TyErd's questions

[TyErd]

Let the length of the shorter side of the rectangle be x
Let the perimeter be P(x)
Let the area be A(x)

Then the other side length is 3x

So



From the linear approximation formula, the change in perimeter is

So if the change in perimeter is 2%, then





Because

Therefore the change in area is





So the change is 4%

I hope that made some sense

How did you get that last line to equal 4/100

[kamil9876]

use the chain rule to prove , where n is a negative number. I dont get it.

Hmm not sure about the "where n is a negative number" but I would just do it normally like lol thats basically it I guess...

but it says using chain rule

I'm not sure how the chain rule plays into it, but here's how I would do it. Since is a negative number, it's usually convenient to 'bring the negative out'. To do this, set , then is a positive number.

Then , so

Differentiating implicitly,





So we have proved differentiation works for negative exponents. Notice that nowhere in the proof did we actually differentiate a negative exponent, since is positive.

If you want to use the chain rule, you can prove it for n=-1 first (from first principles, quite easy). Then use the chain rule as follows:



And since v is a positive integer, you can do it using and what you already know for the n=-1 case and the n=positive integer case.


Also: /0, you assumed y is differentiable without proof, this way shows it is (a technical point that I'm sure doesnt matter in VCE, heck i never remember this being required back in the days, but you might benefit from it for Analysis).

[the.watchman]

Let the length of the shorter side of the rectangle be x
Let the perimeter be P(x)
Let the area be A(x)

Then the other side length is 3x

So



From the linear approximation formula, the change in perimeter is

So if the change in perimeter is 2%, then





Because

Therefore the change in area is





So the change is 4%

I hope that made some sense

How did you get that last line to equal 4/100

3x^2/25 = 4% x 3x^2, right?

[TyErd]

oh yeah thnx! i get it now

[kenhung123]

Just wondering can (e^{3x}-e^{2x})/e^{2x} be cancelled to (e^2x-e^{x})/e^{x}

[Cthulhu]

Just wondering can (e^{3x}-e^{2x})/e^{2x} be cancelled to (e^2x-e^{x})/e^{x}

It can be Simplified to

[kenhung123]

Thanks

Also when you have like common denominator, but the numerator has negative power do I multiply the numerator with denominator to bring it down like:

[the.watchman]

Well the definition of is
So yes, you do that

[moekamo]

Thanks

Also when you have like common denominator, but the numerator has negative power do I multiply the numerator with denominator to bring it down like:

except the 4 has to stay on the numerator in that second fraction since it is not being raised to a negative power...

[kenhung123]

Got it, thanks

[the.watchman]

Thanks

Also when you have like common denominator, but the numerator has negative power do I multiply the numerator with denominator to bring it down like:

except the 4 has to stay on the numerator in that second fraction since it is not being raised to a negative power...

Oops, good point

[TyErd]

Find two positive numbers who sum is 4 and such that the sum of the cube of the first and the square of the seconds is small as possible. I have no idea where to start.

These minima and maxima problems are hard to understand! I dont really understand how differentiation works in some problems. Any tips ?

[kamil9876]

you want to minimize

to turn it into a one variable problem: using the first equation.

[Yitzi_K]

x+y=4
y=4-x

So you want  x^3 + (4-x)^2 to be as small as possible.

Find the derivative, then solve for 0.

[TyErd]

Okay thanks , Find the point on the parabola that is closest to the point (3,0)