3.15 Use the previous three exercises to estimate Ꙥ. (i) Show that 2 < Ꙥ < 4. (ii) Show that 2.8284427125 < Ꙥ < 3.828427
Posted: Mon May 02, 2022 4:09 pm
3.15 Use the previous three exercises to estimate Ꙥ. (i) Show
that 2 < Ꙥ < 4. (ii) Show that 2.8284427125 < Ꙥ <
3.828427125. (Hint: Use Exercise 3.12 with the estimate Ꙥ < 4 of
part (i) in the upper bound.) (iii) Show that s3 = 0.765366864,
area (P3) = 3.061147456, and 3.061147456 < Ꙥ < 3.5611467456.
(Hint: Use Exercise 3.12 with the estimate Ꙥ < 4 from part (i).)
(iv) Show that s2 = 0.390180643, area (P4) = 3.121445144, and
3.121445144 < Ꙥ < 3.371445144. (Hint: Use Exercise 3.12 with
the estimate Ꙥ < 4 from part (i).) (v) Repeat this procedure
until you can estimate Ꙥ well enough to see that its first digits
are 3.14 (Hint: it suffices to show there are inequalities 3.140
< Ꙥ < 3.149.)
that 2 < Ꙥ < 4. (ii) Show that 2.8284427125 < Ꙥ <
3.828427125. (Hint: Use Exercise 3.12 with the estimate Ꙥ < 4 of
part (i) in the upper bound.) (iii) Show that s3 = 0.765366864,
area (P3) = 3.061147456, and 3.061147456 < Ꙥ < 3.5611467456.
(Hint: Use Exercise 3.12 with the estimate Ꙥ < 4 from part (i).)
(iv) Show that s2 = 0.390180643, area (P4) = 3.121445144, and
3.121445144 < Ꙥ < 3.371445144. (Hint: Use Exercise 3.12 with
the estimate Ꙥ < 4 from part (i).) (v) Repeat this procedure
until you can estimate Ꙥ well enough to see that its first digits
are 3.14 (Hint: it suffices to show there are inequalities 3.140
< Ꙥ < 3.149.)