Find the degree of percision of the following expression integral from -1 to 1 f
ID: 2963103 • Letter: F
Question
Find the degree of percision of the following expression integral from -1 to 1 f(x) d(x):a) f(1) + f(-1)
b) 2/3 [ f(-1) + f(0) + f(1) ]
c) f(-1/sqrt(3)) + f(1/sqrt(3))
Find the degree of percision of the following expression integral from -1 to 1 f(x) d(x):
a) f(1) + f(-1)
b) 2/3 [ f(-1) + f(0) + f(1) ]
c) f(-1/sqrt(3)) + f(1/sqrt(3))
Find the degree of percision of the following expression integral from -1 to 1 f(x) d(x):
a) f(1) + f(-1)
b) 2/3 [ f(-1) + f(0) + f(1) ]
c) f(-1/sqrt(3)) + f(1/sqrt(3))
Find the degree of percision of the following expression integral from -1 to 1 f(x) d(x):
a) f(1) + f(-1)
b) 2/3 [ f(-1) + f(0) + f(1) ]
c) f(-1/sqrt(3)) + f(1/sqrt(3))
Explanation / Answer
integral from -1 to 1 f(x) d(x) approximation has precision n if the approximation is exact for f(x) = x^a 0<=a<=n
or
degree of precision = n if integral from -1 to 1 f(x) d(x) = the approximation for f(x)=1,x,....x^n
a.) f(x)=1
integral from -1 to 1 f(x) d(x)=2
f(1) + f(-1)=2
f(x)=x
integral from -1 to 1 f(x) d(x)=0
f(1) + f(-1)=0
f(x)=x^2
integral from -1 to 1 f(x) d(x)=2/3
f(1) + f(-1)=2
failed at x^2
So degree of precision = 1
b.)
f(x)=1
integral from -1 to 1 f(x) d(x)=2
2/3 [ f(-1) + f(0) + f(1) ] =2
f(x)=x
integral from -1 to 1 f(x) d(x)=0
2/3 [ f(-1) + f(0) + f(1) ] =0
f(x)=x^2
integral from -1 to 1 f(x) d(x)=2/3
2/3 [ f(-1) + f(0) + f(1) ] =4/3
failed at x^2
So degree of precision = 1
c.)
f(x)=1
integral from -1 to 1 f(x) d(x)=2
f(-1/sqrt(3)) + f(1/sqrt(3)) =2
f(x)=x
integral from -1 to 1 f(x) d(x)=0
f(-1/sqrt(3)) + f(1/sqrt(3)) =0
f(x)=x^2
integral from -1 to 1 f(x) d(x)=2/3
f(-1/sqrt(3)) + f(1/sqrt(3)) =2/3
f(x)=x^3
integral from -1 to 1 f(x) d(x)=0
f(-1/sqrt(3)) + f(1/sqrt(3)) =0
f(x)=x^4
integral from -1 to 1 f(x) d(x)=2/5
f(-1/sqrt(3)) + f(1/sqrt(3)) =2/9
failed at x^4
So degree of precision = 3
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