Question 1

Typically, ____ equations involve trigonometric, exponential, or logarithmic functions and cannot be reduced to any polynomial equation in x .

Accepted Answer

Transcendental equations involve functions that cannot be expressed as a finite combination of algebraic functions, therefore they cannot be reduced to any polynomial equation in x. Trigonometric, exponential, and logarithmic functions are all examples of transcendental functions. Quadratic equations can be reduced to polynomial equations of degree 2, while linear equations involve only linear functions of x. The term "real" is not applicable in this context as it refers to the domain or range of a function, not its type. Transcendental equations involve functions like trigonometric, exponential, or logarithmic which cannot be represented by any polynomial equation. Quadratic and linear equations are types of polynomial equations. Real equations simply refer to equations involving real numbers.

Question 2

In the ____ method, you can predict with some precision the number of iterations required to find a function s root to a certain accuracy.

Accepted Answer

The bisection method is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a bracketing method and guarantees convergence, allowing prediction of the number of iterations needed to achieve a desired accuracy.

Question 3

A root-finding method that uses a minimum of finesse is known as a ____ method.

Accepted Answer

A brute-force method is a straightforward approach that systematically evaluates every possible solution without any advanced optimization or algorithmic techniques. This method can be time-consuming, but it guarantees finding the solution accurately. Therefore, a brute-force method is the best choice when a minimum of finesse is required. The term "brute-force" in computer science refers to a straightforward approach that involves trying all possible solutions without using a sophisticated algorithm. In the context of root-finding, a brute-force method would involve repeatedly plugging in values into a function until a root is found. While this method may be less efficient than more refined algorithms, it can still be effective for simple functions or when an initial guess is already reasonably close to the root. Therefore, the best choice is D.

Question 4

Many of the theorems you learned for roots of polynomials don t apply to transcendental equations.

Accepted Answer

The theorems for roots of polynomials only apply to algebraic equations, whereas transcendental equations involve transcendental functions (e.g. trigonometric functions, exponential functions) and do not satisfy the properties of algebraic equations. Therefore, many of the theorems for roots of polynomials do not apply to transcendental equations. Transcendental equations involve functions that cannot be expressed as a finite combination of algebraic operations and constants, while polynomials involve only algebraic operations. Therefore, many of the theorems that apply to polynomial roots do not apply to transcendental equations.

Question 5

The regula falsi procedure can be made to converge more rapidly if the interval can be made to collapse from only one direction.

Accepted Answer

The answer of The regula falsi procedure can be made...

Question 6

A polynomial of degree n has precisely ____ roots.

Accepted Answer

A polynomial of degree n has exactly n roots, counting multiplicity. This is known as the Fundamental Theorem of Algebra. According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots, including any repeated roots. Therefore, choice B is the correct answer.

Question 7

Using the ____ root-finding method, the interval is made to collapse from both directions.

Accepted Answer

The modified regula falsi method adjusts the interval from both sides to ensure convergence, unlike the standard regula falsi method.

Question 8

The ____ method presents the basics on which most root-finding methods are constructed.

Accepted Answer

The bisection method is the most basic and straightforward root-finding method upon which many other methods are built. It involves repeatedly dividing an interval in half and determining which half must contain a root until a sufficiently small interval is obtained. The bisection method is the most basic root-finding method and is used as a starting point for the development of more advanced methods. It is a simple and reliable method that is commonly used in mathematics, engineering, and science.

Question 9

With the regula falsi method of root-finding, success is based on the size of the function, not the size of the interval.

Accepted Answer

The regula falsi method of root-finding is based on the sign of the function at the endpoints of the interval, not the size of the function or the interval.

Question 10

The real roots of polynomials are usually classified as being either ____ numbers or irrational.

Accepted Answer

Rational numbers are numbers that can be expressed as a ratio of two integers, meaning they can be written in the form p/q where p and q are integers and q is not equal to zero. Since the coefficients of polynomials are usually integers, their rational roots will also be expressed as ratios of integers. However, it is possible for polynomials to have irrational roots that cannot be expressed as a ratio of integers. These can include roots such as square roots or cube roots of non-perfect squares or non-perfect cubes, as well as transcendental numbers like pi or e. Therefore, the real roots of polynomials are usually classified as either rational or irrational.

Question 11

An approximation to the area under a complicated curve is obtained by assuming the function can be replaced by simpler functions over a limited range.

Accepted Answer

This is the basic idea behind numerical integration techniques like the trapezoidal rule and Simpson's rule. This is the basic idea behind numerical integration methods such as the trapezoidal rule and Simpson's rule.

Question 12

Simpson s rule converges nicely in most instances and it is easy to use.

Accepted Answer

Simpson's rule is a precise method that converges nicely in most instances and is simple to use. Simpson's rule is a very effective method for numerical integration and has a high degree of accuracy for smooth functions, making it an easy and efficient approach for many applications.

Question 13

The regula falsi root-finding method is essentially the same as bisection, except instead of using the interval s midpoint at each step of the calculation, you use a(n) ____ value for the root.

Accepted Answer

The regula falsi method uses an interpolated value for the root, which is calculated based on a linear approximation between the endpoints of the interval, unlike the bisection method that simply uses the midpoint.

Question 14

The secant method of root-finding does not pose divergence problems.

Accepted Answer

The secant method may pose convergence problems if the initial interval for the search does not contain the root, or if the function has a flat slope or behaves nonlinearly in the vicinity of the root. In such cases, the iterations may fail to converge or oscillate around the root. The secant method of root-finding can pose divergence problems if the starting points are not chosen carefully or if the function being evaluated has multiple roots or sharp curves. This is because the secant method is a quasi-linear method that relies on linear approximations of the function and can diverge if the approximations are poor.

Question 15

By using information about the ____ of a functional value in addition to its sign, you can speed up the convergence of your root-finding scheme.

Accepted Answer

The size of a functional value indicates how far from zero the function value is, which can help in adjusting the step size or method to speed up convergence in root-finding algorithms.

Question 16

One of the most common tasks in science and engineering is finding the roots of equations.

Accepted Answer

Finding the roots of equations is a common task in many fields, including mathematics, physics, engineering, and computer science. Finding the roots or solutions of equations is a fundamental problem in various scientific and engineering fields, including mathematics, physics, chemistry, and computer science. It involves finding values of the variable that make the equation true. This problem is typically solved using analytical or numerical methods, depending on the complexity of the equation and the available computational resources.

Question 17

Numerical integration is a stable process.

Accepted Answer

The answer of Numerical integration is a stable process....

Question 18

A change in a function s ____, indicating the vicinity of a root has been located, forms the key element in the computer code for locating roots.

Accepted Answer

A change in sign indicates that the function has switched from positive to negative or vice versa, which is a strong indication of the presence of a root in the vicinity. Therefore, the change in sign is the key element used in many numerical methods for root-finding. A change in the sign of a function indicates that a root has been crossed. This is the key element in many root-finding algorithms.

Question 19

A straight line is the simplest approximation to a function.

Accepted Answer

A straight line is a linear function and is often used to approximate other functions, making it the simplest form of approximation. A straight line, also known as a linear function, is the simplest type of function that can be used to approximate a more complex function. It has a constant rate of change and can be described by an equation in the form y = mx + b, where m is the slope and b is the y-intercept.

Question 20

Simpson s rule is a popular method of evaluating integrals of functions that are smooth.

Accepted Answer

Simpson's rule is a well-known method used to numerically approximate the value of an integral. It is particularly useful for smooth functions, where the integral is taken over a closed interval. Simpson's rule is a numerical method for approximating the value of a definite integral. It is particularly useful for integrals of smooth functions, as it involves fitting parabolic curves to the function's graph.

Quiz 14: Numerical Methods