IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Example

The title page is missing key administrative details such as candidate name, number, and date. Ensure the title page follows IB formatting guidelines with complete candidate information.

The table of contents lacks page numbers, which reduces navigability. Include page references for each section to help the reader locate content efficiently.

Empty page-number columns in the TOC create ambiguity. Populate the second column consistently or remove it to maintain a clear, well-structured contents page.

The “Abstract” heading is nested within the Introduction, which disrupts structure. Present the Abstract as a standalone section before the Introduction.

The transition from introducing osculating circles to applying the formula is abrupt; insert a brief roadmap sentence to guide the reader through your methodology.

Equation numbering resets mid-document, which disrupts logical progression. Continue numbering sequentially to preserve clarity.

Figure 4 is shown without an explicit in-text reference or discussion of features; introduce and interpret the figure in the body text.

The conclusion merges limitations, future work, and evaluation in one block. Structure with subheadings (Limitations, Future Work) for clearer organization.

References are inconsistently formatted and missing publisher details or DOI. Adopt a consistent citation style (e.g., APA) and ensure completeness.

The Wikipedia citation lacks a full URL and accessed date formatting per IB standards. Provide complete source details for transparency.

The MathWorld (Weisstein) reference shows a truncated URL. Include the full web address and date accessed to ensure reproducibility.

Use of inline math for the 300 m constraint is clear and accurate, demonstrating appropriate mathematical notation to communicate FIA requirements.

Figure 1 is a low-resolution screenshot with no axis labels or scales; enhance clarity by redrawing or annotating axes, units, and key points.

The osculating circle formula alternates between “yr” and y′; choose a consistent prime notation and define y′ and y″ before using them.

Equation 2’s caption refers to the turn approximating function, but the function itself is not shown. Include the explicit formula for clarity.

The downforce formula mixes SI units inconsistently and approximates 6000 N without derivation; ensure unit consistency and show substitution steps.

The friction force calculation mixes kilonewtons and newtons (“6 kN”); convert consistently (e.g., to newtons) and clarify μ’s dimensionless nature.

Figure 3’s axes and units are unlabeled, hindering interpretation of maximum speed variations. Add clear axis labels and unit markers.

The derivation for wₘₐₓ is presented in a single inline fraction “(7−4)/2” followed by “1.5 km”, which is confusing. Break down the calculation in steps.

Equation 12’s caption incorrectly labels a second-derivative calculation as “first derivative.” Revise to reflect the correct sequence.

The curvature formula includes (1+cot²t)^(3/2), but since 1+cot²t=csc²t, simplify to csc³t to enhance clarity and reduce algebraic bulk.

The factor of 2 in the average speed integral (Equation 15) is unexplained. Clarify why the integral is doubled to compute average speed over a full lap.

The opening paragraph effectively conveys personal engagement through the student’s lifelong passion for Formula 1, establishing strong context for the investigation.

Proposed expansion to 3D aerodynamics and racing line calculation shows good forward thinking and personal initiative. Strengthen this by outlining concrete steps.

The argument that an ellipse behaves like a circle in minimizing total curvature is asserted without proof. Provide at least a sketch or numerical comparison.

Concluding that an ellipse minimizes curvature by analogy with the circle needs deeper critical reflection. Compare total curvature numerically for ellipse vs circle.

The conclusion cites “real-world data” matching results but does not specify datasets or benchmarks. Include specific comparisons with existing track speeds.

In Equation 3 for osculating radii, R can be negative when y″ is negative; clarify domain restrictions or use |R| to convey radius magnitude.

The logical deduction from FIA rules to track dimensions is terse. Expand each inference step to show how constraints define track height and width.

The second derivative of the circle is miscalculated: the formula for d²y/dx² contains an unmotivated b term and incorrect sign. Re-derive step by step.

The integral limits in Equation 14 are given as 0 to 2aπ, but arc parameter t should run from 0 to 2π regardless of a. Adjust bounds accordingly.

Lap time calculation uses s=4.54586 km in the numerator without prior definition of lap length; explicitly introduce and justify this track length.