Maths 3A/3B

[anotherworld2b]

Leaving this post unfinished for you to do some work yourself. Come back later if you tried and still couldn't finish

Is there a particular way to prove and make a conjecture? I never seem to have an idea how to start 

[anotherworld2b]

Loosely, I'd say yes, but only someone who is doing/has done the WACE would know for sure you should consult your syllabus!

Be sure to clarify this sort of stuff with your teacher in the first instance too! Like, you deserve to know what topics to study for your test, aha

Will do 

[RuiAce]

Is there a particular way to prove and make a conjecture? I never seem to have an idea how to start 

In general, conjecturing something is hard and requires the mindset of a third year uni student at the least.

For this question, however, try to figure out why instead of 12, if we used 6 the statement would be true...
(If you can figure out why it's true, then you can figure out my inspiration as to why I picked 6)


Proving a conjecture is just a fancy way of saying "proving something I made up myself"

[anotherworld2b]

Hi i was just wandering if my answers to these two questions are correect because the answer is different in my book.

[MightyBeh]

Hi i was just wandering if my answers to these two questions are correect because the answer is different in my book.

seems fine to me, but don't forget your f'(x) in your working out. I wasn't sure what you were doing for a sec.

[RuiAce]

All that's not ok is your setting out for reasons MightyBeh stated. Your final answer is correct and can be confirmed on WolframAlpha which you should always use to check your work if you disagree with the answers

[anotherworld2b]

Could i get help with q35 and 36 please?

[jamonwindeyer]

Could i get help with q35 and 36 please?

Sure thing! So 35 is in two parts; first, what is the gradient of that line?



So in gradient intercept form, it is easier to see that \(m=\frac{41}{16}\) now, we simply use the derivative to find the gradient:



For what values of \(x\) is that derivative equal to \(\frac{41}{16}\)?

So Question 36 is more of a test of algebraic manipulation than anything else, it's using the definition that you've written and that's all! It can be tough though, I'll show you the first one for \(f(x)=\frac{1}{x}\)



Notice that I used cross multiplication in the middle there to express the numerator as a single fraction! See if that algebra makes sense, and have a go at manipulating the second one!

HINT: For Q36(b), there is a not so obvious trick. You'll arrive at an expression containing:



Try multiplying the top and bottom by \(\sqrt{x+h}+\sqrt{x}\), and see where it gets you

[anotherworld2b]

I will try to redo them later 
I was also wondering could i get help to understand these two questions? ive tried to understand the answers for these 2 questions but i still dont understand it

Sure thing! So 35 is in two parts; first, what is the gradient of that line?



So in gradient intercept form, it is easier to see that \(m=\frac{41}{16}\) now, we simply use the derivative to find the gradient:



For what values of \(x\) is that derivative equal to \(\frac{41}{16}\)?

So Question 36 is more of a test of algebraic manipulation than anything else, it's using the definition that you've written and that's all! It can be tough though, I'll show you the first one for \(f(x)=\frac{1}{x}\)



Notice that I used cross multiplication in the middle there to express the numerator as a single fraction! See if that algebra makes sense, and have a go at manipulating the second one!

HINT: For Q36(b), there is a not so obvious trick. You'll arrive at an expression containing:



Try multiplying the top and bottom by \(\sqrt{x+h}+\sqrt{x}\), and see where it gets you

[RuiAce]

I will try to redo them later 
I was also wondering could i get help to understand these two questions? ive tried to understand the answers for these 2 questions but i still dont understand it

Is there something about induction in itself that you find difficult to grasp?

[jamonwindeyer]

This guide might be worth a read for induction!

[anotherworld2b]

basically everything honestly 

Is there something about induction in itself that you find difficult to grasp?

[anotherworld2b]

This guide might be worth a read for induction!

Thank you for the guide
I will read and see if i understand the questions now. Are there guides for other proofs such as proof by exhaustion?

I did parts a and b for q13 but im not sure what to do for c and d. I am also unsure how to do part b for q14. Could i get some advice please?

[jamonwindeyer]

Thank you for the guide
I will read and see if i understand the questions now. Are there guides for other proofs such as proof by exhaustion?

Proof by exhaustion just means proving every single possible case! For example, proving every year that the Olympics are held is divisible by four, by going through every year and checking that it is divisible by 4. It's (as you'd expect), fair exhausting

I did parts a and b for q13 but im not sure what to do for c and d. I am also unsure how to do part b for q14. Could i get some advice please?

Sure! Here are some rough guides

For Part C, the total units produced after \(t\) hours is:



Note that the amount of units produced in the final hour is just the number produced after 8 hours, minus the number produced after 7 hours. That is:



For Part D, the production rate is given by the derivative (remember, a derivative is a rate of change!). So, the derivative being:



We just evaluate that for \(t=1,2,3\) as the question requires

Question 14 is similar, we are given a function telling us how much water has leaked after \(t\) minutes:



For Part A, we just substitute the given values for \(t\) into the equation! Remember that the second value is given in hours, so you'll need to convert it into minutes first!

Part B is identical to 13D, we find the derivative as a means for finding the rate of leakage (remember, derivatives are rate of change, if you take one thing from these examples THIS must be it):



Again, substitute the given values for \(t\) to obtain your solutions

[anotherworld2b]

Thank you for you help.
Could i please get help with part e and f?

Proof by exhaustion just means proving every single possible case! For example, proving every year that the Olympics are held is divisible by four, by going through every year and checking that it is divisible by 4. It's (as you'd expect), fair exhausting

Sure! Here are some rough guides

For Part C, the total units produced after \(t\) hours is:



Note that the amount of units produced in the final hour is just the number produced after 8 hours, minus the number produced after 7 hours. That is:



For Part D, the production rate is given by the derivative (remember, a derivative is a rate of change!). So, the derivative being:



We just evaluate that for \(t=1,2,3\) as the question requires

Question 14 is similar, we are given a function telling us how much water has leaked after \(t\) minutes:



For Part A, we just substitute the given values for \(t\) into the equation! Remember that the second value is given in hours, so you'll need to convert it into minutes first!

Part B is identical to 13D, we find the derivative as a means for finding the rate of leakage (remember, derivatives are rate of change, if you take one thing from these examples THIS must be it):



Again, substitute the given values for \(t\) to obtain your solutions