Expert: Paul Klarreich - 1/23/2007

Question
I have found a line tangent to the curve y = 1/x. (my line equation is y-(squareroot of two, divided by two)=-.5(x-squareroot of two)).  

I am supposed to use this line to show that ln 2 is greater than 0.66.

Find the line equation was easy. I am just unsure how to proceed....  

Answer
Questioner:   Aaron
Category:  Calculus
 
Question:  I have found a line tangent to the curve y = 1/x.

(my line equation is y-(sqrt(2)/2)=-.5(x- sqrt(2)).  

I am supposed to use this line to show that ln 2 is greater than 0.66.

Finding the line equation was easy. I am just unsure how to proceed....  
............................
Hi, Aaron,

[I will write  S for sqrt(2) from now on.]

Here's your line:

y - 1/S = -1/2(x - S)

We will construct a trapezoid and compute its area.  Recall that the area of a trapezoid is  h/2(b1 + b2)

In this case, our 'height' is lying flat on the x-axis and the two 'bases' are vertical.

Base on the left:  This is at  x = 1 and we substitute that to find y:

y - 1/S = -1/2(1 - S)

y - 1/S = -1/2 +  S/2
y - 1/S = -1/2 +  1/S  << S = sqrt(2), so S/2 = 1/S

y = -1/2 + 2/S

y = -1/2 + S
   2S - 1
y = ------
      2

The base on the right has x = 2.  {Why? because we are going to compare this with ln 2.  Explanation comes soon.}

y - 1/S = -1/2(2 - S)

y - 1/S = -1 + 1/S

y = -1 + 2/S

y = -1 + S

y = S - 1

OK, now, for our area, we have
    2S - 1
b1 = ------
      2

b2 = S - 1

h = 1
       1  2S - 1
Area = ---[------- + S - 1]
       2     2

       1  2S - 1 + 2S - 2
Area = ---[---------------]
       2         2

      4S  - 3
Area = -------
         4

My calculator gives about  0.664 for that value.

So what does this have to do with  ln 2, you ask?  Draw and shade the following area:

Left:  x = 1
Right: x = 2
Bottom:  y = 0, the x-axis.
Top:  y = 1/x

You will see this is the 'area under the curve' for  y = 1/x from x = 1 to x = 2, and is equal to

{2  1
|  --- dx = ln 2 - ln 1 = ln 2
}1  x

And you will see that this 'area....' totally contains that trapezoid, therefore must be bigger.