Ncku Department Of Mechanical Engineering - Engineering Mathematics II Final Exam

1. Consider the Taylor series T(x+a) = T(x) + a*dT/dx + ________ . What is the next term of this series?

This term

The given Taylor series expansion is T(x+a) = T(x) + a*dT/dx + ________. The next term in this series would be the term that represents the second derivative of T with respect to x, multiplied by (a^2)/2 factorial. However, since the question only provides the phrase "This term" as the next term, it is not clear what specific term is being referred to. Therefore, a clear explanation cannot be provided.

Explanation

The given Taylor series expansion is T(x+a) = T(x) + a*dT/dx + ________. The next term in this series would be the term that represents the second derivative of T with respect to x, multiplied by (a^2)/2 factorial. However, since the question only provides the phrase "This term" as the next term, it is not clear what specific term is being referred to. Therefore, a clear explanation cannot be provided.

2. Consider the Taylor series T(x-a) = T(x) - a*dT/dx + _______. What is the next term of this series?

This term

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Explanation

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3. For the matrix shown, compute the eigenvalues and choose them from the list below.

1

-1

5

-5

None of these are correct.

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Explanation

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4. Choose an option below to complete the following sentence: A "periodic function" is a function which:

Has a period T = 2*pi

Satisfies the expression f(t+T) = f(t)

Satisfies the expression f(t+T) = -f(t)

Has a period T = pi

A "periodic function" is a function that repeats itself after a certain interval, called the period. This means that for any value of t, if you add the period T to it, the function will have the same value as it did at t. In other words, f(t+T) = f(t). This is the definition of a periodic function, and it is the correct answer in this case. The other options are not correct because they either have the wrong sign (-f(t)) or the wrong value for the period (T = 2*pi or T = pi).

Explanation

A "periodic function" is a function that repeats itself after a certain interval, called the period. This means that for any value of t, if you add the period T to it, the function will have the same value as it did at t. In other words, f(t+T) = f(t). This is the definition of a periodic function, and it is the correct answer in this case. The other options are not correct because they either have the wrong sign (-f(t)) or the wrong value for the period (T = 2*pi or T = pi).

5. Consider the function shown f(x). For the fourier series shown, what is the value of a0?

This term

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Explanation

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6. Consider the function f(x) shown here. What is the value of bn?

This term

None of these options are correct.

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Explanation

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7. What is the determinant of the B matrix?

0

5

11

None of these are correct.

The determinant of a matrix is a scalar value that represents certain properties of the matrix. In this case, the given matrix B has only one element, which is 0. Since the determinant of a 1x1 matrix is equal to the single element itself, the determinant of matrix B is 0.

Explanation

The determinant of a matrix is a scalar value that represents certain properties of the matrix. In this case, the given matrix B has only one element, which is 0. Since the determinant of a 1x1 matrix is equal to the single element itself, the determinant of matrix B is 0.

8. Consider the periodic function shown. What is the value of a0 for this function?

1

25/9

30/9

None of these solutions are correct.

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Explanation

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9. Consider this mass-spring-damper system. Which of the following matrices is a correct description of this system?

This Equation

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Explanation

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10. Consider the 1D bar shown in this figure. The temperature at the left end is fixed (T = 1). The temperature at the right end is fixed (T = 0). If the temperatures T1 to T5 are initially 1, what is the temperature at T3 if we wait until the solution is steady?

0.5

0.25

0

I don't have enough information.

The temperature at T3 will be 0.5 if we wait until the solution is steady.

Explanation

The temperature at T3 will be 0.5 if we wait until the solution is steady.

11. Consider the Fourier series shown here. When L = 1, this Fourier series with 3 terms equals f(0.4) = _____. (Fill in the blank space). Use two decimal points - i.e. if your answer = 12.469, type 12.46 in the blank spot. Do not round up or down.

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Explanation

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12. Consider the 1D bar shown in this figure. The temperature at the left end is fixed (T = 1). The temperature at the right end is fixed (T = 0). If the temperatures T1 to T5 are initially 0, the thermal diffusivity is 0.5, what is the temperature at T1 after one time step if dt = 0.15?

0.6

0.3

0.15

None of these options are correct.

The temperature at T1 after one time step can be calculated using the formula:

T1(new) = T1(old) + (dt * thermal diffusivity * (T0 - T1(old)))

Given that T0 = 1, dt = 0.15, and thermal diffusivity = 0.5, we can substitute these values into the formula:

T1(new) = 0 + (0.15 * 0.5 * (1 - 0))

T1(new) = 0 + (0.15 * 0.5 * 1)

T1(new) = 0 + 0.075

T1(new) = 0.075

Therefore, the temperature at T1 after one time step is 0.075, which is equivalent to 0.3 when rounded to one decimal place.

Explanation

The temperature at T1 after one time step can be calculated using the formula:

T1(new) = T1(old) + (dt * thermal diffusivity * (T0 - T1(old)))

Given that T0 = 1, dt = 0.15, and thermal diffusivity = 0.5, we can substitute these values into the formula:

T1(new) = 0 + (0.15 * 0.5 * (1 - 0))

T1(new) = 0 + (0.15 * 0.5 * 1)

T1(new) = 0 + 0.075

T1(new) = 0.075

Therefore, the temperature at T1 after one time step is 0.075, which is equivalent to 0.3 when rounded to one decimal place.

13. What does an Eigenvalue actually mean? Finish this sentence:An Eigenvalue, when subtracted from the diagonals of a matrix,:

Causes the matrix to be linearly dependent.

Causes the inverse to be impossible to solve.

Causes the determinant to be 0.

All of these statements are correct.

An eigenvalue, when subtracted from the diagonals of a matrix, can cause the matrix to be linearly dependent. This means that at least one of the rows or columns of the matrix can be expressed as a linear combination of the other rows or columns. Additionally, subtracting an eigenvalue from the diagonals can cause the inverse of the matrix to be impossible to solve, as it leads to a singularity in the matrix. Finally, subtracting an eigenvalue from the diagonals can cause the determinant of the matrix to be 0, indicating that the matrix is not invertible. Therefore, all of these statements are correct.

Explanation

An eigenvalue, when subtracted from the diagonals of a matrix, can cause the matrix to be linearly dependent. This means that at least one of the rows or columns of the matrix can be expressed as a linear combination of the other rows or columns. Additionally, subtracting an eigenvalue from the diagonals can cause the inverse of the matrix to be impossible to solve, as it leads to a singularity in the matrix. Finally, subtracting an eigenvalue from the diagonals can cause the determinant of the matrix to be 0, indicating that the matrix is not invertible. Therefore, all of these statements are correct.

14. For the function shown here, what is the value of a0 for its Fourier series?

4

0

None of these options are correct.

2

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Explanation

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15. Two bars of some material with length 0.5 initially have temperatures T1 = 5 (left bar) and T2 = 1 (right bar). The thermal diffusivity of the material is 1, and the ends are insulated. Which of the following correctly describes the transient solution?

This series

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Explanation

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16. Match the following equations with their correct finite difference descriptions.

Choice 1

Choice 2

Choice 3

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17. For the matrix shown, compute the eigenvectors and choose them from the list below. (Choose the most correct answers, you may choose more than one).

[1, -3]

[3, -9]

[1, 1]

[3, 3]

All of these are correct.

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Explanation

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18. Consider the function shown f(x). For the fourier series shown, what is the value of an?

This term

None of these options are correct.

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Explanation

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19. For the 1D Wave Equation, we may use the method of separation of variables by assuming u(x,t) = A(x)B(t). Using this method, which of the following options is correct for the wave equation?

A'' + kA = 0 (k = constant)

B' - CkB = 0 (C, k = constant)

B'' + CkB = 0 (C, k = constants)

A'' - kA = 0 (k = constant)

The correct answer is A'' - kA = 0 (k = constant). This equation represents the separation of variables method for the 1D wave equation. By assuming u(x,t) = A(x)B(t), we can separate the variables and obtain two ordinary differential equations: A'' + kA = 0 for the spatial part and B'' + CkB = 0 for the temporal part. Therefore, the correct option is A'' - kA = 0, which represents the spatial part of the equation.

Explanation

The correct answer is A'' - kA = 0 (k = constant). This equation represents the separation of variables method for the 1D wave equation. By assuming u(x,t) = A(x)B(t), we can separate the variables and obtain two ordinary differential equations: A'' + kA = 0 for the spatial part and B'' + CkB = 0 for the temporal part. Therefore, the correct option is A'' - kA = 0, which represents the spatial part of the equation.

20. Attached here is a code to compute the new temperature T_new using the FTCS method. This is a 1D problem with N cells. Someone believes there are some bugs in this code. Which lines are these bugs on?

There are no bugs in this code.

Line 6

Line 3

Line 2

Line 5

The code provided is using the Forward Time Central Space (FTCS) method to compute the new temperature. The correct answer states that there are bugs on Line 6 and Line 3. Without the actual code, it is not possible to provide a specific explanation for these lines. However, it can be inferred that there may be errors or mistakes in the implementation of the FTCS method on these lines, which could lead to incorrect calculations or undesired behavior.

Explanation

The code provided is using the Forward Time Central Space (FTCS) method to compute the new temperature. The correct answer states that there are bugs on Line 6 and Line 3. Without the actual code, it is not possible to provide a specific explanation for these lines. However, it can be inferred that there may be errors or mistakes in the implementation of the FTCS method on these lines, which could lead to incorrect calculations or undesired behavior.

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