Modeling Population Growth with Logistic Function and Constraints
A population grows following a logistic model given by \(P(t) = \frac{L}{1 + Ae^{-kt}}\), where \(L\) is the carrying capacity, \(A\) is a constant determined by initial conditions, \(k\) is a positive growth rate, and \(t\) is time in years. (a) Define the logistic function and explain the significance of each parameter. (b) Given a […]
Part b
Use a sign diagram to solve the inequality \((x+3)(x+2) > 0\), considering the critical values \(x = -3\) and \(x = -2\). Remember that \(x eq 3\) due to the original denominator.
Solving Logarithmic Inequalities
Solve the inequality \(\log_2(x-1) > 3\). State the domain of the logarithmic function and ensure your solution is consistent with it.
Solving Inequalities Involving Piecewise Functions
Consider the piecewise function \( f(x)= \begin{cases} 2x+1 & \text{if } x<3 \\ -x+7 & \text{if } x\ge 3 \end{cases}\). Solve the inequality \( f(x) \le 7 \) by addressing each interval separately and then combining the valid ranges.