Examine the function f(x)=x+(1/x)
a) Find the points on the curve at which the tangent lines pass through the point (1,1)
b) Prove that no tangent line to the curve passes through the origin
c) Prove that there is one tangent line passing through the point (A,A), A (does not equal)0, and it occurs at the point with x-coordinate (A/2).

Question

anonymous · Answer

$$f(x)=x+\frac{1}{x}$$
$$f'(x)=1-\frac{1}{x^2}$$

anonymous · Answer

part B set 
$$1-\frac{1}{x^2}=0$$ and try to solve for x

anonymous · Answer

oh yey its you again

anonymous · Answer

I assume its 1

anonymous · Answer

hold on maybe i made a mistake

anonymous · Answer

yeah these questions are for the lose

anonymous · Answer

ok first one

anonymous · Answer

point on the curve is $$(x,x+\frac{1}{x})$$ and if it goes through (1,1) the slope is 
$$\frac{x+\frac{1}{x}-1}{x-1}$$ and you know this has to equal the derivative so set 
$$\frac{x+\frac{1}{x}-1}{x-1}=1-\frac{1}{x^2}$$and solve

anonymous · Answer

actually easier than it looks get $$x=\frac{1}{2}$$

anonymous · Answer

I just did it. :) and got it

gecko · Answer

Aahh can't believe I found this! Thanks so much in advance! The answer to (a) is supposed to be (1/2, 5/2), but I can't figure out how to get there.

anonymous · Answer

UF?

gecko · Answer

Yep lol (:

anonymous · Answer

you got 1/2 yes?

anonymous · Answer

yep.

anonymous · Answer

now repeat w/ (0,0)  insted of (1,1)  and see the trouble

anonymous · Answer

is it because you can't do (0, 0+1/0)?

anonymous · Answer

write$$\frac{x+\frac{1}{x}}{x}=1-\frac{1}{x^2}$$ and you will not be able to solve

anonymous · Answer

left hand side is slope at (0,0), right hand side is derivative.  you cannot solve this

anonymous · Answer

ok right.

anonymous · Answer

and for part c do I plug in A where the x's were, so (A+(1/A))/A=1-(1/A^2)

anonymous · Answer

?

anonymous · Answer

no for part c you plug in A in the formula for the slope, just like we plugged in (1,1)

anonymous · Answer

$$\frac{x+\frac{1}{x}-A}{x-A}=1-\frac{1}{x^2}$$ solve for x

anonymous · Answer

ok I did that and got A/2

anonymous · Answer

how do I prove that that's the only tangent line.

anonymous · Answer

and the point is (a/2,1)

anonymous · Answer

you did it when you solved and got a/2 !  you are done

anonymous · Answer

:) now what's the best way to sketch this curve and illustrate those ideas?