{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.


a. Find L′(1) and interpret the meaning of this value.

Verified step by step guidance

1. Identify the function L(t) = 11.94 / (1 + 4e^(-1.65t)) and recognize that you need to find the derivative L′(t) to determine the rate of change of the culmen length with respect to time.
2. Apply the quotient rule for differentiation, which states that if you have a function in the form of f(t) = g(t) / h(t), then f′(t) = (g′(t)h(t) - g(t)h′(t)) / (h(t))^2.
3. In this case, let g(t) = 11.94 and h(t) = 1 + 4e^(-1.65t). Calculate g′(t) and h′(t). Since g(t) is a constant, g′(t) = 0. For h(t), use the chain rule to find h′(t).
4. Substitute g(t), g′(t), h(t), and h′(t) into the quotient rule formula to find L′(t).
5. Evaluate L′(1) by substituting t = 1 into the derivative you found, and interpret the result in the context of the problem, which indicates the rate of change of the culmen length at t = 1 week.