Problem 73 - Calculus for Biology and Medicine

Question

Suppose that the concentration of nitrogen in a lake exhibits periodic behavior. That is, if we denote the concentration of nitrogen at time \(t\) by \(c(t)\), then we assume that $$c(t)=2+\sin \left(\frac{\pi}{2} t\right)$$ (a) Find \(\frac{d c}{d t}\). (b) Use a graphing calculator to graph both \(c(t)\) and \(\frac{d c}{d t}\) in the same coordinate system. (c) By inspecting the graph in (b), answer the following questions: (i) When \(c(t)\) reaches a maximum, what is the value of \(d c / d t ?\) (ii) When \(d c / d t\) is positive, is \(c(t)\) increasing or decreasing? (iii) What can you say about \(c(t)\) when \(d c / d t=0 ?\)

Step-by-Step Solution

Answer

(i) \(\frac{d c}{d t} = 0\); (ii) \(c(t)\) is increasing; (iii) \(c(t)\) is at a peak or trough.

Step 1: Differentiate the Function

To find \( \frac{d c}{d t} \), differentiate \( c(t) = 2 + \sin \left( \frac{\pi}{2} t \right) \). The derivative of a constant is 0, and the derivative of \( \sin(x) \) is \( \cos(x) \). Thus, the derivative of \( \sin \left( \frac{\pi}{2} t \right) \) is \( \frac{\pi}{2} \cos \left( \frac{\pi}{2} t \right) \) using the chain rule.

Step 2: Write the Derived Result

The derivative of \( c(t) \), \( \frac{d c}{d t} \), is: \[ \frac{d c}{d t} = \frac{\pi}{2} \cos \left( \frac{\pi}{2} t \right) \]

Step 3: Graph the Functions

Use a graphing calculator to graph both \( c(t) = 2 + \sin \left( \frac{\pi}{2} t \right) \) and its derivative \( \frac{d c}{d t} = \frac{\pi}{2} \cos \left( \frac{\pi}{2} t \right) \) on the same coordinate system. Observe how these functions relate.

Step 4: Analyze the Graph (i)

When \( c(t) \) reaches a maximum, it is at the peak of its periodic cycle. This occurs when the slope of the tangent line is zero, i.e., when \( \frac{d c}{d t} = 0 \). At these points, \( \cos \left( \frac{\pi}{2} t \right) = 0 \).

Step 5: Analyze the Graph (ii)

When \( \frac{d c}{d t} \) is positive, \( c(t) \) is increasing because the slope of the tangent line is positive.

Step 6: Analyze the Graph (iii)

When \( \frac{d c}{d t} = 0 \), it indicates that \( c(t) \) is at a turning point, either a maximum or a minimum. For the sine function, these are at the peaks and troughs.

Key Concepts

Periodic Functions

Periodic functions are those that repeat their values at regular intervals. They can be visualized as waves that go up and down consistently. These functions are prevalent in nature, particularly where cycles or repeated patterns occur — like the concentration of certain chemicals in a lake. For example, in our exercise, the concentration of nitrogen, denoted as \( c(t) = 2 + \sin \left( \frac{\pi}{2} t \right) \), is a periodic function due to the presence of the sine term.

The sine function itself is inherently periodic with a fundamental period of \( 2\pi \). However, in this context, the period is modified by the factor \( \frac{\pi}{2} \), resulting in periodic behavior that repeats every 4 units of \( t \). Understanding this repeating pattern is essential for predicting any long-term behavior of the system tied to the function.

Derivative of Trigonometric Functions

In differential calculus, finding the derivative of trigonometric functions is one of the core tasks. Derivatives measure the rate at which a function's value changes as its input changes — in simple terms, it tells us how steep the graph is at a particular point. For the function \( c(t) = 2 + \sin \left( \frac{\pi}{2} t \right) \), we need to compute its derivative with respect to \( t \).

• Start by noting that the derivative of a constant (like 2) is zero, since constants do not change.
• The derivative of \( \sin(x) \) is \( \cos(x) \).
• Applying the chain rule for \( \sin \left( \frac{\pi}{2} t \right) \), the derivative becomes \( \frac{\pi}{2} \cos \left( \frac{\pi}{2} t \right) \).

By finding the derivative, \( \frac{d c}{d t} = \frac{\pi}{2} \cos \left( \frac{\pi}{2} t \right) \), we now understand how the concentration of nitrogen changes at any given time. This information is crucial for examining how quickly the concentration is increasing or decreasing.

Graphing Derivatives

Graphing derivatives alongside their original functions provides valuable insight into the behavior of both functions. When you graph \( c(t) \) and \( \frac{d c}{d t} \) together, you can visually assess how the rate of change (derivative) impacts the function.

• Where \( \frac{d c}{d t} = 0 \), \( c(t) \) is at a maximum or a minimum. This is because the slope of the tangent line at these points is flat, indicating no change at that instant.
• When \( \frac{d c}{d t} > 0 \), \( c(t) \) is increasing — the graph is moving upwards.
• Conversely, when \( \frac{d c}{d t} < 0 \), \( c(t) \) is decreasing — the graph is moving downwards.

Visually interpreting the graph helps one to identify critical points and appreciate how the peaks and valleys of \( c(t) \) correspond to zero crossings of the derivative \( \frac{d c}{d t} \). Understanding these relations is fundamental in grasping the nature of trigonometric functions and their derivatives.