Problem 50 - Calculus for Biology and Medicine

Question

Fungal Growth As a fungus grows, its rate of growth changes. Young fungi grow exponentially, while in larger fungi growth slows, and the total dimensions of the fungus increase as a linear function of time. You want to build a mathematical model that describes the two phases of growth. Specifically if \(R(t)\) is the rate of growth given as a function of time, \(t\), then you model $$ R(t)=\left\\{\begin{array}{ll} 2 e^{t} & \text { if } 0 \leq t \leq t_{c} \\ a & \text { if } t>t_{c} \end{array}\right. $$ where \(t_{c}\) is the time at which the fungus switches from exponential to linear growth and \(a\) is a constant. (a) For what value of \(a\) is the function \(R(t)\) continuous at \(t=t_{c}\) ? (Your answer will include the unknown constant \(t_{c}\) ). (b) Assume that \(t_{c}=2 .\) Draw the graph of \(R(t)\) as a function of \(t\)

Step-by-Step Solution

Answer

(a) \( a = 2e^{t_c} \). (b) For \( t_c = 2 \), the graph has an exponential part up to \( t=2 \) and a constant \( 2e^2 \) for \( t>2 \).

Step 1: Understanding Continuity at a Point

To ensure that the function \( R(t) \) is continuous at \( t = t_c \), the left-hand limit as \( t \) approaches \( t_c \) from the left (\( t \to t_c^- \)) must equal the right-hand limit as \( t \) approaches \( t_c \) from the right (\( t \to t_c^+ \)) and the function value at \( t = t_c \).

Step 2: Calculate Left-Hand Limit

The expression for \( R(t) \) for \( 0 \leq t \leq t_c \) is \( 2e^{t} \). Therefore, the left-hand limit as \( t \to t_c^- \) is \( \lim_{t \to t_c^-} 2e^t = 2e^{t_c} \).

Step 3: Determine Right-Hand Limit

For \( t > t_c \), \( R(t) = a \), so the right-hand limit as \( t \to t_c^+ \) is \( \lim_{t \to t_c^+} a = a \).

Step 4: Setting Limits Equal for Continuity

To be continuous at \( t = t_c \), the left-hand limit must equal the right-hand limit: \( 2e^{t_c} = a \). This equation provides the value of \( a \) in terms of \( t_c \).

Step 5: Substitute Specific Value for \( t_c \) to Plot Graph

With \( t_c = 2 \), substitute into the expression to find \( a \):\[ a = 2e^2. \] This allows us to sketch the graph by evaluating \( R(t) \) at each phase.

Step 6: Sketch the Graph of \( R(t) \)

For \( 0 \leq t \leq 2 \), \( R(t) = 2e^t \) is an exponential curve. At \( t = 2 \), \( R(t) = 2e^2 \). For \( t > 2 \), \( R(t) \) is a constant function with value \( 2e^2 \). The graph transitions smoothly at \( t = 2 \), confirming continuity.

Key Concepts

Exponential Growth

Exponential growth describes a situation where a quantity increases at a rate proportional to its current size. This type of growth is commonly observed in young fungi as they have abundant resources and space to expand rapidly. Mathematically, in our fungal growth model, this growth phase is represented by the function \( R(t) = 2e^t \). Here, \( e \) is the base of the natural logarithm, approximately equal to 2.718. The presence of \( e^t \) indicates that as time \( t \) increases, the rate of growth exponentially rises.
This means that for any small increment in time, the quantity grows by a constant percentage. It results in a rapid increase, and that's why young fungi exhibit exponential growth when they are small, as they can divide and multiply without restrictions. This phase of growth continues until a certain time \( t_c \), after which the growth slows down.

Linear Growth

Linear growth is characterized by a constant rate of increase over time. In the context of the fungus, after the initial exponential growth phase, resources become constrained, leading to slower growth. This change is modeled with a constant rate function: \( R(t) = a \) for \( t > t_c \). Here, \( a \) is a constant determined based on the transition point \( t_c \), ensuring smooth transition between the growth phases.
Unlike exponential growth, where the rate changes based on size, linear growth adds equal increments over equal time intervals. In simple terms, the bigger the fungus gets, the more demands it places on its environment, resulting in evened-out, steady growth. It's the point where the rate of growth isn't influenced by the current size of the fungus but remains constant, creating a linear relationship over time.

Continuity in Calculus

In calculus, a function is continuous at a point if there is no interruption in its value at that location. For our fungal model, ensuring continuity at \( t = t_c \) is crucial to prevent any sudden jumps or drops in the growth rate function \( R(t) \). Continuity means the left-hand limit (approaching from the left of \( t_c \)), right-hand limit (approaching from the right), and the actual function value at \( t_c \) are all equal.
To achieve this in the fungus model, set the left limit \( 2e^{t_c} \) equal to the right limit \( a \), resulting in \( a = 2e^{t_c} \). For example, if \( t_c = 2 \), then \( a = 2e^2 \) ensures there is no disruption in \( R(t) \)'s value at \( t = 2 \). This seamless transition allows the growth model to accurately reflect how fungi grow and shift from exponential to linear growth smoothly.