Help with Michael Spivak's fourth edition calculus book

[solar nebula]

Hi,
I am having difficulties doing the problems in Michael Spivak's fourth edition calculus book.
I am not sure how to approach the problems. I am able to understand the contents chapters but when I start to do the chapter problems, I have no idea how to do them.Any advice will be appreciated.By the way I am a first year student.

Thanks.

 

[DivisionByZro]

You're a first year student in what? Music? Physics? It would be helpful if you could provide more details. It's impossible to properly help you when we have nothing to work with. What courses have you taken before? Do you have experience with proofs and proof techniques? How much calculus do you already know?

 

[micromass]

Spivak does indeed have quite difficult exercises. They're very worthwhile to solve, but they can be hard. I suggest getting perhaps a Schaum's outline and make exercises from that book first. These are the ones you actually need.

 

[arpeggio]

I have the same question

The book's first chapter is very vague. Not that the information is vague but I don't really know what they are asking for. For example, they ask something like
prove that
lXYl = lXl x lYl

...what am I supposed to say, in what kind of form?
In high school, the only thing that was similar to this was trig identity where you had to say left side equals right side

 

[solar nebula]

You're a first year student in what? Music? Physics? It would be helpful if you could provide more details. It's impossible to properly help you when we have nothing to work with. What courses have you taken before? Do you have experience with proofs and proof techniques? How much calculus do you already know?

I am a computer science student.The only proofs I have done is trigonometric proofs, when you show one side is equal to the other side. I have done high school calculus and functions, I know them well. Spivak's kind of math is the kind of math I want to learn, because it's no longer just plugging in, but using your brain.

Thanks.

 

[solar nebula]

Spivak does indeed have quite difficult exercises. They're very worthwhile to solve, but they can be hard. I suggest getting perhaps a Schaum's outline and make exercises from that book first. These are the ones you actually need.

Can you please let me know the title of the Schaum's outline book? Thanks.

 

[intwo]

You should check out a book on proofs before trying the problems in Spivak. It's not necessary, but it would be beneficial.

Try How to Prove It: A Structured Approach by Velleman.

 

[solar nebula]

You should check out a book on proofs before trying the problems in Spivak. It's not necessary, but it would be beneficial.

Try How to Prove It: A Structured Approach by Velleman.

Thanks, I'll check it out.

 

[diligence]

I have the same question

The book's first chapter is very vague. Not that the information is vague but I don't really know what they are asking for. For example, they ask something like
prove that
lXYl = lXl x lYl

...what am I supposed to say, in what kind of form?
In high school, the only thing that was similar to this was trig identity where you had to say left side equals right side

Don't worry, these seemingly trivial type problems were also confusing to me not long ago. I imagine many students have similar troubles. The more you are exposed to proof based math the more you will begin to understand what he's asking for.

To answer your question, you could approach this this of problem in two ways. The first is to show that the left side is less than or equal to the right side, and also that the right side is less than or equal to the left side. Thus you can then conclude that they must be equal. Not to say this is the best approach for this specific problem, but this is a very common method for showing equality.

The second would be to break it down on a case by case basis and show equality always holds. For example, case 1: when both x,y nonnegative. Case 2: x nonnegative, y negative...etc etc etc

Also, an important thing to take from this problem is to make sure you know the definition of the absolute value function. Can you state the absolute value function from R to R?

 

[arpeggio]

Don't worry, these seemingly trivial type problems were also confusing to me not long ago. I imagine many students have similar troubles. The more you are exposed to proof based math the more you will begin to understand what he's asking for.

To answer your question, you could approach this this of problem in two ways. The first is to show that the left side is less than or equal to the right side, and also that the right side is less than or equal to the left side. Thus you can then conclude that they must be equal. Not to say this is the best approach for this specific problem, but this is a very common method for showing equality.

The second would be to break it down on a case by case basis and show equality always holds. For example, case 1: when both x,y nonnegative. Case 2: x nonnegative, y negative...etc etc etc

Also, an important thing to take from this problem is to make sure you know the definition of the absolute value function. Can you state the absolute value function from R to R?

Hopefully I will get it too. Thanks for your response
What is R to R ?

 

[Deveno]

R, or more properly $\mathbb{R}$ is a common abbreviation for the set of real numbers.

half the battle in proving something like:

|xy| = (|x|)(|y|)

is knowing "what it is" you are proving.

so you need to be clear about what |x| MEANS.

what you are proving is:

path a: take two real numbers x,y ----> multiply them together----->take the absolute value

path b: take two numbers x,y----->take the absolute value of each one---->multiply the two absolute values together

start at end "at the same places".

with absolute value, the DEFINITION is:

|x| = x, if x ≥ 0
|x| = -x, if x < 0.

since the definition of |x| is in "two cases", you will have FOUR cases (one for each possible combination of the cases for x and y) for your proof:

a) x, y ≥ 0
b) x ≥ 0, y < 0
c) x < 0, y ≥ 0
d) x,y < 0

here is "one" possible way to handle case (b):

if x ≥ 0, y < 0, then either x = 0, and thus |xy| = |0y| = |0| = 0 = 0|y| = |0|*|y| = |x|*|y|, or:

xy < 0, so that |xy| = -xy = x(-y) = |x|*(-y) = |x|*|y|.

it may prove troublesome to make a separate "case within a case" when either x or y (or both) ≥ 0, so you might prefer to make the following five cases:

a) x,y > 0
b) x > 0, y < 0
c) x < 0, y > 0
d) x,y < 0
e) either x or y, or both = 0

since "case by case" proofs are awkward (and rather long), anything we can prove about absolute values that let's us "forget about the individual cases", is helpful, as it simplifies further work later on.

 

[arpeggio]

^Thanks, kind of get the idea now
proving is pretty intense

 

[dudebroIII]

Spivak is a great book ... I really appreciate his weird easter eggs. need to buy a new copy.

what's your purpose for reading it as a 1st yr in the middle of Dec? i assume you are following it on your own since o/w you would have asked in Sep. if you want to study math at this level, now (as a 1st year ug) is the time to take courses not self study ... any (competent) introductory level proof-based calculus class will teach you how to do proofs etc. i'd be surprised if you could pick up spivak with no proof experience and no guidance and get much out of it. why not just take a class instead of embarking on a massive self study mission?

 

[diligence]

Thanks for elaborating Deveno

proving is pretty intense

Yep, tell me about it ;) I just finished taking graduate analysis and it was by far the most challenging academic experience of my career.

 

[solar nebula]

Spivak is a great book ... I really appreciate his weird easter eggs. need to buy a new copy.

what's your purpose for reading it as a 1st yr in the middle of Dec? i assume you are following it on your own since o/w you would have asked in Sep. if you want to study math at this level, now (as a 1st year ug) is the time to take courses not self study ... any (competent) introductory level proof-based calculus class will teach you how to do proofs etc. i'd be surprised if you could pick up spivak with no proof experience and no guidance and get much out of it. why not just take a class instead of embarking on a massive self study mission?

Well, I had taken a course in first semester(in which we used Spivak's book).Then dropped it and will be taking it in the second semester . Now that I have bit of time before second semester, I just want get a bit comfortable with proofs. Intwo said to try "How to Prove It: A Structured Approach by Velleman". So I am currently going through it.