These are explanations and solutions for IB past papers, not the official version. For official papers, you can go to IB Follet or access them through your school.
These are explanations and solutions for IB past papers, not the official version. For official papers, you can go to IB Follet or access them through your school.
These are explanations and solutions for IB past papers, not the official version. For official papers, you can go to IB Follet or access them through your school.
These are explanations and solutions for IB past papers, not the official version. For official papers, you can go to IB Follet or access them through your school.
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IB MATH AA HL, Paper 3, May, 2023, TZ1, Solved Past Paper
Master the 2023 IB May for Paper 3 Mathematics AA HL with examiner tailored solutions and comments for TZ1
Question 1 [Explained]
Let's delve into a group of mathematical functions represented by \(( f_n(x) = x^n e^{-x} )\). Here, (x) is any number that's zero or more, and (n) is a whole number that's one or greater.
As an example, if we set (n) to 1, the specific function ( f1(x)) simplifies to \(( x e^{-x} )\), with the condition that (x) cannot be negative.
Question 1 [a] [Explanation]
Create a diagram of the function ( y = f1(x)), and identify the point on the graph where the function reaches its highest value before decreasing again. Provide the exact location of this peak in terms of its coordinates.
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Question 1 [b] [Explanation]
Demonstrate that for a function ( f (x)), the space enclosed between its curve, the x-axis, and a vertical boundary at (x = b) (assuming (b) is a positive value) can be quantified. This area is represented by the expression \(( \frac{e^b - b - 1}{e^b} )\), which emerges from the integration process within the defined bounds.
The task involves applying integral calculus to establish a relationship between the exponential function's behavior and the geometrical area it encompasses under certain conditions. Here, (b) acts as a limit to this region, setting a finite boundary for the calculation.
Video Solution by an IB Examiner - Coming soon
Question 1 [c] [Explanation]
Let's think about a curve given by the equation ( y = fn(x)), where (n) is a natural number. The area under this curve from where (x = 0) to a certain point along the positive x-axis is denoted by (An). We find this area, (An), by performing an integration of ( fn(x)) with respect to (x), starting at 0 and ending at (x).
To capture the total area beneath the curve as we extend indefinitely along the x-axis, we consider it as the limit of the integral of ( fn(x)) as (x) tends towards infinity.
Video Solution by an IB Examiner - Coming soon
Question 1 [c] [i] [Explanation]
Consider the function \(( \frac{e^{b} - b - 1}{e^{b}} )\) and examine its behavior as the variable (b) increases without bound. To evaluate this limit, we employ l'Hôpital's rule, which is particularly useful when the straightforward evaluation yields an indeterminate form such as \(( \frac{\infty}{\infty} )\).
Understand that the conditions for applying l'Hôpital's rule are met in this case, and we are interested in finding out the approaching value of this expression as (b) goes to infinity.
Question 1 [c] [ii] [Explanation]
The question asks for the determination of A1, which is a variable associated with a particular sequence or function, given a specific formula or set of conditions. This involves applying known mathematical principles or calculations to find the value of A1 based on the information provided.