1.
A good way of solving a polynomial equation involving numerator and denominator is to apply partial fraction, what is partial fraction?
Correct Answer
A. The decomposition process of starting with simplified answer
Explanation
Partial fraction decomposition is a method used to break down a rational function into simpler fractions. It involves starting with a simplified answer and then decomposing the final expression into its initial polynomial fractions. This process allows us to solve polynomial equations involving numerators and denominators more easily.
2.
Calculus question of Laplace transform or inverse Laplace to solve them depending on the problem given, what is the partial fraction in integration?
Correct Answer
A. Partial fraction decomposition
Explanation
Partial fraction decomposition is the process of breaking down a rational function into simpler fractions. It is used in integration to simplify the integration of complex functions by expressing them as a sum of simpler fractions. This allows for easier integration and can be especially useful when dealing with functions that involve polynomial factors. Therefore, the correct answer is partial fraction decomposition.
3.
Laplace transform has always been applied to partial fraction question and its table can be used to solve equations using any of the formulae that applies to the problem, describe inverse Laplace transform?
Correct Answer
D. All of the above
Explanation
The given answer "All of the above" is correct because all three statements are true. If a function f(s) has an inverse Laplace transform f(t), then f(t) is uniquely determined, meaning there is only one possible inverse Laplace transform for a given function. Additionally, when considering functions that differ on a point set having Lebesgue measure zero, they are considered the same. This means that even if two functions have different values at a single point, they are still considered equal in the context of the inverse Laplace transform.
4.
In mathematics, there are rational numbers, irrational numbers, and rational functions, when is a function said to be a rational function?
Correct Answer
D. All of the above
Explanation
A rational function is defined as a function that can be expressed as a quotient of two polynomials, where both the numerator and denominator are polynomials. Therefore, any function that can be defined by a rational fraction, as well as any algebraic function, can be considered a rational function. Hence, the correct answer is "All of the above".
5.
Solving a partial fraction involves steps and in decomposition, you start with a proper rational expression, factor the bottom into linear factors, then?
Correct Answer
D. All of the above
Explanation
When solving a partial fraction, the correct approach involves writing out a partial fraction for each factor, multiplying the whole equation by the bottom, and then solving for the coefficients by substituting zeros of the bottom. Therefore, the correct answer is "All of the above."
6.
Solving partial fraction always involves substitution down the line in the process of integration and differentiation, what is integration by substitution?
Correct Answer
A. U-substitution for finding integrals
Explanation
Integration by substitution, also known as u-substitution, is a technique used to simplify integrals by making a substitution of variables. It involves substituting a new variable, usually denoted as u, in order to transform the integral into a simpler form that can be easily evaluated. This method is particularly useful when dealing with integrals that involve composition of functions or when the integrand is a product of functions. By substituting variables, the integral can be rewritten in terms of the new variable, making it easier to calculate the antiderivative and find the solution. Therefore, the correct answer is "U-substitution for finding integrals".
7.
In the process of solving partial fraction, the need to decompose equation by parts arises for easy problem solving, what is integration by parts?
Correct Answer
A. Used when two functions multiplied together
Explanation
Integration by parts is a method used when two functions are multiplied together. It allows us to find the antiderivative of the product of two functions by breaking it down into simpler components. This technique is often used to solve integrals that involve products of functions, and it involves choosing one function to differentiate and another function to integrate. By applying the integration by parts formula, we can simplify the integral and solve it more easily.
8.
The most basic form of substitution when an integral contains some function and derivative, that is, for an integral of the form, what substitution is this?
Correct Answer
A. U-substitution
Explanation
U-substitution is the most basic form of substitution when an integral contains some function and its derivative. In U-substitution, a new variable "u" is introduced, which is equal to the function inside the integral. This allows us to simplify the integral by replacing the function and its derivative with the new variable. U-substitution is a commonly used technique in calculus to evaluate integrals and solve problems involving functions and their derivatives.
9.
Several rules apply when solving a partial fraction and this can only be used when the equation can be taken apart using a certain mathematical rule, what is chain rule?
Correct Answer
B. For differentiating compositions of functions
Explanation
The chain rule is a rule in calculus that allows us to find the derivative of a composition of functions. It states that if we have a function that is composed of two or more functions, then the derivative of the composition is equal to the derivative of the outer function multiplied by the derivative of the inner function. This rule is used when we need to find the derivative of a function that is made up of nested functions, such as f(g(x)). Therefore, the correct answer is "For differentiating compositions of functions."
10.
How do you in integrate LN X? if: In(x) dx. set u = In(x), dv=dx, then we find. du = (1/x)dx, v=x.
Correct Answer
D. All of the above
Explanation
The given answer "All of the above" is correct because it includes all the steps mentioned in the explanation. The integration by parts method is used, where u is set as In(x) and dv is set as dx. The values of du and v are found accordingly. Then, the formula for integration by parts, uv - ∫v du, is applied by substituting the values of u, v, and du. Therefore, all the mentioned steps are required to integrate LN X.