Applied Calculus with Python Short Course

Lesson 1 • Setting Up the Python Environment

Install Python, Jupyter, NumPy, SymPy, and Matplotlib. This toolchain underpins every computational exercise in the course.

Lesson 2 • Python Syntax for Mathematical Work

Cover variables, loops, functions, and list comprehensions as applied to numeric computation. These constructs translate mathematical notation into executable code.

Lesson 3 • Algebraic Expressions with SymPy

Use SymPy symbols to build, simplify, and factor algebraic expressions. Symbolic manipulation is the gateway to analytic calculus in Python.

Lesson 4 • Numerical Arrays with NumPy

Create and operate on NumPy arrays to evaluate functions over intervals. Array-based computation is the numerical backbone of applied calculus workflows.

Lesson 1 • Continuity and Discontinuities

Define continuity formally and classify removable, jump, and infinite discontinuities. Python scripts detect and visualise discontinuities in real functions.

Lesson 2 • The Concept of a Limit

Introduce the informal and formal definitions of a limit using tables and graphs. Numerical exploration in Python builds intuition before symbolic methods.

Lesson 3 • Computing Limits with SymPy

Apply SymPy's limit function to evaluate analytic limits including indeterminate forms. Symbolic results are verified against numerical approximations.

Lesson 1 • Derivatives of Special Functions

Differentiate trigonometric, exponential, and logarithmic functions. SymPy handles these analytically while NumPy evaluates them numerically.

Lesson 2 • Basic Differentiation Rules

Apply power, constant, sum, and difference rules to differentiate polynomials. SymPy's diff function automates and verifies hand calculations.

Lesson 3 • Implicit and Numerical Differentiation

Differentiate implicitly defined relations and apply finite-difference schemes numerically. These techniques handle functions that cannot be solved explicitly.

Lesson 4 • Product, Quotient, and Chain Rules

Extend differentiation to products, quotients, and composite functions. These rules handle the majority of functions encountered in applied problems.

Lesson 1 • Optimisation

Find absolute and local extrema using critical points and boundary analysis. SymPy solves the derivative equations while SciPy's minimise validates results.

Lesson 2 • L'Hôpital's Rule and Asymptotic Analysis

Resolve indeterminate limits using L'Hôpital's rule and analyze asymptotic behaviour. SymPy automates repeated application and confirms graphical asymptotes.

Lesson 3 • Increasing, Decreasing, and Concavity

Use first and second derivatives to determine function behaviour and concavity. Python plots annotated with critical features reinforce analytical findings.

Lesson 4 • Related Rates Problems

Differentiate multi-variable equations with respect to time to link changing quantities. Python simulates dynamic scenarios and plots rate relationships.

Lesson 1 • The Fundamental Theorem of Calculus

State and apply both parts of the Fundamental Theorem to connect differentiation and integration. SymPy demonstrates the theorem symbolically on concrete examples.

Lesson 2 • Riemann Sums and the Definite Integral

Approximate area under a curve using left, right, and midpoint Riemann sums. Python loops and NumPy vectorization make convergence behaviour visible.

Lesson 3 • Basic Integration Rules and Formulas

Apply power, exponential, and trigonometric antiderivative rules. SymPy validates each result and highlights common pitfalls such as missing constants.

Lesson 4 • Integration by Substitution

Apply u-substitution to transform integrals into simpler forms. SymPy's substitution tools automate the technique for complex composite integrands.

Lesson 1 • Average Value and Mean Value Theorem

Compute the average value of a function and locate the guaranteed mean-value point. These concepts appear in signal processing and statistical averaging.

Lesson 2 • Volumes of Solids of Revolution

Apply disk, washer, and shell methods to compute volumes of rotated solids. Three-dimensional Matplotlib plots make the geometry concrete.

Lesson 3 • Arc Length and Surface Area

Derive and evaluate arc length and surface area integrals for smooth curves. SciPy's quad handles integrands that resist symbolic antiderivatives.

Lesson 4 • Accumulation and Net Change

Interpret the integral as accumulated change in velocity, population, and cost models. Python simulates time-series accumulation and compares to analytic results.

Lesson 1 • Partial Derivatives and Gradients

Compute partial derivatives symbolically with SymPy and assemble the gradient vector. The gradient's geometric meaning as steepest ascent is visualised with quiver plots.

Lesson 2 • Functions of Several Variables

Define and visualise scalar fields using contour plots and 3D surface plots. Understanding the geometry of multivariable functions precedes analytic techniques.

Lesson 3 • Directional Derivatives and the Chain Rule

Compute directional derivatives and apply the multivariable chain rule to composite functions. These tools model sensitivity along arbitrary directions in parameter space.

Lesson 4 • Optimisation of Multivariable Functions

Find critical points using the gradient and classify them with the second derivative test. SciPy's minimise extends optimisation to constrained and high-dimensional problems.

Lesson 1 • Numerical ODE Solvers

Implement Euler's method and use SciPy's solve_ivp for accurate numerical solutions. Comparing methods reveals trade-offs between simplicity and precision.

Lesson 2 • Separable and Linear First-Order ODEs

Solve separable ODEs by variable separation and linear ODEs with integrating factors. SymPy's dsolve automates both methods and returns general solutions.

Lesson 3 • Systems of Differential Equations

Model coupled dynamic systems such as predator-prey and compartmental models. SciPy solve_ivp handles vector-valued ODEs and phase-plane plots reveal behaviour.

Lesson 4 • Calculus in Applied Modelling

Integrate calculus tools into end-to-end applied models from data to differential equation to insight. Capstone mini-projects consolidate all course competencies.