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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 96
Chapter 3, Problem 96

{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.

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First, identify the function given: L(t) = \(\frac{11.94}{1 + 4e^{-1.65t}\)}. This is a rational function where the numerator is a constant and the denominator is a function of t.
To find L'(t), we need to differentiate L(t) with respect to t. This requires using the quotient rule for differentiation, which states that if you have a function \(\frac{u(t)}{v(t)}\), its derivative is \(\frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2}\).
In our case, u(t) = 11.94 and v(t) = 1 + 4e^{-1.65t}. The derivative of u(t), u'(t), is 0 because it is a constant. For v(t), apply the chain rule to find v'(t). The derivative of 4e^{-1.65t} is -6.6e^{-1.65t} (using the chain rule).
Substitute these derivatives into the quotient rule formula: L'(t) = \(\frac{0 \cdot (1 + 4e^{-1.65t}\)) - 11.94 \(\cdot\) (-6.6e^{-1.65t})}{(1 + 4e^{-1.65t})^2}. Simplify the expression to find L'(t).
Finally, evaluate L'(1) by substituting t = 1 into the derivative expression. This value represents the rate of change of the culmen length at t = 1 week, indicating how quickly the length is increasing at that specific time.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. In this context, L'(t) represents the instantaneous rate of change of the culmen length L with respect to time t. Calculating L'(1) will provide insight into how quickly the beak length is growing at t = 1 week.
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Exponential Function

The function L(t) includes an exponential term, e^(-1.65t), which influences the growth behavior of the beak length over time. Exponential functions are characterized by their rapid increase or decrease, depending on the sign of the exponent. Understanding this concept is crucial for interpreting how the beak length evolves as the owlet ages.
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Interpretation of Derivatives

Interpreting the value of a derivative involves understanding its practical significance in a real-world context. For L'(1), this means analyzing what the calculated rate of change indicates about the growth of the owlet's beak at one week old, such as whether it is growing rapidly, slowly, or not at all, which can have implications for the bird's development.
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Related Practice
Textbook Question

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.

c. g'(π)

Textbook Question

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a. Use the Product Rule (twice) to find a formula for d/dx (f(x)g(x)h(x)).

Textbook Question

Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

b. y = f(x) / g(x)

Textbook Question

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

Textbook Question

Composition containing sin x Suppose f is differentiable for all real numbers with f(0)=−3,f(1)=3,f′(0)=3, and f′(1)=5. Let g(x)=sin(πf(x)). Evaluate the following expressions.

b. g'(1)

Textbook Question

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.

a. g(1)