Calculus I (Track B)
Course Syllabus
1
Course Information
Course title:
Level:
Credits:
Teaching hours:
Course type:
Prerequisites:
Course coordinator:
Instructors:
2
Calculus I (Track B)
Undergraduate, first semester of the Calculus B sequence
5
80
Foundational calculus course
High School Mathematics in China
Yunzhi Zou
Yunzhi Zou, Bai Lin
Course Description
This course provides a systematic introduction to single-variable calculus and its applications.
It begins with functions and elementary models, then develops limits and continuity as the conceptual foundation of differentiation and integration. The course studies derivatives, standard
differentiation techniques, and applications such as curve sketching, optimization, related rates,
and linear approximation. It then introduces definite and indefinite integrals, the Fundamental
Theorem of Calculus, techniques of integration, improper integrals, and geometric and physical
applications of integration.
The final part of the course broadens the scope of single-variable calculus through elementary
differential equations, parametric and polar curves, and an introduction to sequences, infinite
series, power series, and Taylor series. Throughout the semester, emphasis is placed on the connection between graphical, numerical, algebraic, and analytic viewpoints. Students are expected
not only to carry out standard calculations accurately, but also to understand the main ideas
that justify the methods being used and to apply calculus to elementary modelling problems.
3
Course Aims
The course aims to help students:
• develop a coherent understanding of the central ideas of single-variable calculus;
• use limits and continuity to describe local and asymptotic behaviour of functions;
• understand derivatives as rates of change and integrals as accumulated quantities;
• become fluent in the principal techniques of differentiation and integration;
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• use calculus to analyse graphs, extrema, geometric quantities, and simple models;
• gain an introductory understanding of ordinary differential equations, parametric curves,
polar curves, sequences, infinite series, power series, and Taylor series;
• connect computational methods with their conceptual and geometric interpretations;
• prepare for further study in multivariable calculus and other quantitative subjects.
4
Learning Outcomes
By the end of the course, students should be able to:
1. work with elementary functions and interpret functions through formulas, graphs, tables,
and models;
2. evaluate limits by algebraic, graphical, and standard asymptotic methods;
3. determine continuity and describe the behaviour of functions near finite points and at infinity;
4. compute derivatives using the standard differentiation rules, including the chain rule and
implicit differentiation;
5. use derivatives to study monotonicity, extrema, concavity, approximation, related rates, and
optimization;
6. compute definite and indefinite integrals and apply the Fundamental Theorem of Calculus;
7. use substitution, integration by parts, partial fractions, and other standard integration techniques;
8. analyse improper integrals and use numerical or approximate integration methods in standard examples;
9. compute areas, volumes, work, arc lengths, and surface areas using integration;
10. solve elementary separable and first-order linear differential equations and interpret simple
population models;
11. work with parametric and polar curves and compute their basic geometric quantities;
12. determine convergence or divergence of elementary sequences and series using standard tests,
determine intervals of convergence of power series, and construct basic Taylor and Maclaurin
series.
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Course Content
5.1
Functions and Elementary Models
Main topics. Functions and their representations, domains and ranges, transformations of
graphs, elementary functions, inverse functions, exponential and logarithmic functions, and simple mathematical models.
Important definitions. Function, domain, range, graph, inverse function, composition, polynomial function, rational function, trigonometric function, exponential function, and logarithmic
function.
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Important ideas and results.
• A function may be represented algebraically, graphically, numerically, or verbally, and these
representations should be interpreted consistently.
• Composition and inverse functions provide basic tools for constructing and analysing models.
• Exponential and logarithmic functions are inverse to one another and play a central role in
growth and decay models.
5.2
Limits and Continuity
Main topics. Limits of functions, one-sided limits, algebra of limits, limits involving infinity,
infinite limits, continuity, and evaluation of indeterminate forms in standard examples.
Important definitions. Limit, one-sided limit, infinite limit, limit at infinity, continuous function,
and asymptote.
Important results.
• Algebraic rules for limits.
• Squeeze theorem.
• Intermediate Value Theorem for continuous functions.
• Standard limits underlying the derivatives of elementary functions.
• L’Hospital’s rule for appropriate indeterminate forms.
5.3
Differentiation
Main topics. Derivative as a rate of change, tangent lines, differentiation rules, derivatives of
elementary functions, chain rule, implicit differentiation, higher derivatives, and linearization.
Important definitions. Derivative, differentiability, tangent line, higher derivative, implicit
derivative, differential, and linear approximation.
Important results.
• Product and quotient rules.
• Chain rule:
d
f (g(x)) = f ′ (g(x))g ′ (x).
dx
• Differentiability implies continuity.
• Linear approximation near a point:
f (x) ≈ f (a) + f ′ (a)(x − a).
5.4
Applications of Derivatives
Main topics. Related rates, local and global extrema, the Mean Value Theorem, monotonicity,
concavity, curve sketching, optimization, Newton’s method, and asymptotic behaviour.
Important definitions. Critical point, local extremum, global extremum, increasing function,
decreasing function, concavity, inflection point, and stationary point.
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Important results.
• Fermat’s theorem on interior extrema.
• Rolle’s theorem and the Mean Value Theorem:
f ′ (c) =
f (b) − f (a)
b−a
for some appropriate c ∈ (a, b).
• First- and second-derivative tests for extrema in standard situations.
• Newton iteration:
xn+1 = xn −
5.5
f (xn )
.
f ′ (xn )
Definite Integrals and the Fundamental Theorem of Calculus
Main topics. Accumulated quantities, Riemann sums, definite integrals, antiderivatives, the
Fundamental Theorem of Calculus, and substitution.
Important definitions. Riemann sum, definite integral, antiderivative, indefinite integral, and
net area.
Important results.
• Basic linearity and interval-additivity properties of the definite integral.
• Fundamental Theorem of Calculus:
d
dx
∫︂ x
f (t) dt = f (x)
a
for continuous f , together with
∫︂ b
f (x) dx = F (b) − F (a)
a
whenever F ′ = f .
• Substitution rule for integrals.
5.6
Techniques and Applications of Integration
Main topics. Areas between curves, volumes, work, integration by parts, partial fractions,
improper integrals, numerical approximation, arc length, and surface area.
Important definitions. Improper integral, arc length, surface of revolution, and work as an
accumulated quantity.
Important results.
• Integration by parts:
∫︂
u dv = uv −
∫︂
v du.
• Decomposition of suitable rational functions into partial fractions.
• Improper integrals are interpreted through limits of proper integrals.
• Standard integral formulas for areas, volumes, arc lengths, and surfaces of revolution.
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5.7
Elementary Differential Equations
Main topics. Introduction to ordinary differential equations, Euler’s method, separable equations, population growth models, first-order linear equations, and the predator–prey model as
an example of a dynamical system.
Important definitions. Differential equation, solution, initial condition, separable equation, firstorder linear equation, equilibrium solution, and numerical approximation.
Important ideas and results.
• Separable equations can be reduced to integration when the variables can be separated.
• First-order linear equations have the form
y ′ + p(x)y = q(x),
and can be treated using an integrating factor.
• Euler’s method gives a basic numerical approximation to an initial-value problem.
5.8
Parametric and Polar Curves
Main topics. Parametric equations, derivatives of parametric curves, polar coordinates, polar
curves, and geometric interpretation.
Important definitions. Parametric curve, parameter, polar coordinate, radial coordinate, and
polar graph.
Important results.
• For a parametrized curve x = x(t), y = y(t),
dy
dy/dt
=
dx
dx/dt
whenever dx/dt ̸= 0.
• Cartesian and polar coordinates are related by
x = r cos θ,
5.9
y = r sin θ.
Sequences, Infinite Series, Power Series, and Taylor Series
Main topics. Sequences, convergence, infinite series, partial sums, standard convergence tests,
power series, radius and interval of convergence, Taylor polynomials, Taylor series, and Maclaurin series.
Important definitions. Sequence, convergent sequence, divergent sequence, infinite series, partial
sum, absolute convergence, conditional convergence, power series, radius of convergence, interval
of convergence, Taylor polynomial, Taylor series, and Maclaurin series.
Important ideas and results.
• A series
∑︁∞
n=1 an converges precisely when its sequence of partial sums converges.
• A necessary condition for convergence is an → 0.
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• Standard comparison, ratio, integral, and alternating-series tests are used in representative
examples.
• A power series centered at a has the form
∞
∑︂
cn (x − a)n ,
n=0
and converges on an interval determined by its radius of convergence, with the endpoints
considered separately.
• For a sufficiently differentiable function, the Taylor polynomial of degree N centered at a is
N
∑︂
f (n) (a)
TN (x) =
n=0
n!
(x − a)n .
• When the Taylor series represents the function, one has
f (x) =
∞
∑︂
f (n) (a)
n!
n=0
(x − a)n ,
with the Maclaurin series corresponding to the special case a = 0.
• Standard Taylor and Maclaurin series for functions such as ex , sin x, cos x, and (1 − x)−1
are used to obtain local approximations and to generate related power-series expansions.
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Assessment
The assessment scheme is:
• learning process (attendance, participation, group work, and related activities): 10%;
• assignments and quizzes, including online and offline work: 30%;
• two closed-book midterm examinations: 30% in total;
• one closed-book final examination: 30%.
Assessment should reward both accurate computation and clear mathematical reasoning. Students are expected to show the main steps of their work, use notation correctly, and explain the
interpretation of their answers when appropriate.
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Student Expectations
Students are encouraged to:
• review definitions, formulas, and worked examples regularly rather than only before examinations;
• practise both routine computational exercises and problems requiring interpretation or modelling;
• sketch graphs and geometric configurations whenever they help clarify a problem;
• check whether an answer is reasonable by considering signs, units, scales, and limiting cases;
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• ask questions whenever a concept, notation, or step in an argument is unclear;
• work steadily, since later topics depend strongly on earlier ideas about functions, limits,
derivatives, and integrals.
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Textbook
James Stewart, Calculus: Early Transcendentals, Ninth Edition, Metric Version, Cengage, 2019.
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Linear Algebra I (Track B)
Course Syllabus
1
Course Information
Course title:
Level:
Credits:
Teaching hours:
Course type:
Prerequisites:
Course coordinator:
Instructor:
2
Linear Algebra I (Track B)
Undergraduate, first semester of the Linear Algebra B sequence
2
32
Foundational linear algebra course
High school mathematics in China
Yunzhi Zou
Yunzhi Zou, Bai Lin
Course Description
This course is the first part of the Linear Algebra B sequence. It introduces the basic language,
computational methods, and geometric ideas of linear algebra through systems of linear equations, matrices, linear transformations, vector subspaces of Rn , bases, dimension, coordinates,
similarity of matrices, and determinants.
The course begins with systems of linear equations and Gaussian–Jordan elimination, then
develops matrix algebra and the interpretation of matrices as linear transformations. Particular
attention is paid to invertibility, composition of linear transformations, kernels and images, and
the geometric meaning of linear maps. The second half of the course introduces subspaces, linear
independence, spanning sets, bases, dimension, coordinate representations, changes of basis, and
similarity of matrices. The course concludes with determinants, their algebraic properties, their
geometric interpretation, and Cramer’s rule.
Throughout the course, computational techniques are developed together with conceptual understanding. Students are expected to move comfortably between systems of equations, matrix
calculations, vector-space language, and geometric interpretations, and to explain why the principal algorithms and structural results work in standard finite-dimensional settings.
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Course Aims
The course aims to help students:
• develop fluency in solving systems of linear equations by systematic elimination;
• understand matrices both as computational objects and as representations of linear transformations;
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• connect matrix operations with composition and inversion of linear maps;
• understand subspaces of Rn , linear combinations, spanning, linear independence, bases, and
dimension;
• use kernels, images, and coordinates to describe linear transformations and subspaces;
• understand change of basis and similarity as different matrix representations of the same
linear transformation;
• understand determinants algebraically and geometrically;
• develop the ability to interpret linear-algebra calculations rather than treating them as isolated algorithms;
• prepare for the study of eigenvalues, orthogonality, least squares, and abstract vector spaces
in Linear Algebra II.
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Learning Outcomes
By the end of the course, students should be able to:
1. represent a system of linear equations by an augmented matrix and solve it using Gaussian
or Gauss–Jordan elimination;
2. determine whether a linear system has no solution, a unique solution, or infinitely many
solutions;
3. perform basic matrix operations and compute matrix products and inverses when they exist;
4. interpret a matrix as a linear transformation of Rn and describe elementary geometric transformations by matrices;
5. determine the kernel and image of a linear transformation in standard finite-dimensional
examples;
6. recognize and verify subspaces of Rn ;
7. test linear independence and determine spanning sets and bases;
8. compute the dimension of a subspace and express vectors in different coordinate systems;
9. recognize similar matrices and interpret the relation B = P −1 AP as a change of basis for a
linear transformation;
10. compute determinants and use their main algebraic properties efficiently;
11. interpret determinants in terms of invertibility, orientation, area, and volume;
12. use Cramer’s rule in appropriate small-dimensional examples;
13. explain the main ideas behind the algorithms and structural results introduced in the course.
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Course Content
5.1
Systems of Linear Equations and Elimination
Main topics. Linear equations, systems of linear equations, augmented matrices, elementary row
operations, echelon forms, Gaussian elimination, Gauss–Jordan elimination, pivot variables, free
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variables, and solution sets.
Important definitions. Linear equation, linear system, augmented matrix, elementary row operation, row-echelon form, reduced row-echelon form, pivot position, free variable, consistent
system, and homogeneous system.
Important ideas and results.
• Elementary row operations preserve the solution set of a linear system.
• Every matrix can be reduced to a unique reduced row-echelon form.
• A consistent linear system may have either a unique solution or infinitely many solutions,
depending on the presence of free variables.
• A homogeneous system
Ax = 0
always has the trivial solution, and nontrivial solutions occur precisely when free variables
are present.
5.2
Matrices, Vectors, and Matrix Algebra
Main topics. Vectors in Rn , linear combinations, matrix addition and scalar multiplication,
matrix multiplication, identity matrices, transpose, inverse matrices, and algebraic manipulation
of matrix equations.
Important definitions. Vector, matrix, row vector, column vector, linear combination, matrix
product, identity matrix, transpose, invertible matrix, and inverse matrix.
Important ideas and results.
• Matrix multiplication is determined by row–column products:
(AB)ij =
∑︂
aik bkj .
k
• Matrix multiplication is associative and distributive but is generally not commutative.
• If A and B are invertible, then
(AB)−1 = B −1 A−1 .
• Matrix equations provide a compact representation of systems of linear equations and compositions of linear operations.
5.3
Linear Transformations and Geometry
Main topics. Linear transformations from Rn to Rm , matrix representations, image and kernel,
geometric transformations in the plane and space, composition, and inverse transformations.
Important definitions. Linear transformation, standard matrix, image, range, kernel, one-to-one
linear transformation, onto linear transformation, composition, and inverse transformation.
Important ideas and results.
• A map T : Rn → Rm is linear when
T (u + v) = T (u) + T (v),
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T (cu) = cT (u).
• Every linear transformation T : Rn → Rm is represented by a matrix A such that
T (x) = Ax.
• The kernel and image of a linear transformation are subspaces.
• Composition of linear transformations corresponds to multiplication of their representing
matrices.
• A linear transformation is invertible exactly when its representing matrix is invertible.
5.4
Invertibility and Matrix Products
Main topics. Matrix products, inverse matrices, invertible linear transformations, solving systems by inverses, and relations among invertibility, kernels, and uniqueness of solutions.
Important definitions. Invertible matrix, inverse transformation, nonsingular matrix, and singular matrix.
Important ideas and results.
• If A is invertible, then the system
Ax = b
has the unique solution x = A−1 b for every b.
• For a square matrix, invertibility is equivalent to the homogeneous equation Ax = 0 having
only the trivial solution.
• Invertibility can be tested by row reduction to the identity matrix.
5.5
Subspaces, Linear Independence, Bases, and Similarity
Main topics. Subspaces of Rn , spanning sets, linear dependence and independence, bases, column spaces, null spaces, construction of bases from generating families, matrix representations
relative to a basis, change of basis, and similarity of matrices.
Important definitions. Subspace, span, generating set, linearly dependent family, linearly independent family, basis, column space, null space, change-of-basis matrix, and similar matrices.
Important ideas and results.
• The span of a family of vectors is the smallest subspace containing those vectors.
• Pivot information obtained by row reduction can be used to identify linear dependence and
construct bases in standard matrix problems.
• A basis combines the two properties of spanning and linear independence.
• Every vector in a vector space has a unique representation as a linear combination of the
vectors of a basis.
• The matrix representing a linear transformation depends on the choice of basis. If A and B
represent the same linear transformation in two different bases, then there is an invertible
change-of-basis matrix P such that
B = P −1 AP.
In this case, A and B are called similar matrices.
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• Similarity expresses the fact that two matrices may describe the same linear transformation
in different coordinate systems. Similar matrices have, in particular, the same determinant
and the same invertibility properties; further invariants of similarity will be studied in Linear
Algebra II.
5.6
Dimension, Coordinates, and Change of Basis
Main topics. Dimension of a subspace, coordinates relative to a basis, change of coordinates
in elementary examples, change-of-basis matrices, similarity, and the relation between kernels,
images, and dimension.
Important definitions. Dimension, coordinate vector, ordered basis, rank, and nullity.
Important ideas and results.
• All bases of a finite-dimensional vector space have the same number of vectors; this number
is its dimension.
• For a linear transformation T : Rn → Rm ,
dim(ker T ) + dim(Im T ) = n.
• Coordinates allow an abstract vector relation to be converted into a matrix calculation once
a basis has been chosen.
• Changing the basis changes the representing matrix by similarity: if P converts coordinates
from one basis to another, then the corresponding matrices are related by B = P −1 AP .
5.7
Determinants
Main topics. Definition and computation of determinants, cofactor expansion, effects of elementary row operations, determinant of a product, invertibility, geometric interpretation, and
Cramer’s rule.
Important definitions. Determinant, minor, cofactor, orientation, signed area, signed volume,
and Cramer’s rule.
Important results.
• A square matrix is invertible if and only if
det(A) ̸= 0.
• Determinants are multiplicative:
det(AB) = det(A) det(B).
• Elementary row operations have predictable effects on the determinant and can be used to
compute it efficiently.
• In dimensions two and three, | det(A)| gives the area or volume scaling factor of the corresponding linear transformation.
• For an invertible square system, Cramer’s rule expresses each component of the solution as
a ratio of determinants.
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6
Assessment
The assessment scheme is:
• learning process (attendance, participation, group work, and related activities): 10%;
• assignments and quizzes, including online and offline work: 30%;
• two closed-book midterm examinations: 30% in total;
• one closed-book final examination: 30%.
Assessment should reward both accurate computation and clear mathematical reasoning. Students are expected to show the essential steps of elimination and matrix calculations, use notation
correctly, and explain geometric or structural interpretations when appropriate.
7
Student Expectations
Students are encouraged to:
• practise row reduction and matrix calculations regularly until the basic algorithms become
reliable;
• connect calculations with the underlying geometric and structural meaning;
• draw simple pictures for transformations in R2 and R3 whenever they clarify an idea;
• distinguish carefully among spanning, independence, basis, dimension, image, and kernel;
• check computations by substitution, dimension counts, or inverse operations whenever possible;
• work steadily, since later topics depend strongly on the language and techniques developed
earlier in the course.
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Textbook
Otto Bretscher, Linear Algebra with Applications, International Edition, Pearson, 2014. ISBN
9780321796974.
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