Math Answers

You are given the following integral to evaluate: ∫(𝟑𝒙 − 𝟐)𝒅𝒙.

Without performing any calculations, which numerical method of the following gives the

best accuracy: (1) Trapezoidal rule, (2) Simpson’s 1/3 rule, and (3) Simpson’s 3/8 rule?

Justify your answer

Evaluate the following integral: 𝑰 = ∫ 𝒙𝐬𝐢𝐧(𝒙)𝒅𝒙 𝝅/𝟒

𝟎

(1) analytically, (2) using single application of the trapezoidal rule, (3) using composite

trapezoidal rule with n = 2 and 4. For the numerical estimates (2) and (3), determine the true

percent relative error based on (1).

Determine if the following sets are linearly dependent, or independent.

(i) {1,sin(x),cos(x)}

(ii) {sin²(x),cos(2x),cos²(x)}

Given two bases

B={1−x,2+x,3−x+x2}

and

C={1,2+x,1+x−x2}

of P2, the vector space of polynomials of degree ≤2,

(i) find p(x)∈P2 whose coordinates with respect to B is [p(x)]B=⎡⎣⎢1 −1 3⎤⎦⎥,

(ii) find the transition (change of coordinates) matrix CMB∈R3×3 from B to C,

(iii) calculate the coordinates [p(x)]C∈R3 of p(x)∈P2 with respect to C.

Let W⊆R

5

W⊆R5 be the set of solutions of the linear homogeneous system given by

x1−x2+4x3−x4−x5=0

−x1+x2−x3+2x4+x5=0

x3−3x4+x5=0

Accordingly,

(i) show that W⊆R5 is a subspace,

(ii) find a basis for W

(iii) dim(W)=?

Solve the following ordinary differential equation over the interval from x = 0 to 1 using a

step size of 0.25 where y(0) = 1.

𝒅𝒚

𝒅𝒙 = (𝟏 + 𝟐𝒙)𝒚

𝟐

(1) Analytically.

(2) Using Euler’s method.

(3) Using Heun’s method without iteration.

(4) Using the fourth-order RK method.

If x is positive show that. x>log(1+x)>x-(x^2/2)

Using Taylor's theorem prove that. x-(x^3/6)<sinx<x-(x^3/6)+(x^5/120)

Let be a real number. Let x~2.5 be an approximate value of with absolute error

at most 0.01. The function ( )

is evaluated at instead of . Estimate the absolute error

For the population, (2,5,3,4,2). Construct the sampling distribution of sample mean x' of size "3" without replacement.