Mathematics for Engineering (MAE101)
Vietnamese title: Toán cho ngành kỹ thuật
Credits: 3

Description:
Upon finishing the course, students must acquire:
1. the following knowledge: (ABET a1)
• The basic concepts of single variable calculus: limit, derivative, integral.
• Concepts of linear system of equations, matrix and their applications
• The concepts of vector spaces, basis and dimension, linear transformations and their applications.
• The range of application of calculus and algebra in science, technology, economics & finance...
2. the following skills: (ABET a2)
• Find limit, derivative and integral of functions of single variables;
• Find solutions to linear systems by using row-reduced echelon form, Cramer's Rule, inverse matrix
• Find determinants and inverse of a matrix
• Find linear transformations in R^2 and R^3.
• Find dimensions and bases of vector spaces .
• Apply the concepts of calculus and algebra in solving problems in science, technology, economics & finance, ...

Learning outcomes:
Describe basic properties and find the limit of a function LO1.1 Find the domain, range, zeros and analyze the symmetry of the graph; create new functions from old functions. LO1.2 Find the slope and write an equation of a line; describe how a function changes under graph translations. LO1.3 Compute the limit of a function using graphs and limit laws, including the Squeeze's Theorem. LO1.4 Find limits at infinity and infinity limits, and apply in finding asymptotes of a function. LO1.5 Identify where a function is continuous; use the Intermedia Value Theorem to show that an equation has a solution.
Calculate derivatives of a function LO2.1 Find the average rate of change; identify derivative as the limit of a difference quotient. LO2.2 Recognize derivative as the instantaneous velocity and the slope of tangent lines. LO2.3 Find derivatives using basic formulas and the sum, difference, product, quotient, chain rules.
Use derivatives to analyze a function and to solve applications LO3.1 Use implicit differentiation to find derivatives and to solve related rates problems. LO3.2 Find linear approximation of a function; calculate differentials and apply in calculating errors of approximation. LO3.3 Apply Rolle's Theorem and Mean Value Theorem to a function; find critical numbers and absolute extrema of a function. LO3.4 Identify local extrema, and describe the monotonicity of a function; solve optimization problems. LO3.5 Recognize acceleration as second derivative; determine inflection points and describe the concavity of a function. LO3.6 Use Newton method to approximate solutions of an equation.
Calculate integrals and find area under the graph of a function LO4.1 Use Riemann sums to approximate area; recognize area as a definite integral which is the limit of Riemann sums. LO4.2 Apply the Fundamental Theorems of Calculus to find derivatives of integrals, and to evaluate definite integrals. LO4.3 Recognize indefinite integrals as anti-derivatives; find integrals using basic formulas and the techniques of substitution and integration by parts. LO4.4 Use integration to solve applications and to compute the average value of a function. LO4.5 Approximate definite integrals using Midpoint, Trapezoidal and Simpson rules. LO4.6 Determine if an improper integral converges.
Solve a system of linear equations using Gaussian elimination LO5.1 Identify if a matrix is in echelon forms; carry a matrix into an echelon using row operations; calculate the rank of a matrix. LO5.2 Use echelon matrices to solve/describe the solution set of a linear system.
Perform basic operations on matrices and manipulate matrix transformations LO6.1 Perform addition, scalar multiplication and transpose of matrices; verify if a matrix is symmetric. LO6.2 Multiply two matrices of appropriate sizes; transform between a linear system and a matrix equation. LO6.3 Determine the matrix of a linear transformation; use properties of linear transformations in solving problems.
Find the determinant, inverse and diagonalization of a matrix LO7.1 Compute the cofactors and the determinant of a matrix; analyze determinants in relation with row/column operations. LO7.2 Check if a matrix is invertible; find the adjugate and the inverse of a matrix; use properties of inverses and determinants in solving problems. LO7.3 Solve a system of linear equations using inverse matrices or Cramer's rule. LO7.4 Find the characteristic polynomial, eigenvalues and eigenvectors to diagonalize a matrix; check if a matrix is diagonalizable.
Manipulate vector operations and linear transformations in R^3. LO8.1 Manipulate vector operations; find equations of lines. LO8.2 Compute the dot product and the angle between two vectors; find the projection of a vector on another vector and manipulate equations of planes. LO8.3 Compute the cross product; analyze areas and volumns in relation with dot and cross products. LO8.4 Determine linear transformations in R^3 and discuss applications in computer graphic.
Find the dimension and bases of a subspace LO9.1 Determine if a subset of vectors is a subspace; describe the subspace spanned by a set of vectors. LO9.2 Check if a set of vectors are independent; determine if a set of vectors form a basis. LO9.3 Find a basis and the dimension of a subspace; analyze the rank in relation with the dimension of the column, row and the null space of a matrix. LO9.4 Compute dot product and length of vectors; check if a set of vectors are orthogonal or orthonormal; write a vector as a linear combination of an orthogonal basis.

Student tasks:
Students must attend at least 80% of contact hours in order to be accepted to the final examination.

Tools:

Materials:
description: Calculus 1
note: https://openstax.org/details/books/calculus-volume-1
description: Calculus 2
note: https://openstax.org/details/books/calculus-volume-2
description: powerpoint in CD + Powerpoint prepared by previous lecturers (PP).
description: Student Video Skillbuilder in CD (Video).
description: Animation in CD (Animation).
description: Computerized tools (for example Maxima http://maxima.sourceforge.net/) (CT)
description: Essential Calculus
author: James Stewart
publisher: Thomson Brooks/Cole
isbn: 978-0-495-10860-3
description: Calculus for engineers
author: Donald Trim
publisher: Prentice Hall,
isbn: 0-13-085603-7
description: Calculus
author: Stanley I. Grossman
isbn: 0-03-096420-2
description: Linear Algebra with applications
author: W.Keith Nicholson,
publisher: Mc Graw Hill Education
isbn: 987-1-259-07299-4.
description: Linear Algebra with Applications
author: W. Keith Nicholson
publisher: Lyryx Online Resources
published_date: 2021
edition: 2021-A
note: https://lyryx.com/linear-algebra-applications/
description: Coursera - FPT University Private Authored Courses: Linear transformation with python (Guided Project)
note: Optional practice

Assessments:
category: Assignments/Exercises
type: on-going
weight: 30.0
duration: 3 hours
question_type: LO1, LO2, LO3, LO4, LO5, LO6, LO7, LO8, LO9
no_question: OP1: Open // OP2: (For Constructivism Approach only): Follow lecturer's proposal
knowledge_and_skill: OP1: At least 3/chapter // OP2: (For Constructivism Approach only): Follow lecturer's proposal
grading_guide: Assignment 1 covers Chapters 1, 2, 3, 4 (Vol.1), 1, 3 (Vol.2) - Calculus ; Assignment 2 covers Chapters 1, 2, 3 (Linear algebra) ; Assignment 3 covers Chapters 4, 5 (Linear algebra)
note: OP1: Each assignment is divided into two parts: the take-home component counts for no more than 50% of the total grade, and the in-class component counts for at least 50% (this may take the form of a presentation, essay, problem-solving, etc.).
OP2: (For Constructivism Approach only): Follow lecturer's proposal
category: Progress Test
type: on-going
weight: 30.0
duration: OP1: Test 1: 40'; Test 2&3: 30' // OP2: (For Constructivism Approach only): Follow lecturer's proposal
question_type: LO1, LO2, LO3, LO4, LO5, LO6, LO7, LO8, LO9
no_question: OP1: MC or a suitable format // OP2: (For Constructivism Approach only): Follow lecturer's proposal
knowledge_and_skill: OP1: Test 1: 20 Test 2&3: 15+ At least 5*n questions where n is the number of Chapters covered in the test.+ Structure:• level 1 (knowledge & comprehension): 20%; • level 2 (application & analysis): 60%; • level 3 (synthesis & evaluation): 20% // OP2: (For Constructivism Approach only): Follow lecturer's proposal
grading_guide: Test 1 covers Chapters 1, 2, 3, 4 (Vol.1), 1, 3 (Vol.2) - Calculus ; Test 2 covers Chapters 1, 2, 3 (Linear algebra) ; Test 3 covers Chapters 4, 5 (Linear algebra)
note: OP1: by instructor, and by computer
OP2: (For Constructivism Approach only): Follow lecturer's proposal
category: Final Exam
type: Final exam
weight: 40.0
duration: 60'
no_question: Computer gradable
knowledge_and_skill: 50
Structure:
• level 1: 10%
• level 2: 80%
• level 3: 10%
grading_guide: All studied chapters. Each studied Chapter contributes 2-8 questions, depending on the number of topics in the Chapter.
note: by Exam Board

Schedule:
session: 0 Introduction.
1.1 Review of Functions.
1.2 Basic Classes of Functions
learning_outcomes: ITU
session: 2.1 A Preview of Calculus.
2.2 The Limit of a Functions
learning_outcomes: TU
session: 2.3 The Limit Laws
2.4 Continuity
learning_outcomes: TU
session: 3.1 Defining the Derivative
3.2 The Derivative as a Function
learning_outcomes: IT
session: 3.3 Differentiation Rule.
3.4 Derivatives as Rates of Change
3.5 Derivatives of Trigonometric Functions
learning_outcomes: TU
session: 3.6 The Chain Rule. 3.8 Implicit Differentiation
learning_outcomes: TU
session: 3.9 Derivatives of Exponential and Logarithmic Functions
learning_outcomes: TU
session: 4.1 Related Rates
learning_outcomes: TU
session: 4.2 Linear Approximations and Differentials
learning_outcomes: TU
session: 4.3 Maxima and Minima
4.4 The Mean Value Theorem
learning_outcomes: TU
session: 4.5 Derivatives and the Shape of a Graph
4.6 Limits at Infinity and Asymptotes
learning_outcomes: TU
session: 4.7 Applied Optimization Problems
learning_outcomes: TU
session: 4.9 Newton's Method
learning_outcomes: TU
session: 4.10 Antiderivatives
learning_outcomes: TU
session: 1.1 Approximating Areas
learning_outcomes: IT
session: 1.2 The Definite Integral
1.3 The Fundamental Theorem of Calculus
learning_outcomes: IT
session: 1.4 Integration Formulas and the Net Change Theorem
learning_outcomes: TU
session: 1.5 Substitution
learning_outcomes: TU
session: 3.1 Integration by Parts 3.6 Numerical Integration
learning_outcomes: TU
session: 3.7 Improper integrals
learning_outcomes: TU
session: Review
learning_outcomes: U
session: Progress Test 1
learning_outcomes: U
session: 1.1 Solutions of Systems of Linear Equations
learning_outcomes: IT
session: 1.1 Elementary Operations
learning_outcomes: TU
session: 1.2 Gaussian Elimination
learning_outcomes: TU
session: 1.3 Homogeneous Equations
learning_outcomes: TU
session: 2.1 Matrix Addition, Scalar Multiplication, and Transposition
learning_outcomes: TU
session: 2.2 Matrix-Vector Multiplication
learning_outcomes: TU
session: 2.2 Matrix Transformations
learning_outcomes: TU
session: 2.3 Matrix multiplication
learning_outcomes: TU
session: 2.4 Matrix Inverses
learning_outcomes: TU
session: 2.4 Properties of Inverses
learning_outcomes: TU
session: 2.6 Linear Transformations
learning_outcomes: TU
session: 3.1 The Cofactor Expansion
learning_outcomes: TU
session: 3.1 Properties of Determinants
learning_outcomes: TU
session: 3.2 Determinants and Matrix Inverses
learning_outcomes: TU
session: 3.2 Cramer's Rule
learning_outcomes: TU
session: 3.3 Eigenvalues and Eigenvectors
learning_outcomes: TU
session: 3.3 Diagonalization
learning_outcomes: TU
session: Review
learning_outcomes: U
session: Progress Test 2
learning_outcomes: U
session: 4.1 Vectors
learning_outcomes: TU
session: 4.1 Lines in Space
learning_outcomes: TU
session: 4.2 Projections
learning_outcomes: TU
session: 4.2 Planes
learning_outcomes: TU
session: 4.2 Cross Product
learning_outcomes: TU
session: 4.3 More on the Cross Product
learning_outcomes: TU
session: 4.4 Linear Operations on R^3
learning_outcomes: TU
session: 4.4 Transformations of Areas and Volumes
learning_outcomes: TU
session: 4.5 An application in Computer Graphics
learning_outcomes: IT
session: 5.1 Subspaces of R^n
learning_outcomes: TU
session: 5.1 Spanning Sets
learning_outcomes: TU
session: 5.2 Linear Independence
learning_outcomes: TU
session: 5.2 Dimension
learning_outcomes: TU
session: 5.3 Dot Product, Length, and Distance
learning_outcomes: TU
session: 5.3 Orthogonal Sets and the Expansion Theorem
learning_outcomes: TU
session: 5.4 Rank of a Matrix
learning_outcomes: TU
session: 5.4 Rank, Nullity, and Linear Systems
learning_outcomes: TU
session: Review
learning_outcomes: U
session: Progress Test 3
learning_outcomes: U