Question

#5. Evaluate ∫e^a(theta) sin(b(theta)) d(theta) where a,b ∈ℝ

Answer #1

Suppose theta is an acute angle in a right triangle. Given
tan(\theta )=(3)/(5), evaluate:1-sin^(2)(\theta ).

use residues to evaluate the definite integral
integral (0 to 2 pi) ( d theta/ ( 5 +4 sin theta))

Given random variable Y,
E(Y) = theta
Var(Y) = (theta^2)/50
theta hat = bY where b< 1
MSE(theta hat) = (11 * theta^2)/512
Find b

Evaluate by Green’s theorem ∮(cos?sin? − ??)?? + sin?cos???
where ? is the circle ?^2 + ?^2 = 1

Evaluate the triple integral.
2 sin (2xy2z3) dV, where
B
B =
(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 2, 0 ≤ z ≤ 1

(a) Is the vector field F = <e^(−x) cos y, e^(−x) sin y>
conservative?
(b) If so, find the associated potential function φ.
(c) Evaluate Integral C F*dr, where C is the straight line path
from (0, 0) to (2π, 2π).
(d) Write the expression for the line integral as a single
integral without using the fundamental theorem of calculus.

Let B = { f: ℝ → ℝ
| f is continuous } be the ring of all continuous functions from
the real numbers to the real numbers. Let a be any real number and
define the following function:
Φa:B→R
f(x)↦f(a)
It is called the evaluation homomorphism.
(a) Prove that the evaluation homomorphism is a ring
homomorphism
(b) Describe the image of the evaluation homomorphism.
(c) Describe the kernel of the evaluation homomorphism.
(d) What does the First Isomorphism Theorem for...

Use integration by parts to evaluate the integral: ∫ e^8r sin (
− 3r )dr

differentiate.
a. e^xtan(x)
b. sin(1/sqrtx)
c. ln(e^x/sqrt(x^2)+3)
d. subscriptx tan(x)
e. f(secx) where f'(x)= x/ln(x)

Perform Monte-Carlo estimation where theta = E(e^U) where U is a
continuous uniform random variable between zero and one a. Estimate
!using 1000 data points and obtain a 95% confidence interval for
this case. b. Perform part a) 100 times. Check how often the true
!actually does fall within the 100 resulting confidence intervals.
Please solve in R.

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