IB Maths AA HL Sl 5 1 Limits and Derivative Concept Questions

Practise IB Mathematics HL 5.1 by applying limits and derivative concept methods to questions from the Question Bank.

Exam points

interpret f′(x) as an instantaneous rate of change or tangent gradient and evaluate it at a stated point
use derivative values and limiting behaviour to determine rates such as height, distance or population change
evaluate derivative limits, including behaviour as the independent variable approaches infinity, and state the result with context

[Maximum number: 2]

A function f is defined by $f(x)=\arcsin \left(\frac{x^{2}-1}{x^{2}+1}\right), x \in \mathbb{R}$ .

By considering limits, show that the graph of y=f(x) has a horizontal asymptote and state its equation.

as $x \rightarrow \pm \infty, f(x) \rightarrow \arcsin 1\left(\rightarrow \frac{\pi}{2}\right)$
so the horizontal asymptote is $y=\frac{\pi}{2}$

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