Math Is Fun Forum

Hello again mathisfun and welcome to fun with circles:

Question 19 in my textbook has me a little bit stuck. It goes like this:

I've done the first part of the question, but on the second part, I get an answer that disagrees with the book's and I'm not sure who's right.

This is what I've done:

The gradient of the radius to A is:

Therefore, the equation of the tangent to C at A is:

Therefore, at the point of intersection:

I chose to eliminate y:

Sub x into (2)

But, my textbook has the opposite sign:

So: Who's right? If I'm right - well, that's nice If not: what've I done wrong? Get confused with signs? I wouldn't be too surprised, but I've checked several times!

Another tricky little question :S

I wasn't really sure where I was going with this so I decided to just dive right in and differentiate:

As for the second part of the first proof, I'm pretty stumped. We know that k > 0, so for k < 1, k - 1 will be negative and an element of the set of real numbers between 0 and 1 (exclusive). In other words:

So you will have a fraction and the exponent of x will not be an integer. Whereas, for k > 1 you will have a positive exponent, which may be an integer and may, in fact, be any positive real number. That's the closest thing to a brainwave I've had really - do you have any ideas?

Another quick question.

The question I've just completed reads:

I completed the proof and then got a little bit stuck on the second half of the question. What I was minded to do was square root both sides in order to obtain an expression in terms of plain old V and then use the chain rule. This turned out to be a bit problematic and, in the end, I had to resort to what I thought was a bit of a cheat, but it was the only way I could get the right answer. Instead, I did:

Solved

for x:

Found that

And solved for V: V = 36.

Question: Is this legitimate? More to the point, is it generally legitimate, cause I've got things like implicit differentiation in the back of my mind and something feels wrong about differentiating V² but I couldn't think how else to do it!

Just a quick one to ask what is the best way to solve the following equation:

My approach was to say:

And then square both sides:

However, -3/4 obviously isn't a solution to:

If you take the principal value of the square root function and it is also not given as a solution by wolfram alpha.

So I was wondering if I went about this question the right way, or should I have done something different?