Master Numerical Integration Techniques: Newton-Cotes Formulas

Question 1

State the formula for approximating a definite integral using the trapezoidal rule.

Accepted Answer

(b-a) * (f(a)+f(b))/2

Answer

(b-a) * f((a+b)/2)

Answer

(b-a) * (f(a)-f(b))/2

Answer

(a-b) * (f(a)+f(b))/2

Question 2

Which of the following accurately describes an advantage of Simpson's rule over the trapezoidal rule?

Accepted Answer

Enhanced accuracy due to higher order of precision

Answer

Reduced computational cost

Answer

Simplified application process

Answer

Increased resilience to function irregularities

Question 3

Which formula is used for Simpson's rule, a higher-order method for numerical integration?

Accepted Answer

∫[a,b] f(x) dx ≈ (b-a)/6 * [f(a) + 4f((a+b)/2) + f(b)]

Answer

∫[a,b] f(x) dx ≈ (b-a)/8 * [f(a) + 3f((a+b)/2) + 3f((3a+b)/4) + f(b)]

Answer

∫[a,b] f(x) dx ≈ (b-a)/2 * [f(a) + f(b)]

Answer

∫[a,b] f(x) dx ≈ (b-a)/3 * [f(a) + 2f((a+b)/2) + f(b)]

Question 4

What is the order of convergence of Simpson's rule?

Accepted Answer

4

Answer

2

Answer

3

Answer

1

Question 5

Compared to the Trapezoidal rule, what is an advantage of Simpson's rule?

Accepted Answer

It has a higher order of convergence, leading to improved accuracy.

Answer

It requires fewer function evaluations, reducing computational cost.

Answer

It is simpler to apply, making it easier to use.

Question 6

For the Trapezoidal rule to yield an exact result, what condition must the function f(x) satisfy?

Accepted Answer

f(x) is a linear function.

Answer

f(x) is a trigonometric function.

Answer

f(x) is a cubic function.

Answer

f(x) is a quadratic function.

Question 7

Which of the following is the general error term for Simpson's rule?

Accepted Answer

-b^5 f''''(ξ)/180n^4

Answer

-b^7 f''''''(ξ)/180n^6

Answer

-b^3 f'''(ξ)/180n^3

Question 8

Which of the following is the formula for the composite Trapezoidal rule?

Accepted Answer

∫[a,b] f(x) dx ≈ h/2 [f(x0) + 2(f(x1) + f(x2) + ... + f(xn-1)) + f(xn)]

Answer

∫[a,b] f(x) dx ≈ h/2 [f(x0) + f(x1) + f(x2) + ... + f(xn)]

Question 9

Which of the following is the formula for the composite Simpson's rule?

Accepted Answer

∫[a,b] f(x) dx ≈ h/3 [f(x0) + 4(f(x1) + f(x3) + ... + f(xn-1)) + 2(f(x2) + f(x4) + ... + f(xn-2)) + f(xn)]

Answer

∫[a,b] f(x) dx ≈ h/3 [f(x0) + 2(f(x1) + f(x2) + ... + f(xn-1)) + f(xn)]

Question 10

The Trapezoidal rule approximates the integral of a function over an interval by:

Accepted Answer

Using the average of the function values at the endpoints of the interval.

Answer

Using the function value at the midpoint of the interval.

Answer

Using the definite integral of the function over the interval.

Question 11

Simpson's rule approximates the integral of a function over an interval by:

Accepted Answer

Using a quadratic polynomial that interpolates the function values at three equally spaced points in the interval.

Answer

Using a linear polynomial that interpolates the function values at two equally spaced points in the interval.

Question 12

Which of the following functions would have the smallest error when approximated using the Trapezoidal rule?

Accepted Answer

A linear function.

Answer

A cubic function.

Answer

A quadratic function.

Question 13

Which of the following is an advantage of Newton-Cotes formulas?

Accepted Answer

They are easy to implement and computationally efficient.

Answer

They are always accurate.

Answer

They are the most efficient numerical integration techniques.

Answer

They can be used to approximate integrals of any function.

Question 14

Which of the following is a disadvantage of Newton-Cotes formulas?

Accepted Answer

They can be inaccurate for functions with high-order derivatives.

Answer

They can only be used to approximate integrals of polynomial functions.

Answer

They are difficult to implement.

Question 15

The error term of the Simpson's 1/3 rule is given by:

Accepted Answer

-f''''(c)(b-a)^5/180n^4

Answer

-f''''(a)(b-a)^5/180n^4

Answer

-f'''''(c)(b-a)^5/180n^4

Answer

-f''''(b)(b-a)^5/180n^4

Question 16

The order of convergence of the Simpson's 3/8 rule is:

Accepted Answer

4

Answer

3

Answer

5

Answer

2

Question 17

Which method is NOT classified as a Newton-Cotes formula?

Accepted Answer

Gaussian quadrature

Answer

Trapezoidal rule

Answer

Romberg integration

Answer

Simpson's rule

Question 18

For the error term of a Newton-Cotes formula to be valid, what assumption must be met?

Accepted Answer

The function should have continuous derivatives up to a certain order.

Answer

The function should be continuous.

Answer

The function should be differentiable.

Question 19

What is the primary advantage of utilizing Newton-Cotes formulas?

Accepted Answer

Simplicity and ease of application.

Answer

Robustness under various conditions

Answer

Unparalleled accuracy

Answer

Exceptional efficiency

Question 20

A potential limitation of Newton-Cotes formulas is:

Accepted Answer

Accuracy may be insufficient for certain function types.

Answer

High computational cost

Answer

Challenging implementation

Question 21

Which of the following is the Trapezoidal rule formula for approximating the integral of a function f(x) over the interval [a, b]?

Accepted Answer

∫[a, b] f(x) dx ≈ (b - a) * (f(a) + f(b)) / 2

Answer

∫[a, b] f(x) dx ≈ (b - a) * (f(a) - f(b)) / 2

Answer

∫[a, b] f(x) dx ≈ (b - a) * (f(a) + f(b) / 4

Answer

∫[a, b] f(x) dx ≈ (b - a) * (f(a - f(b)) / 4

Question 22

Which of the following is a disadvantage of using Newton-Cotes formulas?

Accepted Answer

They can be unstable for certain functions.

Answer

They are accurate for most functions.

Answer

They are computationally efficient.

Answer

They are easy to implement.

Question 23

Which of the following is NOT a Newton-Cotes formula?

Accepted Answer

Gauss-Legendre quadrature

Answer

Simpson's rule

Answer

Trapezoidal rule

Answer

Midpoint rule

Question 24

What is the error term for the Trapezoidal rule for a function with continuous second derivative?

Accepted Answer

-h^2/12 * f''(ξ)

Answer

h^2/12 * f''(x)

Answer

-h^2/2 * f''(ξ)

Answer

h^2/2 * f'''(ξ)

Question 25

What is the composite Trapezoidal rule?

Accepted Answer

A method for integrating a function over an interval by dividing the interval into subintervals and applying the Trapezoidal rule to each subinterval.

Answer

A method for integrating a function over an interval using a single application of the Trapezoidal rule.

Question 26

Which of the following is an advantage of using Newton-Cotes formulas?

Accepted Answer

They are easy to implement.

Answer

They can be used to integrate functions with singularities.

Answer

They are always the most accurate numerical integration method.

Answer

They are suitable for parallel computation.

Question 27

What is the disadvantage of using Simpson's rule?

Accepted Answer

It requires evaluations of the function at equally spaced points.

Answer

It is more difficult to implement than the Trapezoidal rule.

Answer

It is less accurate than the Trapezoidal rule.

Question 28

Which Newton-Cotes formula uses linear polynomials to approximate the integral of a function?

Accepted Answer

Trapezoidal Rule

Answer

Simpson's Rule

Answer

Rectangular Rule

Answer

Midpoint Rule

Question 29

Which of the following is NOT a characteristic of Newton-Cotes formulas?

Accepted Answer

They always provide the most accurate numerical integration results.

Answer

They utilize equally spaced nodes within the integration interval.

Answer

They rely on polynomial interpolation of the integrand function.

Question 30

The Trapezoidal rule tends to be more accurate when:

Accepted Answer

The function being integrated resembles a straight line within each subinterval.

Answer

A large number of subintervals is used regardless of the function's behavior.

Answer

The function exhibits rapid fluctuations within the integration interval.

Question 31

What is the main advantage of using Simpson's rule over the Trapezoidal rule for numerical integration?

Accepted Answer

Simpson's rule generally achieves higher accuracy with the same number of function evaluations.

Answer

Simpson's rule is computationally simpler to implement.

Question 32

When using Newton-Cotes formulas, increasing the number of subintervals typically leads to:

Accepted Answer

Reduced error in the approximation, but increased computational cost.

Answer

Increased error due to rounding errors becoming more significant.

Answer

No change in the accuracy of the approximation.

Question 33

Which field of study frequently utilizes Newton-Cotes formulas for practical applications?

Accepted Answer

Numerical analysis and scientific computing

Answer

Economics and financial modeling

Answer

Theoretical physics and cosmology

Answer

Abstract algebra and group theory

Question 34

Which of the following is NOT a Newton-Cotes formula?

Accepted Answer

Lagrange formula

Answer

Trapezoidal rule

Answer

Simpson's rule

Answer

Midpoint rule

Question 35

Which of the following is a possible error term for Simpson's rule?

Accepted Answer

$\frac{h^4}{180}f^{(4)}(c)$

Answer

$\frac{h^2}{2}f''(c)$

Answer

$\frac{h^3}{6}f'''(c)$

Answer

$\frac{h^5}{120}f^{(5)}(c)$

Question 36

Which of the following is a disadvantage of using Newton-Cotes formulas for numerical integration?

Accepted Answer

They can be inaccurate for functions with high derivatives.

Answer

They are difficult to apply.

Answer

They can only be used to integrate functions with continuous derivatives.