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(Solved): (a) Find the approximations T_(10) and M_(10) for \int_1^2 31e^((1)/(x))dx. (Round your answers to s ...



(a) Find the approximations

T_(10)

and

M_(10)

for

\int_1^2 31e^((1)/(x))dx

. (Round your answers to six decimal places.)

T_(10)= M_(10)=

q,

(b) Estimate the errors in the approximations of part (a) using the smallest possible value for

K

according to the theorem about error bounds for trapezoidal and midpoint rules. (Round your answers to six decimal places.)

|E_(T)|<= |E_(M)|<=

q,

(c) Using the values of

K

from part (b), how large do we have to choose

n

so that the approximations

T_(n)

and

M_(n)

to the integral in part (a) are accurate to within 0.0001 ? For

T_(n^(')),n=

x

. For

M_(n^('))n=

x

. Do the following with the given information.

\int_0^1 27cos(x^(2))dx

(a) Find the approximations

T_(8)

and

M_(8)

for the given integral. (Round your answer to six decimal places.)

T_(8)=,✓ Great work. M_(8)=,✓ Great jobt

(b) Estimate the errors in the approximations

T_(8)

and

M_(8)

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.)

|E_(T)|<= |E_(M)|<=

q,

q,

(c) How large do we have to choose

n

so that the approximations

T_(n)

and

M_(n)

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>=

q,

\times

for

T_(n)

n>=

\times

for

M_(n)


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