A research laboratory is studying the growth and decay of a particular bacterial population under varying environmental conditions. They have developed a model that combines exponential and logarithmic functions to represent the population size, $P(t)$ (in thousands), at time $t$ (in hours) after the experiment begins. The general form of their model is given by $$P(t) = A e^{kt} + B \log_e(t+1)$$, where $A$, $B$, and $k$ are constants that depend on the specific experimental conditions.
b. For a different set of conditions, the researchers observe that the bacterial population initially increases, reaches a maximum, and then starts to decline. Given that $A = 2$ and $B = 1$, deduce the range of possible values for $k$ that would result in this behaviour. Justify your reasoning by considering the derivatives of the exponential and logarithmic terms and their impact on the overall population growth rate.
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Create Free Account Log inThis is a free VCE Units 3 & 4 Mathematical Methods practice question worth 6 marks, testing your understanding of Graphs of Power, Exponential, Log, Circular Functions. It falls under Functions, relations and graphs in Unit 4: Mathematical Methods Unit 4. Submit your answer above to receive instant AI-powered marking and personalised feedback.
Continues the study of functions, algebra, calculus, and introduces probability and statistics.
Covers transformations, polynomial functions, power functions, exponential functions, logarithmic functions, circular functions, and combinations of these.
graphs of the following functions: power functions, $y=x^{n}, n \in Q$; exponential functions, $y=a^{x}, a \in R^{y}$, in particular $y=e^{x}$; logarithmic functions, $y=\log _{x}(x)$ and $y=\log _{(x)}(x)$; and circular functions, $y=\sin (x), y=\cos (x)$ and $y=\tan (x)$ and their key features
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