1. What is the difference between the Riemann integral and the
Darboux Integral?
2. Give the same characteristics between Riemann integrals and
Darboux integrals !.
Use this theorem to find the inverse of the given matrix or show
that no inverse exists. (If an answer does not exist, enter DNE in
any cell.)
1
2
5
1
−1
0
2
1
2
1
−5
0
1
1
2
1
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
2. Let f(x) ≥ 0 on [1, 2] and suppose that f is integrable on
[1, 2] with R 2 1 f(x)dx = 2 3 . Prove that f(x 2 ) is integrable
on [1, √ 2] and √ 2 6 ≤ Z √ 2 1 f(x 2 )dx ≤ 1 3 .
Explain the difference between the actual definition of a
Riemann Integral of function f on the interval [a,b] and the
conclusion of the FTOC Part 2.(Fundamental Theorem of Calculus Part
2)
Explain the difference between the actual definition of a
Riemann Integral of function f on the interval [a,b] and the
conclusion of the FTOC Part 2.(Fundamental Theorem of Calculus Part
2)