IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Exemplar: Exploring the volume of glass used to make a perfume bottle, which is a solid of revolution.

The student uses the solid-of-revolution formula correctly and keeps the notation aligned with the modelling goal. This is a clear example of appropriate mathematical language because the formula is introduced in context and then used consistently in later sections.

The derivative of the transformed arctan function is shown step by step, which strengthens the communication of the method. The chain rule is made visible rather than hidden inside calculator output, so the reader can follow the algebraic structure of the model.

The derivative of the inner expression is written incorrectly here: $$\frac{d}{dx}(2.6x-29.4)=2.6$$, not $$13.52x-152.88$$. This matters because a small notation slip in an intermediate line can make the later derivative appear more complicated than it really is.

The statement that the third and fourth derivatives were taken directly from Desmos weakens the mathematical communication because the reader cannot see how those values were obtained or verified. The student should either show the derivative work or explain the exact computational step used to confirm the values.

The Maclaurin formula is written with $$f^k(0)$$, which is non-standard notation. The student should use $$f^{(k)}(0)$$ consistently, because the standard derivative notation makes the series term unambiguous and mathematically precise.

The student could improve clarity by naming the approximation variable explicitly, for example writing $$P_4(x)$$ for the 4th-degree Maclaurin polynomial. That would help distinguish the original function from its approximation throughout the rest of the exploration.

The polynomial approximation is written out explicitly after the derivative values are found. This is good mathematical communication because it turns several separate computations into one readable model that can be graphed and compared directly with the original function.

What would change in the final volume if the join point were moved slightly left or right? Thinking about this would help the student test whether the continuity adjustment is just visually convenient or numerically important to the model.

It would be helpful to state the model as a final piecewise function before substituting into the volume formula. That small addition would make the notation easier to track and would reduce the amount of interpretation the reader must do.

The final piecewise volume expression is communicated clearly and in a compact form. This is effective because the reader can see immediately how each interval contributes to the total solid without needing to reconstruct the model from earlier pages.

Choosing to split the bottle into distinct regions shows clear independent thinking. The student is not forcing one generic model onto the shape, but adapting the method to the object’s actual geometry.

The fitting process is still heavily dependent on slider tuning, so the personal mathematical decision-making is limited. The student should explain why the chosen parameters are the best fit, rather than leaving the selection to visual adjustment alone.

The use of a customised arctan fit reflects a personal modelling decision rather than a purely textbook example. This is a good sign of initiative because the student is selecting a function based on the observed profile, not just using the first available formula.

Writing the transformed arctan explicitly before differentiating shows initiative, because the student is working with the actual fitted model rather than a simplified template. This makes the investigation feel genuinely tied to the chosen bottle.

The student could strengthen personal engagement by briefly justifying why a 4th-degree Maclaurin polynomial was chosen instead of a lower or higher degree. That would show more intentional control over the modelling trade-off.

The new rational function is introduced through technology-based adjustment rather than a clearly explained analytic choice. To show stronger independence, the student should justify why this family was suitable and what geometric feature it was designed to capture.

The translation used to remove the gap shows creative initiative, because the student identifies a mismatch and then repairs it mathematically. This is stronger than simply accepting the graph as-is and demonstrates active refinement of the model.

The extension ideas are relevant, but the student could make them more original by specifying one concrete method, such as how 3D scanning or numerical integration would be carried out. That would turn a general future idea into a more distinctive plan.

The use of the solid-of-revolution formula is directly relevant to the aim and applied in an authentic modelling context. This is a strong example of mathematics being used purposefully rather than added artificially.

The claim that the squared arctan integral is too difficult is understandable, but the student should justify the approximation choice more carefully. Without a stated error estimate, the reader cannot judge how much accuracy is being traded for convenience.

Relying on Desmos for the third and fourth derivative values leaves a gap in demonstrating full mathematical understanding. The student should show at least one of those derivative calculations manually or verify the outputs algebraically.

The polynomial approximation is a good HL-level response to the integration difficulty, because it converts an intractable expression into one that can be handled analytically. This shows the student is using the mathematics to solve a real modelling problem, not just to reproduce a known technique.

Does making the two functions meet exactly at the join point improve the overall model enough to justify the extra adjustment? Asking this would help the student think about whether visual continuity is also the best numerical choice.

The continuity translation is a mathematically sensible refinement that improves the model’s usefulness. It shows the student responding to a modelling gap with an appropriate adjustment rather than ignoring the mismatch.

The cost calculation uses the upper end of the quoted price range without discussing why that value was chosen. The student should either justify the selection or calculate a range of costs, since a single point estimate is less informative for real-world evaluation.

Including the individual integral results alongside the final total would make the mathematical application easier to audit. That would let the reader see how much each region contributes to the volume and would improve transparency.

The model would be mathematically stronger if the student included a brief error discussion, such as how sensitive the final volume is to the join point or to the Maclaurin truncation. That would turn the reflection into a more rigorous evaluation of the mathematics used.