculus and algebra, but it will use material you learned in Math 151 and Math 152.

Most of you will eventually see how calculus is applied in your chosen field of study, even if you don’t major in math. For this reason, we aim for ability to solve application problems using calculus. Some of the homework problems are quite lengthy and building up your “mathematical problem solving stamina” is just one of the aims of this course. It means that a large number of “word problems” (“story problems”) or “multi-step problems” are encountered in the course. This is one key place Math 253 will differ from a typical high school course. In addition, it is important to note that the ability to apply calculus requires more than computational skill; it requires conceptual understanding. As you work through the homework, you will find two general types of problems: calculation/skill problems and multi-step/word problems. A good rule of thumb is to work enough of the skill problems to become proficient, then spend the bulk of your time working on the longer multi-step problems.

Five common misconceptions

Misconception #1: Theory is irrelevant and the lectures should be aimed just at showing you how to do the problems.

The issue here is that we want you to be able to do ALL problems – not just particular kinds of problems – to which the methods of the course apply. For that level of command, the student must attain some conceptual understanding and develop judgment. Thus, a certain amount of theory is very relevant, indeed essential. A student who has been trained only to do certain kinds of problems has acquired very limited expertise.

Misconception #2: The purpose of the classes and assignments is to prepare the student for the exams.

The real purpose of the classes and homework is to guide you in achieving the aspiration of the course: command of the material. If you have command of the material, you should do well on the exams.

Misconception #3: It is the teacher’s job to cover the material.

As covering the material is the role of the textbook, and the textbook is to be read by the student, the instructor should be doing something else, something that helps the student grasp the material. The instructor’s role is to guide the students in their learning: to reinforce the essential conceptual points of the subject, and to show their relation to the solving of problems.

Misconception #4: Since you are supposed to be learning from the book, there’s no need to go to the lectures.

The lectures, the reading, and the homework should combine to produce true comprehension of the material. For most students, reading a math text won’t be easy. The lectures should serve to orient the student in learning the material.

Misconception #5: Since I did well in math, even calculus, in a good high school, I’ll have no trouble with math at PLU.

There is a different standard at the college level. Students will have to put in more effort in order to get a good grade than in high school (or equivalently, to learn the material sufficiently well by college standards).

How do I succeed?

Most people learn mathematics by doing mathematics. That is, you learn it by active participation; it is very unusual for someone to learn calculus by simply watching the instructor perform. For this reason, the homework is THE heart of the course and more than anything else, study time is the key to success in Math 253. We advise an average of 15 hours of study per week, OUTSIDE class. Also, during the first week, the number of study hours will probably be even higher as you adjust to the viewpoint of the course and brush up on precalculus/algebra skills. In effect, this means that Math 253 will be roughly a 20 hour per week effort; the equivalent of a half-time job! It is much better to spread your studying evenly as possible across the week; cramming 15 hours of homework into the day before an assignment is due does not work. Pacing yourself, using a time schedule throughout the week, is a good way to insure success; this applies to any course at the PLU, not just math.

What resources are available to help me succeed?

Calculus is a challenging course and the math department would like to see every one of you pass through with a positive experience. To help, a number of resources are available.

  • Your instructor will be accessible to help you during office hours, which will be announced early the first week of the term. If you are new to the university, you might have the false impression that professors are aloof and hard to approach. I will make myself very accessible to help you and you should not be afraid to ask for advice or help.
  • There is a tutoring service run by the academic advising. It is run by previous students who have succeeded in the course.
  • The internet. By this, I do not mean Slader or simply google the answer of your exercise and copy them down. Look for resources, understand the process of problem solving.
  • YOUR FELLOW STUDENTS! Meet in a small group of fellow students in the class and work through problems together. Explaining solutions to one another is often the best way to learn.

Welcome to Math 151!

Hi all,

Welcome to Math 151! This is going to be the first course in the calculus sequences here at PLU.

(The following is modified from this note.)

What makes this course interesting?

The use of calculus and its consequences cuts across many disciplines, ranging from biology to business to engineering to the social sciences. At the risk of oversimplifying, calculus provides powerful tools to study “the rate of change.” For example, we might want to study how fast a disease is spreading through a population, by studying the “number of diagnosed cases per day”. We hope that seeing how calculus can be used to solve real world problems will be interesting.

What makes this course difficult?

The hardest thing about calculus is precalculus. The hardest thing about precalculus is algebra.

Having a solid grounding in precalculus is much more important to success in Math 151 than is having taken high school calculus. In fact, a student with a very good precalculus background but no earlier course in calculus may do better than a student with a year of high school calculus and serious gaps in their precalculus background. If you feel you have such gaps, please let me know as soon as possible. We can solve the problem efficiently is identified early.

You all know from previous math classes how one course will build upon the next, and calculus is no exception. Math 151 will introduce only one genuinely new idea, the concept of a “limit”. The course then combines precalculus and algebra tools with “limits” to solve new types of problems. We will soon see that some calculations are very unforgiving, as far as algebra or precalculus mistakes are concerned.

Most of you will eventually see how calculus is applied in your chosen field of study, even if it is not mathematics. For this reason, we aim for ability to solve application problems using calculus. Some of the homework problems are quite lengthy and building up your “mathematical problem solving stamina” is just one of the aims of this course. You might see a large number of “word problems” (“story problems”) or “multi-step problems” are encountered in the course. This is one key place Math 151 will differ from a typical high school course. In addition, it is important to note that the ability to apply calculus requires more than computational skill; it requires conceptual understanding. As you work through the homework, you will find two general types of problems: calculation/skill problems and multi-step/word problems. A good rule of thumb is to work enough of the skill problems to become proficient, then spend the bulk of your time working on the longer multi-step problems.

Five common misconceptions

Misconception #1: Theory is irrelevant and the lectures should be aimed just at showing you how to do the problems.

The issue here is that we want you to be able to do ALL problems – not just particular kinds of problems – to which the methods of the course apply. For that level of command, the student must attain some conceptual understanding and develop judgment. Thus, a certain amount of theory is very relevant, indeed essential. A student who has been trained only to do certain kinds of problems has acquired very limited expertise.

Misconception #2: The purpose of the classes and assignments is to prepare the student for the exams.

The real purpose of the classes and homework is to guide you in achieving the aspiration of the course: command of the material. If you have command of the material, you should do well on the exams.

Misconception #3: It is the teacher’s job to cover the material.

As covering the material is the role of the textbook, and the textbook is to be read by the student, the instructor should be doing something else, something that helps the student grasp the material. The instructor’s role is to guide the students in their learning: to reinforce the essential conceptual points of the subject, and to show their relation to the solving of problems.

Misconception #4: Since you are supposed to be learning from the book, there’s no need to go to the lectures.

The lectures, the reading, and the homework should combine to produce true comprehension of the material. For most students, reading a math text won’t be easy. The lectures should serve to orient the student in learning the material.

Misconception #5: Since I did well in math, even calculus, in a good high school, I’ll have no trouble with math at PLU.

There is a different standard at the college level. Students will have to put in more effort in order to get a good grade than in high school (or equivalently, to learn the material sufficiently well by college standards).

How do I succeed?

Most people learn mathematics by doing mathematics. That is, you learn it by active participation; it is very unusual for someone to learn calculus by simply watching the instructor perform. For this reason, the homework is THE heart of the course and more than anything else, study time is the key to success in Math 151. We advise a minimum of at least 10 hours of study per week, OUTSIDE class. Also, during the first week, the number of study hours will probably be even higher as you adjust to the viewpoint of the course and brush up on precalculus/algebra skills. In effect, this means that Math 151 will be at least a 15 hour per week effort; almost the equivalent of a half-time job! In addition, it is much better to spread your studying evenly as possible across the week; cramming 10 hours of homework into the day before an assignment is due does not work. Pacing yourself, using a time schedule throughout the week, is a good way to insure success; this applies to any course at the PLU, not just math.

What resources are available to help me succeed?

Calculus is a challenging course and the math department would like to see every one of you pass through with a positive experience. To help, a number of resources are available.

  • Your instructor will be accessible to help you during office hours. If you are new to the university, you might have the false impression that professors are aloof and hard to approach. I will make myself very accessible to help you and you should not be afraid to ask for advice or help.
  • There is a math workshop operated by academic advising, and they are staffed with students who have previously succeeded in this course.
  • YOUR FELLOW STUDENTS! Meet in a small group of fellow students in the class and work through problems together. Explaining solutions to one another is often the best way to learn.