Mathematical language (Three Precalculus Answers)

[HATLEY1997]

Just looking to have three answers checked to see if I am on the correct page or not

 

[Dr.Peterson]

First problem:




You need to use some words; say what you are proving in each part, and what is assumed, and why each step is valid. You might start like this: "Suppose that x and y are real numbers such that ...".

The first column appears to be abandoned; why is it there? And aren't you assuming that y is not zero, when you divide by it?

The second column is more useful; but you seem to be proving in only one direction, not "if and only if".

I myself would show that each equation is equivalent to the same equation, and then explain how that proves the goal. But they probably expect something that looks more like what you are trying to do.

Second problem:




Again, the first paragraph seems to be unused. Why did you do that part? What does it imply about n?

Also, although the second paragraph starts well, you make a huge leap in claiming you've shown n is odd. What does what you say have to do with the definition of oddness? (And is that really what you want to show? Does it even make sense?)

You may have some good ideas, but they aren't stated.

I'll leave the third problem for others.

 

[Steven G]

#3
You want to show by mathematical induction that if f(x)=e^5x-2, then f⁽ⁿ⁾=5ⁿe^5x-2 where n is a natural number.

1st step is to show that the statement is true for n=1.
f⁽¹⁾= f'(x) = 5e^5x-2 = 5¹e^5x-2 which is what you want.

2nd step is to ASSUME that f⁽ⁿ⁾=5ⁿe^5x-2

3rd step is to SHOW that f⁽ⁿ⁺¹⁾=5ⁿ⁺¹e^5x-2

I'll leave it to you to verify step 3.

 

[Steven G]

I see that you did show us your work with number 3.
I have an issue with your 1st step. It seems that you think that f⁽¹⁾(x) = f'(1)=e³. While it is true that f'(1) = e³, it is not true that f⁽¹⁾(x) = f'(1).
More importantly, e³ is NOT what you want to get when n=1. I also don't know why you mentioned using the product rule to obtain f'(x)
I actually worked out what step 1 should be above.

 

[HATLEY1997]

First problem:
View attachment 38357

View attachment 38356

You need to use some words; say what you are proving in each part, and what is assumed, and why each step is valid. You might start like this: "Suppose that x and y are real numbers such that ...".

The first column appears to be abandoned; why is it there? And aren't you assuming that y is not zero, when you divide by it?

The second column is more useful; but you seem to be proving in only one direction, not "if and only if".

I myself would show that each equation is equivalent to the same equation, and then explain how that proves the goal. But they probably expect something that looks more like what you are trying to do.

Second problem:
View attachment 38358

View attachment 38359

Again, the first paragraph seems to be unused. Why did you do that part? What does it imply about n?

Also, although the second paragraph starts well, you make a huge leap in claiming you've shown n is odd. What does what you say have to do with the definition of oddness? (And is that really what you want to show? Does it even make sense?)

You may have some good ideas, but they aren't stated.

I'll leave the third problem for others.

I’ve had another look at this today, am I right in thinking I need to add 3 to both sides to prove it is odd? Any other guidance would be helpful

 

[Dr.Peterson]

I’ve had another look at this today, am I right in thinking I need to add 3 to both sides to prove it is odd? Any other guidance would be helpful

Add 3 to both sides of what? Please show what you are suggesting you should do. That is, try doing it, and see if it helps!


You need to prove two things:

1. If n+3 is odd, then 5n is a multiple of 10.
2. If 5n is a multiple of 10, then n+3 is odd.

You've proved the first part:

​

(One line has an error that was corrected in the next. And, technically, you should mention that k-1 is an integer.)

Now you're starting the second part:

​

Do something like what you did before; adding 3 will be part of the work, but you don't even have an equation yet.

 

[HATLEY1997]

I have seen the error in the image - thanks for spotting that.

Would I start with ?
n=10k
n+3=10k+3

Or n=10k
5n=50k

Or neither of these?

 

[Dr.Peterson]

Would I start with ?
n=10k
n+3=10k+3

Or n=10k
5n=50k

Or neither of these?

Start with what is assumed in this part of the proof: that 5n is a multiple of 10 (not that n is a multiple of 10). How can you write that in terms of the existence of some integer k?

 

[HATLEY1997]

Start with what is assumed in this part of the proof: that 5n is a multiple of 10 (not that n is a multiple of 10). How can you write that in terms of the existence of some integer k?

5n=10k
n=2k
n+3=2k+3
?

 

[Dr.Peterson]

5n=10k
n=2k
n+3=2k+3
?

Now just show that 2k+3 must be odd. You are only one step away. (How is "odd" defined?)

 

[HATLEY1997]

C

Now just show that 2k+3 must be odd. You are only one step away. (How is "odd" defined?)

It can be written in form 2k+1, for some integer k. So am I only looking at the right side of the equation here?

 

[HATLEY1997]

Is i

C

It can be written in form 2k+1, for some integer k. So am I only looking at the right side of the equation here?

Is it n+3= (2k+1)+2 ?

 

[Dr.Peterson]

Is it n+3= (2k+1)+2 ?

No, that doesn't do what you want.

It can be written in form 2k+1, for some integer k.

You want to show that n+3 is odd, right? That is, that n+3 can be written in the form 2(?) + 1.

So do that. (It won't be the same k you already have; replace the ? with an expression.)

 

[HATLEY1997]

So should 2(?)+1 be equal to 2k+3?

Sorry for all the messages. I don’t know why I am getting myself so confused with this. I really appreciate your help

 

[Dr.Peterson]

So should 2(?)+1 be equal to 2k+3?

You've shown that n+3 can be written as 2k+3, and you want to show that n+3 can be written as 2(?)+1. So, yes, you want to rewrite 2k+3 as 2(...)+1.

Suppose I gave you the equation [imath]2k+3=2z+1[/imath]. Can you solve for z? That's what will go inside the parentheses.

Sorry for all the messages. I don’t know why I am getting myself so confused with this. I really appreciate your help

Can you tell us what you are learning? (What is Unit 9 about? What course is it part of?)

I have assumed that some of what you've learning is either about how to write proofs, or very basic number theory. You don't seem to be well prepared for this problem, so knowing what you are prepared to do might make it easier to help.

Also, are there any examples you've been given, or other problems, that involve proving statements about integers? Are there any theorems? Anything you can show might provide something we can use to help you think about this.