dy/dx - = 1 - y,y(0) = 0 is an example of

[zaasmi]

Power method is applicable if the eigen values are ______________.
realanddistinct
real and equal
positive and distinct
negative and distinct

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Power method is applicable if the eigen values are ______________.
realanddistinct
real and equal
positive and distinct
negative and distinct

Positive and distinct

Explanation:

• Power Method: The power method is designed to find the eigenvalue with the largest magnitude (dominant eigenvalue) and its corresponding eigenvector.

• Requirements: The power method is most effective when the matrix has a single dominant eigenvalue that is larger in magnitude than the other eigenvalues. It is not specifically limited to eigenvalues being positive and distinct, but having a dominant eigenvalue (which can be positive, negative, or zero) significantly larger in magnitude than the others improves convergence.

• Real and Distinct, Real and Equal, Positive and Distinct, Negative and Distinct: While the method can work with various types of eigenvalues, the convergence is most straightforward when there is a clear dominant eigenvalue, which could be positive or negative.

Thus, while the power method is generally applicable, it is especially straightforward when the dominant eigenvalue is distinct and of a significantly larger magnitude compared to others. “Positive and distinct” is often mentioned as a favorable condition but not a strict requirement.

So the most fitting answer from the given options would be:

Positive and distinct

[zaasmi]

Simpson’s rule is a numerical method that approximates the value of a definite
integral by using polynomials.
Quadratic
Linear
Cubic
Quartic

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Simpson’s rule is a numerical method that approximates the value of a definite
integral by using polynomials.
Quadratic
Linear
Cubic
Quartic

Quadratic

Explanation:

• Simpson’s Rule: This numerical method approximates the value of a definite integral by using quadratic polynomials. Specifically, it uses parabolic segments to estimate the area under the curve, effectively fitting a second-degree polynomial (quadratic) to the points.

• Quadratic Polynomial: In Simpson’s rule, the integral is approximated using a quadratic polynomial that interpolates through three points. This polynomial fits the function being integrated and provides a more accurate approximation than linear methods.

Thus, Simpson’s rule is based on using quadratic polynomials.

So the correct option is:

Quadratic

[zaasmi]

In Simpson’s Rule, we use parabolas to approximating each part of the curve. This proves to be very efficient as compared to Trapezoidal rule.
True
False

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

In Simpson’s Rule, we use parabolas to approximating each part of the curve. This proves to be very efficient as compared to Trapezoidal rule.
True
False

True

Explanation:

• Simpson’s Rule: This method uses parabolic segments (quadratic polynomials) to approximate the integral of a function. By fitting a parabola to segments of the curve, Simpson’s rule often provides a more accurate approximation compared to linear methods.

• Trapezoidal Rule: This method approximates the integral by fitting straight lines (trapezoids) to the segments of the curve. While it is simpler, it generally requires more intervals to achieve the same level of accuracy as Simpson’s rule.

• Efficiency: Because parabolas can better capture the curvature of the function compared to straight lines, Simpson’s rule often achieves better accuracy with fewer intervals than the trapezoidal rule.

Therefore, Simpson’s rule is considered more efficient in terms of accuracy compared to the trapezoidal rule.

So, the statement is:

True

[zaasmi]

The predictor-corrector method an implicit method. (multi-step methods)
True
False

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

The predictor-corrector method an implicit method. (multi-step methods)
True
False

False

Explanation:

• Predictor-Corrector Methods: These are a type of multi-step methods used for solving ordinary differential equations. They involve predicting the solution at a future point using an explicit method and then correcting it using an implicit method or vice versa.

• Implicit vs. Explicit: Predictor-corrector methods are not exclusively implicit. The predictor step is usually explicit, while the corrector step can be either implicit or explicit, depending on the specific method used.

Therefore, the statement that the predictor-corrector method is an implicit method is:

False

[zaasmi]

Generally, Adams methods are superior if output at many points is needed.
True
False

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

Generally, Adams methods are superior if output at many points is needed.
True
False

True

Explanation:

• Adams Methods: These are multi-step methods for solving ordinary differential equations. Adams-Bashforth (explicit) and Adams-Moulton (implicit) methods are designed to use information from previous steps to provide higher accuracy.

• Efficiency: Adams methods can be more efficient when many output points are needed because they leverage information from multiple previous points to compute each new point. This can reduce the overall computational effort compared to methods that recalculate at each step without using past information.

• Performance: The performance of Adams methods improves with the number of points because they use past computed values effectively, which makes them advantageous for solving problems over many intervals.

Thus, Adams methods are generally superior when many output points are required.

So, the statement is:

True

[zaasmi]

The Trapezoidal rule is a numerical method that approximates the value of a.______________.
Indefinite integral
Definiteintegral
Improper integral
Function

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

The Trapezoidal rule is a numerical method that approximates the value of a.______________.
Indefinite integral
Definiteintegral
Improper integral
Function

Definite integral

Explanation:

• Trapezoidal Rule: This numerical method approximates the value of a definite integral. It works by dividing the area under a curve into trapezoids and summing their areas to estimate the total integral.

• Indefinite Integral: This represents a family of functions and is not computed using numerical methods like the trapezoidal rule.

• Improper Integral: This type of integral involves infinite limits or unbounded integrands, and while numerical methods can be adapted for improper integrals, the trapezoidal rule itself is typically used for definite integrals.

• Function: This is a broader concept and not directly related to the specific numerical approximation provided by the trapezoidal rule.

Thus, the trapezoidal rule specifically approximates the value of a definite integral.

So, the correct option is:

Definite integral

[zaasmi]

The need of numerical integration arises for evaluating the definite integral of a function that has no explicit ____________ or whose anti derivative is not easy to obtain.

Antiderivative
Derivatives.

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

The need of numerical integration arises for evaluating the definite integral of a function that has no explicit ____________ or whose anti derivative is not easy to obtain.

Antiderivative
Derivatives.

Antiderivative

Explanation:

• Numerical Integration: This technique is used to approximate the definite integral of a function, especially when an explicit antiderivative is difficult or impossible to obtain.

• Antiderivative: The antiderivative (or indefinite integral) of a function is the function whose derivative is the original function. When an antiderivative is not easily obtainable, numerical integration methods are used to estimate the definite integral.

• Derivatives: While derivatives are important in calculus, the issue with finding an explicit antiderivative relates directly to the challenge of evaluating definite integrals.

Thus, the need for numerical integration arises when the antiderivative is not easy to obtain.

So, the correct option is:

Antiderivative

[zaasmi]

An indefinite integral may _________ in the sense that the limit defining it may not exist.

Diverge
Converge

[zaasmi]

@zaasmi said in MTH603 Mid Term Past and Current Solved Paper Discussion:

An indefinite integral may _________ in the sense that the limit defining it may not exist.

Diverge
Converge

Diverge

Explanation:

• Indefinite Integral: This represents a family of functions whose derivatives are the integrand. While the concept of convergence and divergence more commonly applies to definite integrals, the idea of divergence can apply to indefinite integrals in the context of improper integrals or when dealing with limits.

• Diverge: An integral (definite or indefinite) is said to diverge if it does not converge to a finite value. In the context of indefinite integrals, divergence can occur if the integral involves terms or functions that lead to an unbounded result.

Thus, an indefinite integral may diverge if the limits involved lead to an unbounded or non-existent result.

So, the correct option is:

Diverge