Dynamic Calculus
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Population growth
The population \(N(t)\) of a particular town has a growth rate \(\frac{{dN}}{{dt}} = 400t\), where \(t\) is the number of years since 1970. Given the growth rate, how do we estimate the population of the town 10 years from now?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- Given that \(\frac{{dN}}{{dt}} = 400t\), use the process of anti-differentiation to find the population function, \(N(t)\). Confirm your answer by clicking the 'show anti-differentiation' checkbox.
- The primitive function \(N(t)\) represents a family of curves, each member of which corresponds to a unique value of the constant, \(c\). Move the slider 'c' and observe other possible graphs of the population function.
- In the context of this problem, what range of values of \(c\) are valid? Explain.
- In 1970, the population of the town was 1000. Use this information to evaluate the constant, \(c\). Using the slider c, move the graph of the population function so that it correctly shows the population of the town in 1970.
- Calculate the population of the town 10 years from now. Click the 'show calculation of population' checkbox and check your answer by moving the slider.
- Using the applet, determine in what year the population will first exceed 500 000. Confirm your answer by calculation.