This is a question I've had a lot of trouble with. I HAVE solved it, however, with a lot of trouble and with an extremely ugly calculation. So I want to ask you guys (who are probably more 'mathematically-minded' so to say) how you would solve this. Keep in mind that you shouldn't use too advanced stuff, no differential equations or similair things learned in college:
Given are the functions $f_p(x) = \dfrac{9\sqrt{x^2+p}}{x^2+2}$. The line $k$ with a slope of 2,5 touches $f_p$ in $A$ with $x_A = -1$. Get the function of k algebraically.
- I might have used wrong terminology, because English is not my native language, I will hopefully clear up doubts on what this problem is by showing what I did.
First off, I got $[f_p(x)]'$. This was EXTREMELY troublesome, and is the main reason why I found this problem challenging, because of all the steps. Can you guys show me the easiest and especially quickest way to get this derivative?
After that, I filled in $-1$ in the derivative and made the derivative equal to $2\dfrac{1}{2}$, this was also troublesome for me, I kept getting wrong answers for a while, again: Can you guys show me the easiest and especially quickest way to solve this?
After you get p it is pretty straightforward. I know this might sound like a weird question, but it basically boils down to: I need quicker and easier ways to do this. I don't want to make careless mistakes, but because the length of these types of question, it ALWAYS happens. Any tips or tricks regarding this topic in general would be much appreciated too.
Update: A bounty will go to the person with the most clear and concise way of solving this question!
Best Answer
By "touches" I assume you mean that the line is tangent to the graph of $f_p$. You can try implicit differentiation. Start with $$ y = \frac{9\sqrt{x^2 + p}}{x^2 + 2}. $$ Multiply by $x^2 + 2$ to get $$ y(x^2 + 2) = 9\sqrt{x^2 + p}. $$ Squaring, you get $$ y^2 (x^2 + 2)^2 = 81 (x^2 + p). $$ Differentiate both sides implicitly by $x$: $$ 2yy'(x^2 + 2)^2 + y^2 2 (x^2 + 2) 2x = 162x. $$ Now, plug in all the data ($x = -1$, $y'(-1) = 2.5$) to get a quadratic equation for $y(-1) = y$: $$ -12y^2 + 45y = -162. $$ Solutions are $y = 6$ and $y = \frac{-9}{4}$, but notice your function is always positive, so $y(-1) = 6$ and the line is $2.5x + 8.5$.