Quantitative Methods Answers

approximate the real root to four decimal places of x3+5x-3=0 (newton raphson method)

Given that y'=x+y with y(0) = 1 find y(0.1).h=0.05 by Euler's modified method.

Use the composite trapezoidal rule to find approximations to 𝐼 = ∫ 𝐬𝐢𝐧𝒙 𝒅𝒙 𝝅

𝟎

with n = 1, 2, 4, and 8. Then perform Romberg extrapolation on the results.

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).

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.

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

.1.1 Use Euler’s method with step size h = 0.1 to approximate the solution to the initial valueproblem: y'=x y, y(1)=4, atthepointsx=1.1,1.2and1.3,correcttofivedecimal

places.

2.1.2 If the analytical solution to the initial value problem in (2.1.1) is

(4) y = 1 (x2 + 7), determine

4

the %Error in the numerical method where x =1.3

2.2 A chicken cools down from 100 °C to 60 °C within 10