Solution to Exercise 8.4-2

The problem was to approximate

    __
   Ö80

Step 1: Find a number nearby whose square root you know. That would be 81, which is the square or 9.

Step 2: Find the derivative of the square root function.

             _
   f(x)  =  Öx

              1
   f'(x)  =     
             2Öx

Step 3: Find the value of the derivative at the x where you know the square root. That would be at x = 81.

                1       1
   f'(81)  =        =    
              2Ö81     18

Step 4: Find the difference between the number you know the square root of and the number you want to take the square root of. This will be your h. In all the problems we have done so far we subtract the former from the latter to get that difference. In this case we do the same even though the number we know the square root of is greater than the number we want to take the square root of.

   h  =  80 - 81  =  -1

Notice that this time h is negative. This is what the hint referred to.

Step 5: Apply the approximation formula.

    __                         1
   Ö80  »  9 + hf'(9)  =  9 -     =  8.94444444
                              18

The square of this approximation is 80.003086. The actual square root of 80 is 8.9442719.

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