Having alot of trouble with this problem, please help!
Integral limits from (0 to 1) 19cos (x^2) dx
(a) Find the approximations T8 and M8 for the given integral. (Round your answer to six decimal places.)
T8=
M8=
(b) Estimate the errors in the approximations T8 and M8 in part (a). (Use the fact that the range of the sine and cosine functions is bounded by ±1 to estimate the maximum error. Round your answer to seven decimal places.)
|ET| ≤
|EM| ≤
(c) How large do we have to choose n so that the approximations Tn and Mn to the integral are accurate to within 0.0001? (Use the fact that the range of the sine and cosine functions is bounded by ±1 to estimate the maximum error.)
n ≥ ? for Tn
n ≥ ? for Mn
Thanks in advance for the help!
Integral limits from (0 to 1) 19cos (x^2) dx
(a) Find the approximations T8 and M8 for the given integral. (Round your answer to six decimal places.)
T8=
M8=
(b) Estimate the errors in the approximations T8 and M8 in part (a). (Use the fact that the range of the sine and cosine functions is bounded by ±1 to estimate the maximum error. Round your answer to seven decimal places.)
|ET| ≤
|EM| ≤
(c) How large do we have to choose n so that the approximations Tn and Mn to the integral are accurate to within 0.0001? (Use the fact that the range of the sine and cosine functions is bounded by ±1 to estimate the maximum error.)
n ≥ ? for Tn
n ≥ ? for Mn
Thanks in advance for the help!
-
f(x) = 19 cos(x²)
∫₀¹ 19 cos(x²) dx
(1-0)/8 = 1/8 = 0.125
------------------------------
(a)
T8 = 1/8 * 1/2 (f(0) + 2f(0.125) + 2f(0.25) + 2f(0.375) + 2f(0.5) + 2f(0.625) + 2f(0.75) + 2f(0.875) + f(1))
T8 = 1/16 (19cos(0²) + 2*19cos(0.125²) + 2*19cos(0.25²) + 2*19cos(0.375²) + 2*19cos(0.5²) + 2*19cos(0.625²) + 2*19cos(0.75²) + 2*19cos(0.875²) + 19cos(1²))
T8 = 17.144324
M8 = 1/8 (f((0+0.125)/2) + f((0.125+0.25)/2) + f((0.25+0.375)/2) + f((0.375+0.5)/2) + f((0.5+0.625)/2) + f((0.625+0.75)/2) + f((0.75+0.875)/2) + f((0.875+1)/2))
M8 = 1/8 (19cos(0.0625²) + 19cos(0.1875²) + 19cos(0.3125²) + 19cos(0.4375²) + 19cos(0.5625²) + 19cos(0.6875²) + 19cos(0.8125²) + 19cos(0.9375²))
M8 = 17.206779
------------------------------
(b)
f(x) = 19 cos(x²)
f'(x) = -38x sin(x²)
f''(x) = -38 sin(x²) - 38x * 2x cos(x²) = -38 (sin(x²) + 2x² cos(x²))
On interval 0 ≤ x ≤ 1
|f''(x)| ≤ 38 (1 + 2(1)) = 114
E(T) = -(b-a)³/(12N²) * f''(ξ), for some ξ between a and b
|E(T8)| ≤ 1/(12*8²) * 114
|E(T8)| ≤ 0.1484375
E(M) = (b-a)³/(24N²) * f''(ξ), for some ξ between a and b
|E(T8)| ≤ 1/(24*8²) * 114
|E(T8)| ≤ 0.0742188
------------------------------
(c)
From (b)
|E(Tn)| ≤ 1/(12n²) * 114 ≤ 0.0001
9.5/n² ≤ 0.0001
9.5/0.0001 ≤ n²
n² ≥ 95000
n ≥ 308.22
n ≥ 309
|E(Mn)| ≤ 1/(24n²) * 114 ≤ 0.0001
4.75/n² ≤ 0.0001
4.75/0.0001 ≤ n²
n² ≥ 47500
n ≥ 217.94
n ≥ 218
Mαthmφm
∫₀¹ 19 cos(x²) dx
(1-0)/8 = 1/8 = 0.125
------------------------------
(a)
T8 = 1/8 * 1/2 (f(0) + 2f(0.125) + 2f(0.25) + 2f(0.375) + 2f(0.5) + 2f(0.625) + 2f(0.75) + 2f(0.875) + f(1))
T8 = 1/16 (19cos(0²) + 2*19cos(0.125²) + 2*19cos(0.25²) + 2*19cos(0.375²) + 2*19cos(0.5²) + 2*19cos(0.625²) + 2*19cos(0.75²) + 2*19cos(0.875²) + 19cos(1²))
T8 = 17.144324
M8 = 1/8 (f((0+0.125)/2) + f((0.125+0.25)/2) + f((0.25+0.375)/2) + f((0.375+0.5)/2) + f((0.5+0.625)/2) + f((0.625+0.75)/2) + f((0.75+0.875)/2) + f((0.875+1)/2))
M8 = 1/8 (19cos(0.0625²) + 19cos(0.1875²) + 19cos(0.3125²) + 19cos(0.4375²) + 19cos(0.5625²) + 19cos(0.6875²) + 19cos(0.8125²) + 19cos(0.9375²))
M8 = 17.206779
------------------------------
(b)
f(x) = 19 cos(x²)
f'(x) = -38x sin(x²)
f''(x) = -38 sin(x²) - 38x * 2x cos(x²) = -38 (sin(x²) + 2x² cos(x²))
On interval 0 ≤ x ≤ 1
|f''(x)| ≤ 38 (1 + 2(1)) = 114
E(T) = -(b-a)³/(12N²) * f''(ξ), for some ξ between a and b
|E(T8)| ≤ 1/(12*8²) * 114
|E(T8)| ≤ 0.1484375
E(M) = (b-a)³/(24N²) * f''(ξ), for some ξ between a and b
|E(T8)| ≤ 1/(24*8²) * 114
|E(T8)| ≤ 0.0742188
------------------------------
(c)
From (b)
|E(Tn)| ≤ 1/(12n²) * 114 ≤ 0.0001
9.5/n² ≤ 0.0001
9.5/0.0001 ≤ n²
n² ≥ 95000
n ≥ 308.22
n ≥ 309
|E(Mn)| ≤ 1/(24n²) * 114 ≤ 0.0001
4.75/n² ≤ 0.0001
4.75/0.0001 ≤ n²
n² ≥ 47500
n ≥ 217.94
n ≥ 218
Mαthmφm