MAT3701 Exam pack 2024(Linear Algebra III) Questions and answers

MAT3701 EXAM
PACK 2024

QUESTIONS AND
ANSWERS
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CONTENTS Page
MODULE 4
LEARNING UNIT 1: LAPLACE TRANSFORMS
1.1 THE LAPLACE TRANSFORM 2
1.1.1 Definition 2
1.2 PROPERTIES OF THE LAPLACE TRANSFORM 4
1.2.1 Linearity property 4
1.2.2 First translation or shifting property 5
1.2.3 Second translation or shifting property 6
1.2.4 Change of scale property 6
1.2.5 Multiplication by t n 7
1.2.6 Division by t 8
1.3 LAPLACE TRANSFORM OF DERIVATIVES 10
1.4 LAPLACE TRANSFORM OF INTEGRALS 11
1.5 TABLE OF STANDARD LAPLACE TRANSFORMS 13
1.6 POST-TEST 14


LEARNING UNIT 2: INVERSE LAPLACE TRANSFORMS
2.1 THE INVERSE LAPLACE TRANSFORM 16
2.2 PROPERTIES OF THE INVERSE LAPLACE TRANSFORM 18
2.2.1 Linearity property 18
2.2.2 First translation or shifting property 19
2.2.3 Second translation or shifting property 19
2.2.4 Change of scale 20
2.2.5 Multiplication of s n 20
2.2.6 Division by s 21
2.2.7 The Convolution Property 22
2.3 INVERSE LAPLACE TRANSFORMS OF DERIVATIVES 23
2.4 INVERSE LAPLACE TRANSFORMS OF INTEGRALS 24
2.5 POST-TEST 34




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LEARNING UNIT 3: SPECIAL FUNCTIONS
3.1 THE HEAVISIDE UNIT STEP FUNCTION
37
3.2 LAPLACE TRANSFORM OF THE HEAVISIDE UNIT STEP FUNCTION
38
3.3 THE UNIT IMPULSE FUNCTION 41
3.4 LAPLACE TRANSFORM OF THE UNIT IMPULSE FUNCTION
43
3.5 TABLE OF LAPLACE TRANSFORMS OF SPECIAL FUNCTIONS 47
3.6 POST-TEST 48


LEARNING UNIT 4: SOLVING DIFFERENTIAL EQUATIONS USING LAPLACE
TRANSFORMS
4.1 SOLUTION OF DIFFERENTIAL EQUATIONS USING
LAPLACE TRANSFORMS 50
4.2 POST-TEST 61


LEARNING UNIT 5: SOLVING SIMULTANEOUS DIFFERENTIAL EQUATIONS
USING LAPLACE TRANSFORMS
5.1 SOLUTION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS USING
LAPLACE TRANSFORMS 63
5.2 POST-TEST 64


LEARNING UNIT 6: PRACTICAL APPLICATIONS
6.1 PRACTICAL APPLICATIONS OF THE LAPLACE TRANSFORM
66
6.1.1 Application to electric circuits 66
6.1.2 Application to vibrating systems 70
6.1.3 Application to beams 74
6.2 POST-TEST 83




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MODULE 5
LEARNING UNIT 1: GAUSS ELIMINATION
1.1 INTRODUCTION 85
1.2 GAUSS ELIMINATION 85
1.3 POST-TEST 89


LEARNING UNIT 2: EIGENVALUES AND EIGENVECTORS
2.1 INTRODUCTION 91
2.2 EIGENVALUES 91
2.3 EIGENVECTORS 92
2.4 POST-TEST 96


MODULE 6
LEARNING UNIT 1: FOURIER SERIES FOR PERIODIC FUNCTIONS OF
PERIOD 2π
1.1 INTRODUCTION 98
1.2 PERIODIC FUNCTIONS 98
1.3 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD 2π 99
1.4 POST-TEST 103


LEARNING UNIT 2: FOURIER SERIES FOR NON-PERIODIC FUNCTIONS OVER
A RANGE OF 2π
2.1 INTRODUCTION 105
2.2 POST-TEST 105


LEARNING UNIT 3: EVEN AND ODD FUNCTIONS AND HALF-RANGE
FOURIER SERIES
3.1 INTRODUCTION 107
3.2 REVISION: EVEN AND ODD FUNCTIONS 107
3.3 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD 2π 109
3.4 POST-TEST 111




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