The following exercises deal with Fresnel integrals. 248. [T] Use Newton’s approximation of the binomial 1 − x 2 to approximate π as follows. The circle centered at ( 1 2 , 0 ) with radius 1 2 has upper semicircle y = x 1 − x . The sector of this circle bounded by the x -axis between x = 0 and x = 1 2 and by the line joining ( 1 4 , 3 4 ) , corresponds to 1 6 of the circle and has area π 24 . This sector is the union of a right triangle with height 3 4 and base 1 4 and the region below the graph between x = 0 and x = 1 4 . To find the area of this region you can write y = x 1 − x = x × (binomial expansion of 1 − x ) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate π .
The following exercises deal with Fresnel integrals. 248. [T] Use Newton’s approximation of the binomial 1 − x 2 to approximate π as follows. The circle centered at ( 1 2 , 0 ) with radius 1 2 has upper semicircle y = x 1 − x . The sector of this circle bounded by the x -axis between x = 0 and x = 1 2 and by the line joining ( 1 4 , 3 4 ) , corresponds to 1 6 of the circle and has area π 24 . This sector is the union of a right triangle with height 3 4 and base 1 4 and the region below the graph between x = 0 and x = 1 4 . To find the area of this region you can write y = x 1 − x = x × (binomial expansion of 1 − x ) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate π .
The following exercises deal with Fresnel integrals.
248. [T] Use Newton’s approximation of the binomial
1
−
x
2
to approximate
π
as follows. The circle centered at
(
1
2
,
0
)
with radius
1
2
has upper semicircle
y
=
x
1
−
x
. The sector of this circle bounded by the x-axis between x = 0 and
x
=
1
2
and by the line joining
(
1
4
,
3
4
)
, corresponds to
1
6
of the circle and has area
π
24
. This sector is the union of a right triangle with height
3
4
and base
1
4
and the region below the graph between x = 0 and
x
=
1
4
. To find the area of this region you can write
y
=
x
1
−
x
=
x
×
(binomial expansion of
1
−
x
) and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate
π
.
QUESTION 18 - 1 POINT
Jessie is playing a dice game and bets $9 on her first roll. If a 10, 7, or 4 is rolled, she wins $9. This happens with a probability of . If an 8 or 2 is rolled, she loses her $9. This has a probability of J. If any other number is rolled, she does not win or lose, and the game continues. Find the expected value for Jessie on her first roll.
Round to the nearest cent if necessary. Do not round until the final calculation.
Provide your answer below:
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Fundamental Theorem of Calculus 1 | Geometric Idea + Chain Rule Example; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=hAfpl8jLFOs;License: Standard YouTube License, CC-BY