MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
Consider the logistic differential equation
y′=0.03y(300−y).
Two data sets, A and B, record approximate values of a differentiable function y=y(t) at equally spaced times t=0,1,2,3,4,5 (time in days).
Data set A: {(0,15),(1,24),(2,38),(3,60),(4,91),(5,136)}
Data set B: {(0,20),(1,35),(2,60),(3,95),(4,140),(5,195)}
Assume exactly one of the data sets could have been generated by a solution of the differential equation (no explicit solution needed).
(1) Part 1: Identify which data set (A or B) is consistent with the differential equation. Support your choice by discussing the observed growth pattern in relation to the carrying capacity.
(2) Part 2: Using your chosen data set, estimate both the population value and the approximate time at which the growth rate is greatest. Explain the reasoning behind your estimates.
(3) Part 3: Describe, using the table of values, how you can infer the change in concavity of the underlying solution curve without computing derivatives.