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Calculus and Optimisation: Python Implementation of Gradient Descent, Derivatives and Polynomial Regression

This Calculus and Optimisation technical exercise demonstrates the implementation of mathematical and machine-learning concepts using Python. The document is organised as a set of code fragments that must logically work together to perform symbolic calculus, numerical optimisation, physical modelling and polynomial regression. The material uses libraries including NumPy, SymPy, Matplotlib and Scikit-learn, linking mathematical theory with practical computational implementation. Jigsaw_Puzzle_Calculus_Original… A central component is the implementation of gradient descent for linear regression. The supplied function calculates predictions, evaluates prediction errors, computes gradients and updates model parameters iteratively using a configurable learning rate. Mean squared error is recorded during optimisation, and a tolerance-based stopping condition is used to terminate the algorithm when successive cost values change by only a very small amount. Jigsaw_Puzzle_Calculus_Original… Synthetic linear-regression data is generated using a fixed NumPy random seed to support reproducible experimentation. A bias column is added to the feature matrix, initial parameter values are randomly generated, and the custom gradient-descent function is then executed to estimate the intercept and slope of the relationship. Jigsaw_Puzzle_Calculus_Original… The document also demonstrates symbolic differentiation and optimisation using SymPy. A cubic polynomial is defined and differentiated to obtain both its first and second derivatives. Critical points are identified by solving where the first derivative equals zero, while the second derivative is evaluated at each critical point to determine whether the point represents a local minimum, local maximum or saddle point. Jigsaw_Puzzle_Calculus_Original… Jigsaw_Puzzle_Calculus_Original… A further section applies mathematical formulas to projectile motion. Using a specified initial velocity, launch angle and gravitational acceleration, the code calculates both maximum projectile height and horizontal range. This component illustrates how calculus-related mathematical relationships can be translated directly into executable computational models. Jigsaw_Puzzle_Calculus_Original… The final major element explores polynomial regression. Synthetic nonlinear data is generated from a cosine-based function with added random noise. Scikit-learn pipelines are then used to compare polynomial models of degrees 1, 4 and 15. The models are fitted to the synthetic dataset and visualised against the underlying true function, allowing comparison of model complexity and illustrating concepts such as underfitting and overfitting. Jigsaw_Puzzle_Calculus_Original… Jigsaw_Puzzle_Calculus_Original… Overall, the exercise integrates calculus, optimisation, numerical methods and machine-learning modelling through Python. It provides practical experience with differentiation, critical-point analysis, iterative optimisation, mathematical simulation, regression modelling and visualisation. Important: because this file does not identify a university, assessment weighting, academic level, reference style or required word count, those fields should remain Not specified / Not applicable rather than being invented.

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Machine Learning / Data Science

Linear Regression and Stability of the Moore–Penrose Pseudoinverse Using Python

This machine learning practical and assessment activity develops an understanding of linear regression, Ordinary Least Squares and the Moore–Penrose pseudoinverse using Python. The work progresses from generating synthetic regression datasets to implementing regression algorithms manually, applying established machine learning libraries, analysing real datasets and evaluating the stability of estimated regression coefficients. The laboratory component begins with the generation of synthetic linear regression data using NumPy, including explanatory variables, random noise and an outcome variable. Students then implement simple linear regression without relying on machine learning libraries, using the least-squares solution to estimate the intercept and slope. The resulting observations and fitted regression line are visualised using Matplotlib. The work is subsequently extended to multiple linear regression, where several independent variables are used and coefficients are first calculated manually before the same problem is solved using Scikit-learn. The laboratory also introduces application of regression to the Scikit-learn Diabetes dataset, including feature and target standardisation, model fitting, prediction, correlation analysis and interpretation of regression coefficients. It also highlights the importance of residual analysis when assessing whether a linear model is appropriate. The associated weekly challenge focuses on the stability of linear regression solutions estimated using the Moore–Penrose pseudoinverse. Using a house-price dataset containing variables such as property size, number of bedrooms, distance from the city centre and property age, students construct the design matrix, standardise features and the response variable, and calculate regression coefficients using the pseudoinverse. Students then investigate model robustness by repeatedly fitting the regression model to random subsamples of different sizes and analysing the mean and standard deviation of each coefficient. Tables, boxplots or error-bar visualisations can be used to compare coefficient variability. The final discussion considers which variables are most influential, which coefficients are most stable, how sample size affects stability and whether coefficient interpretation remains reliable across different samples. The final work is submitted as a single PDF exported from Jupyter Notebook or Google Colab, combining documented Python code, experimental results, plots and written interpretation in a professionally organised notebook. Overview word count: approximately 360 wor

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