Mathematics Question Thread

[Youssk]

Hey can someone please help me with this question, i'm confused as to where the 1/2 outside the log comes from.

Show that 1/x-1 - 1/x+1 = 2/x^2-1. Hence or otherwise show that ³ ₂ ∫1/x^2 -1 dx = 1/2log_e1.5

Thanks

[RuiAce]

Hey can someone please help me with this question, i'm confused as to where the 1/2 outside the log comes from.

Show that 1/x-1 - 1/x+1 = 2/x^2-1. Hence or otherwise show that ³ ₂ ∫1/x^2 -1 dx = 1/2log_e1.5

Thanks

In the future, please use more bracketing (e.g. 1/(x-1) for \(\frac1{x+1}\)

[Hungry4Apples]

Hey can someone please help me with this question, i'm confused as to where the 1/2 outside the log comes from.

Show that 1/x-1 - 1/x+1 = 2/x^2-1. Hence or otherwise show that ³ ₂ ∫1/x^2 -1 dx = 1/2log_e1.5

Thanks

This is an integration by recognition question. You may not have the ability to integrate the function 1/(x^2+1) but you have been given enough information to do it by means you are already aware of. We found earlier that
2/(x^2+1) = 1/(x-1) - 1/(x+1)

so it would follow that if we divide both sides by two, we have 1/(x^2+1) in a form we can integrate. So the half outside comes from there, since we can only integrate 2/(x^2+1) we do so, while also dividing by a half to keep the same value of the function.

[RuiAce]

This is an integration by recognition question.

A remark that this terminology is not used in the HSC.

[kylesara]

Hi could i please have help with this question.

Find values of a and b if y''= ae^3x cos4x+ be^3x sin 4x, given y=e^3x cos4x

Thanks

[jakesilove]

Hi could i please have help with this question.

Find values of a and b if y''= ae^3x cos4x+ be^3x sin 4x, given y=e^3x cos4x

Thanks

Hey! So, the question tells us that




First, we differentiate our f(x). Note we have to use the chain rule, such that




So, here




Therefore



Then, we differentiate again! We apply the product rule, twice here.



Collecting like terms



From which you can extract your value of a and b

[katnisschung]

hi hi is there any other way to sketch an exponential other than table of values

[RuiAce]

hi hi is there any other way to sketch an exponential other than table of values

You should know the shape of the exponential off by heart.

1. Write down the asymptote and put that in, clearly noting if it's approached as \(x\to \infty\) or \(x \to -\infty\)
2. Sketch any intercepts with the x and/or y axes
3. That's all the information you need. Sketch it.

Nobody says that your graph has to be to a perfect scale

[sophiemacpherso]

In all of the maths past papers, multiple choice is what I struggle with the most... Do you have any tips to improve at it? 

[_____]

This is for a longer integration/volume question from a past paper but there's one part with logarithms that I don't understand.
Write the relation y=log2x in terms of the exponential function (base 2 to the pronumeral x).

The answer is:
y=lnx/ln2
yln2=lnx
lnx=yln2
Therefore x=e^(yln2)

What's going on in this last line? Think I understand the rest. Appreciate your help.

[kiwiberry]

This is for a longer integration/volume question from a past paper but there's one part with logarithms that I don't understand.
Write the relation y=log2x in terms of the exponential function (base 2 to the pronumeral x).

The answer is:
y=lnx/ln2
yln2=lnx
lnx=yln2
Therefore x=e^(yln2)

What's going on in this last line? Think I understand the rest. Appreciate your help.

Remember that

In the same way, replacing y with yln2, we get

[_____]

Remember that

In the same way, replacing y with yln2, we get

Ahhhh now I understand. Thanks!

[katnisschung]

hey hey i have 2 questions

1. the tangent to the curve y=e^x at the point p meets the x-axis at an angle of 45 degrees. Find the coordinates of P
(i remember hearing somewhere the tangent to the point of an exponential curve will be 45 degrees at the y-intercept..is this true
for all exponentials like y=2e^4x)

2. find the maximum value of lnx/x (totally lost here...assuming its something to do with e)

[RuiAce]

hey hey i have 2 questions

1. the tangent to the curve y=e^x at the point p meets the x-axis at an angle of 45 degrees. Find the coordinates of P
(i remember hearing somewhere the tangent to the point of an exponential curve will be 45 degrees at the y-intercept..is this true
for all exponentials like y=2e^4x)

2. find the maximum value of lnx/x (totally lost here...assuming its something to do with e)

\text{Assume that }P\text{ is the point }(x,y)\\ y=e^x \implies \frac{dy}{dx} = e^x\\ \text{So the tangent at }P\text{ has gradient }e^x[/tex]


__________________________

[Kle123]

Could i get help with this question. I asked it a while back but didn't get a response