Evaluate the Riemann sum for f ( x ) = 0.9 x − 1.5 sin (...
Evaluate the Riemann sum for f ( x ) = 0.9 x − 1.5 sin ( 2 x )
over the interval [ 0 , 2.5 ] using five subintervals, taking the
sample points to be right endpoints. step by step and answer
please. can u also show me how to enter in ti 83.
Solutions
Expert Solution
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(a) Find the Riemann sum for
f(x) = 4
sin(x), 0 ≤ x ≤
3π/2,
with six terms, taking the sample points to be right endpoints.
(Round your answers to six decimal places.)
R6 =
(b) Repeat part (a) with midpoints as the sample points.
M6 =
If m ≤ f(x) ≤ M for
a ≤ x ≤ b, where m is the
absolute minimum and M is the absolute maximum of
f on the interval [a, b], then
m(b...
Use Cauchy-Riemann equations to show that the complex function
f(z) = f(x + iy) = z(x + iy) is nowhere differentiable except at
the origin z = 0.6 points) 2. Use Cauchy's theorem to evaluate the
complex integral ekz -dz, k E R. Use this result to prove the
identity 0"ck cos θ sin(k sin θ)de = 0
Evaluate the integral of f(x) below between x=1.5 and x=5.3
using Gauss-Legendre formulas for 2, 3, and 4 points.
Compare with analytical integration, calculate the % error of
the numerical method.
f(x)=4+8x−21x2 +16x3 −5x4
+7x5
17. Integrate the function sin(x) from 0 to π using Riemann and
Simpson integration methods with N subintervals, where N increases
by factors of 2 from 2 to 256 (i.e. N = 2, N = 4, etc.). Plot the
relative error ∆I/I = [| estimated value - true value |/(true
value)], assuming the
highest value of N gives the ‘true’ value. Which method
converges the fastest?
consider f(x) = ln(x)
a) Approximate f(0.9) and f(1.1)
b) Use Taylor remainder to find an error formula for Taylor
polynomial.
Give error bounds for each of the two approximations in (a).
Which of the two approximations in part (a) is closer to correct
value?
c) Compare an actual error in each case with error bound in part
(b).
consider f(x) = ln(x)
a) Approximate f(0.9) and f(1.1)
b) Use Taylor remainder to find an error formula for Taylor
polynomial.
Give error bounds for each of the two approximations in (a).
Which of the two approximations in part (a) is closer to correct
value?
c) Compare an actual error in each case with error bound in part
(b).
Let f ( x , y ) = x^ 2 + y ^3 + sin ( x ^2 + y ^3 ). Determine
the line integral of f ( x , y ) with respect to arc length over
the unit circle centered at the origin (0, 0).