The population of a newly introduced species of insect in a contained ecosystem can be modelled by a twice-differentiable function \(P(t)\), where \(t\) is the time in weeks since introduction. The graph of \(P'(t)\), the rate of change of the population, is shown below.
[Assume graph of P’(t) is provided here: A curve that starts at P’(0) = a (positive value), increases to a maximum, then decreases, crossing the t-axis at t=b and becoming negative, before tending towards zero as t approaches infinity.]
Assume \(P(0) = c\), where c is a positive constant.
c. Sketch a possible graph of an anti-derivative of \(P'(t)\), denoted as \(A(t)\), for \(t \ge 0\). Clearly indicate any key features such as stationary points and concavity. Explain the relationship between \(P'(t)\) and \(A(t)\).
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Create Free Account Log inThis is a free VCE Units 3 & 4 Mathematical Methods practice question worth 3 marks, testing your understanding of Derivative and Anti-derivative Graphs. It falls under Calculus in Unit 3: Mathematical Methods Unit 3. Submit your answer above to receive instant AI-powered marking and personalised feedback.
Extend introductory study of functions, algebra, calculus. Focus on functions, relations, graphs, algebra, and applications of derivatives.
Covers limits, continuity, differentiability, differentiation, and anti-differentiation.
deducing the graph of the derivative function from the graph of a given function and deducing the graph of an anti-derivative function from the graph of a given function
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