Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

The function \(b^{t}\) for \(b=1\) is a special exponential function. Confirm that the derivative equation \(\left[b^{t}\right]^{\prime}=b^{t} \ln b\) is valid for \(b=1\). Draw some graphs of \(b^{t}\) for \(b=1\) and its derivative.

Step-by-Step Solution

Answer

For \(b=1\), the derivative \([b^{t}]' = b^{t} \ln b\) is zero, confirming the equation. Graph is a horizontal line at \(y=1\).

Step 1: Understand the Function

The given function is \(b^t\) with \(b = 1\). Substituting \(b = 1\) into the function gives \(1^t\), which is always 1, regardless of the value of \(t\).

Step 2: Compute the Derivative

To find the derivative of \(b^t\) at \(b=1\), we use the formula \(\left[b^t\right]' = b^t \ln b\). Substitute \(b = 1\) into the formula: \(\left[1^t\right]' = 1^t \ln 1\).

Step 3: Simplify the Derivative Expression

Since \(1^t = 1\) and \(\ln 1 = 0\), the expression simplifies to \(1 \cdot 0 = 0\). This means the derivative of \(1^t\) is 0.

Step 4: Interpretation of the Result

The derivative being 0 indicates that \(1^t\) is a constant function, and the slope of a constant function is zero everywhere. This confirms that the derivative equation holds true.

Step 5: Graph the Functions

Graph \(b^t\) for \(b=1\), which is a horizontal line at \(y=1\). The derivative graph, \(0\), is along the x-axis.

Key Concepts

Derivative

When we talk about the derivative of a function, we are referring to its rate of change. In simple terms, the derivative tells us how a function behaves as we make tiny changes to its input. For exponential functions, which grow or decline at specific rates, the derivative provides key insights into these behaviors.

For the function \(b^t\), where \(b\) is a constant, the derivative is generally given by \(b^t \ln b\). This formula tells us how the output of the function changes with respect to \(t\), the variable.

• If \(b > 1\), the function is increasing, and the derivative is positive.
• If \(0 < b < 1\), the function is decreasing, and the derivative is negative.

But what happens when \(b = 1\)? In this case, \(1^t = 1\) for all \(t\), and the natural logarithm of 1 is 0 (\(\ln 1 = 0\)). Therefore, the derivative becomes \(1^t \ln 1 = 0\).

This result tells us that the function does not change. It confirms that the derivative of a constant function like \(1^t\) is always zero. The derivative equation matches our expectation for a constant function: no change (or slope) at any point along our graph.

Constant Function

A constant function is a function that always returns the same value, no matter what the input is. In our example, the function \(1^t = 1\) is a constant function because it doesn't change as \(t\) changes.

The characteristics of a constant function make it unique:

• It's graph is a horizontal line.
• It has no slope, meaning its derivative is always zero.

This is because differentiation measures the rate at which a function changes. Since a constant function doesn't change, its rate of change is zero. This directly aligns with our calculation that the derivative \(\left[1^t\right]'\) is 0.

Understanding constant functions and their derivatives helps build a foundational grasp of more complex functions. They serve as a simple illustration of how derivatives quantify change, or lack thereof, in functions.

Graph Interpretation

Graphing functions and their derivatives is a useful skill to visualize mathematical concepts. For the function \(b^t\) with \(b = 1\), its graph and that of its derivative provide clear insights.

When we graph \(1^t\), we see a straight, horizontal line at \(y = 1\). Why? Because no matter what the value of \(t\), \(1^t\) always equals 1. This unchanging value is what makes the graph of a constant function so straightforward.

On the same set of axes, graphing its derivative, which is 0, results in a line along the x-axis. The derivative graph being a horizontal line at zero further emphasizes that there is no slope or rate of change in \(1^t\). It serves as a strong visual confirmation of the function's constancy.

• The graph of \(1^t = 1\) illustrates constancy.
• The graph of its derivative \(= 0\) confirms zero slope everywhere.

Graph interpretation thus aids comprehension by visually demonstrating the principles of calculus, tying together the mathematical results with graphical evidence.