Estimation and Limits -- A More Precise Look -- Answers

1. These estimates are very similar to the ones made for the tangent at the point (0.5, 0.25) and are based on the formula

   

   Notice from the graph of the function f(x) = x^2 it is evident that when x is to the right of 0.8 this gives us an overestimate for the slope of the tangent and when x is to the left of 0.8 this gives us an underestimate. The exact answer is 1.6 and your answer should be within 0.01 of this exact answer.

2. These estimates are obtained in a very similar way based on the formula

   

   where f(x) = x (1 - x). In this case it is evident from the graph of the function f(x) that when x is to the right of 0.8 then we get an underestimate and when x is to the left of 0.8 then we get an overestimate. The exact value for the slope of the tangent is -0.6 and your answer should be within 0.01 of this exact value.

3. In this situation estimates are based on the formula

   

   and the exact answer is

   

   where the last line comes from noticing that when x is very close to a then x + a is very close to 2a.

4. Estimates can be found based on the following picture.

   

   leading to a sequence of estimates

   

   where f(x) = cos x. The difference between the estimates and the exact answer is colored red in the figure above. It is evident from this figure that the estimates are underestimates.

   In addition by sliding the red areas over so that they stack up on top of each other we can see that all the underestimates can be contained in a column whose width is h and whose height is 1. Thus, the total area of the underestimates is at most h. Since

   h = pi / (2n)

   if n > 157 the error will be less than 0.01.

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