Choose the letter(s) of the correct answer. Cho0se E if none of the choices corresponds to the correct answer.1. If f(x) - g(x) is to be integrated with respect to x, taking dx as the differentiated variable, then:A. f(x) must be a function of g(x)B. g(x) must be equal to f(g(x))C. f(x) and g(x) can take any form of a functionD. f'x) and gʻ(x) must exist at all costs.2. If f(x) f'(x)dx is the whole function to be integrated, what is the best method to be used?A. usubstitution or change of variablesB. pattern recognitionC. fitting integrand methodD. any of the abovementioned procedure will do3. Given f(x) = x**, how do you solve forthe antiderivative of f (x + n) if the variable of integration is x?A. Substitutex with n and continue to integrate with the appropriate integration procedure.B. Treatnas a variable and proceed with proper integration techniques.C. Assumen is a constant and integrate f (x) using proper integration techniques.D. Do A and C procedures.2/44. What can you conclude from the result of a Riemann sum?A. The exact area under a curve.B. The approximate signed area under (or above) the curve.c. Underestimation is a result of using the left endpoint.D. Overestimation is a result of using the right-endpoint. 5. When does finding the indefinite integral of a certain expression / (x) become impossible?A. When the expression does not converge as x approaches to infinity.B. When the constant of integration is impossible to determine.c. When the graph oscillates fromb to a assuming that b to a falls within the domain of f (x).D. Whenf(x) is a combination of more than two types of elementary function.6. Ifa particular solution of a function f(x) exists and is obtained when x = 0, thenA. the constant of integration is zero.B. F(0) = c.c. this solution may be generalized as F(x) + F(0).D. it is not sufficient to say that F(0) = c.7. If f(x) = 3x", and one wants to find the answer using integration, which of these is/are true?A. x'is a primitiveB. x + cis the antiderivativeC. x + 5is a possible answerD. x' + cis the indefinite integral, rather than the antiderivative8. How many possible antiderivative(a) can a certain functionwith discontinuityhave?A. infinitely manyB. only onec. depends on the equation itselfD. none9. Suppose / (x) is the integrand and F(x) + c is the antiderivative. How can one check if the answer obtained fromintegration is correct?A. Use differentiation such that F' (x) + c is equal to f (x).B. Differentiate Fx) + c and check if the answer is f (x)dx,c. Solve for f"Cx) and it should be equal to F(x) +c.D. If and only if F(x) + c exists at all domain of f (x).10. If the Riemann sums using a leftendpoint produces an underestimated area, which of these is/are most probablyсorect?A. The curve generally has a negative slope(downwards).B. The curve is below the xaxis and increases from left to right.C. The curve is above the x-axis and increases from left to right.D. The curve is above the x-axis and increases from right to left.

The objective is to select the best possible option.

1. If fa) g(x) is to be integrated with respect to x, taking dx as the differentiated variable, then: A. f(x) must be a function of g(x) B. g(x) must be equal to /(g(x) c. f) and g(x) can take any form of a function D. f') and g'(x) must exist at all costs. 2. If f(x) (x)dx is the whole function to be integrated, what is the best method to be used? A. usubstitution or change of variables B. pattern recognition c. fitting integrand method D. any of the abovementioned procedure will do 3. Given f(x) = x+", how do you solve forthe antiderivative of f (x + n) if the variable of integration is x? A. Substitutex with n and continue to integrate with the appropriate integration procedure. B. Treat nas a variable and proceed with proper integration techniques. c. Assumen is a constant and integrate f (x) using proper integration techniques. D. Do A and C procedures. 2/4 4. What can you conclude from the result of a Riemann sum? A. The exact area under a curve. B. The approximate signed area under (or above) the curve. c. Underestimation is a result of using the left-endpoint. D. Overestimation is a result of using the right-endpoint.

1. If fa) g(x) is to be integrated with respect to x, taking dx as the differentiated variable, then: A. f(x) must be a function of g(x) B. g(x) must be equal to /(g(x) c. f) and g(x) can take any form of a function D. f') and g'(x) must exist at all costs. 2. If f(x) (x)dx is the whole function to be integrated, what is the best method to be used? A. usubstitution or change of variables B. pattern recognition c. fitting integrand method D. any of the abovementioned procedure will do 3. Given f(x) = x+", how do you solve forthe antiderivative of f (x + n) if the variable of integration is x? A. Substitutex with n and continue to integrate with the appropriate integration procedure. B. Treat nas a variable and proceed with proper integration techniques. c. Assumen is a constant and integrate f (x) using proper integration techniques. D. Do A and C procedures. 2/4 4. What can you conclude from the result of a Riemann sum? A. The exact area under a curve. B. The approximate signed area under (or above) the curve. c. Underestimation is a result of using the left-endpoint. D. Overestimation is a result of using the right-endpoint.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.

Question

Choose the letter(s) of the correct answer. Cho0se E if none of the choices corresponds to the correct answer.
1. If f(x) - g(x) is to be integrated with respect to x, taking dx as the differentiated variable, then:
A. f(x) must be a function of g(x)
B. g(x) must be equal to f(g(x))
C. f(x) and g(x) can take any form of a function
D. f'x) and gʻ(x) must exist at all costs.
2. If f(x) f'(x)dx is the whole function to be integrated, what is the best method to be used?
A. usubstitution or change of variables
B. pattern recognition
C. fitting integrand method
D. any of the abovementioned procedure will do
3. Given f(x) = x**, how do you solve forthe antiderivative of f (x + n) if the variable of integration is x?
A. Substitutex with n and continue to integrate with the appropriate integration procedure.
B. Treatnas a variable and proceed with proper integration techniques.
C. Assumen is a constant and integrate f (x) using proper integration techniques.
D. Do A and C procedures.
2/4
4. What can you conclude from the result of a Riemann sum?
A. The exact area under a curve.
B. The approximate signed area under (or above) the curve.
c. Underestimation is a result of using the left endpoint.
D. Overestimation is a result of using the right-endpoint.

5. When does finding the indefinite integral of a certain expression / (x) become impossible?
A. When the expression does not converge as x approaches to infinity.
B. When the constant of integration is impossible to determine.
c. When the graph oscillates fromb to a assuming that b to a falls within the domain of f (x).
D. Whenf(x) is a combination of more than two types of elementary function.
6. Ifa particular solution of a function f(x) exists and is obtained when x = 0, then
A. the constant of integration is zero.
B. F(0) = c.
c. this solution may be generalized as F(x) + F(0).
D. it is not sufficient to say that F(0) = c.
7. If f(x) = 3x", and one wants to find the answer using integration, which of these is/are true?
A. x'is a primitive
B. x + cis the antiderivative
C. x + 5is a possible answer
D. x' + cis the indefinite integral, rather than the antiderivative
8. How many possible antiderivative(a) can a certain functionwith discontinuityhave?
A. infinitely many
B. only one
c. depends on the equation itself
D. none
9. Suppose / (x) is the integrand and F(x) + c is the antiderivative. How can one check if the answer obtained from
integration is correct?
A. Use differentiation such that F' (x) + c is equal to f (x).
B. Differentiate Fx) + c and check if the answer is f (x)dx,
c. Solve for f"Cx) and it should be equal to F(x) +c.
D. If and only if F(x) + c exists at all domain of f (x).
10. If the Riemann sums using a leftendpoint produces an underestimated area, which of these is/are most probably
сorect?
A. The curve generally has a negative slope(downwards).
B. The curve is below the xaxis and increases from left to right.
C. The curve is above the x-axis and increases from left to right.
D. The curve is above the x-axis and increases from right to left.

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