Let f ( x ) = cos ( x 2 ) . (a) Use a CAS to approximate the maximum value of | f " ( x ) | on the interval [0, 1]. (b) How large must n be in the midpoint approximation of ∫ 0 1 f ( x ) d x to ensure that the absolute error is less than 5 × 10 − 4 ? Compare your result with that obtained in Example 9. (c) Estimate the integral using the midpoint approximation with the value of n obtained in part (b).
Let f ( x ) = cos ( x 2 ) . (a) Use a CAS to approximate the maximum value of | f " ( x ) | on the interval [0, 1]. (b) How large must n be in the midpoint approximation of ∫ 0 1 f ( x ) d x to ensure that the absolute error is less than 5 × 10 − 4 ? Compare your result with that obtained in Example 9. (c) Estimate the integral using the midpoint approximation with the value of n obtained in part (b).
(a) Use a CAS to approximate the maximum value of
|
f
"
(
x
)
|
on the interval [0, 1].
(b) How large must
n
be in the midpoint approximation of
∫
0
1
f
(
x
)
d
x
to ensure that the absolute error is less than
5
×
10
−
4
?
Compare your result with that obtained in Example 9.
(c) Estimate the integral using the midpoint approximation with the value of
n
obtained in part (b).
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Let F(x) =
sin(4t") dt.
Find the MacLaurin polynomial of degree 7 for F(x).
0.8
sin (Az") da.
Use this polynomial to estimate the value of
Let f(x) be a differentiable function such that
f (0) = 1, f(1) = 4, f(2) = 3, f(3) = 5, f(4) = -7.
-7.
Use the substitution rule for definite integrals to calculate each of the follow-
ing:
3
(a) / VF(#)f'(x)dx.
r1
(b) , F(")
f'(x)
d.x.
f (x)
/
(c)
xf'(x²)dx.
1
pln(3)
(d) /e* f'(e")dx.
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