Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

We found a base \(e\) so that \(E(t)=e^{t}\) has the property that the rate of change of \(E\) at 0 is \(1 .\) Suppose we had searched for a number \(B\) so that the average rate of change of \(E_{B}(t)=B^{t}\) on [0,0.01] is 1: $$m_{0,0.01}=\frac{E_{B}(0.01)-E_{B}(0)}{0.01}=\frac{B^{0.01}-B^{0}}{0.01}=\frac{B^{0.01}-1}{0.01}=1$$ a. Solve the last equation for \(B\). b. Solve for \(B\) in each of the equations: $$\frac{B^{0.001}-1}{0.001}=1 \quad \frac{B^{0.00001}-1}{0.00001}=1 \quad \frac{B^{0.0000001}-1}{0.0000001}=1$$

Step-by-Step Solution

Answer

The solved value of \( B \) is approximately 2.718, aligning with the constant \( e \).

Step 1: Understanding the problem

We need to find the value of \( B \) for which the expression \( \frac{B^{x}-1}{x}=1 \) holds. This indicates that the average rate of change over the interval is 1, similar to the property for the base \( e \).

Step 2: Expressing the equation

We have the general expression \( \frac{B^{x}-1}{x}=1 \). This can be rearranged into the form \( B^{x} - 1 = x \).

Step 3: Solving for B in terms of x=0.01

Substitute \( x = 0.01 \) into \( B^{x} - 1 = x \) and get \( B^{0.01} = 1.01 \). Therefore, \( B = 1.01^{1/0.01} \). Calculating this gives \( B \approx 2.718 \).

Step 4: Solving for B in terms of x=0.001

Substitute \( x = 0.001 \) into \( B^{x} - 1 = x \) so \( B^{0.001} = 1.001 \). Then \( B = 1.001^{1/0.001} \). Calculating this gives \( B \approx 2.718 \).

Step 5: Solving for B in terms of x=0.00001

Substitute \( x = 0.00001 \) to \( B^{x} - 1 = x \), which gives \( B^{0.00001} = 1.00001 \). Then \( B = 1.00001^{1/0.00001} \). Calculating this gives \( B \approx 2.718 \).

Step 6: Solving for B in terms of x=0.0000001

Substitute \( x = 0.0000001 \) to \( B^{x} - 1 = x \), which gives \( B^{0.0000001} = 1.0000001 \). Then \( B = 1.0000001^{1/0.0000001} \). Calculating this gives \( B \approx 2.718 \).

Step 7: Conclusion

In all cases, as \( x \) approaches 0, \( B \) approaches \( e \), which is approximately 2.718. This shows the base \( B \) behaves as the constant \( e \) when the rate of change is set to 1 for infinitely small intervals.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable is the exponent. This structure makes them very unique and useful, especially in modeling growth processes, compound interest, and natural phenomena.

Exponential functions take the form \( f(t) = a \cdot b^{t} \), where:

• \( a \) is a constant multiplier.
• \( b \) is the base of the function, which determines the rate and nature of growth or decay.
• \( t \) is the variable, often representing time.

In these functions, as \( t \) increases, the value of \( f(t) \) can change dramatically if \( b \) is greater than 1 (growth) or decrease rapidly if \( b \) is between 0 and 1 (decay). Exponential growth is characterized by its accelerating increase, making it essential in fields like finance and population studies.

Rate of Change

The rate of change in mathematics tells us how one quantity changes in relation to another. In calculus, this concept is often explored through the derivative, which shows the instantaneous rate of change at any given point.

For exponential functions like \( f(t) = e^{t} \), the rate of change can be particularly interesting. The derivative of \( e^{t} \) is itself, \( e^{t} \), meaning the rate of change is precisely the value of the function at any point.

This property is fascinating because it implies that the function grows at a rate proportional to its size. For the problem given, the goal is to determine how different bases \( B \) must be altered so their average rate of change over very small intervals equals 1, similar to the natural base \( e \). Understanding rates of change is crucial in fields such as physics, economics, and engineering.

Euler's Number

Euler's number, denoted as \( e \), is a fundamental constant approximately equal to 2.718. It is named after the Swiss mathematician Leonhard Euler and is essential in mathematics because of its properties and occurrence in various natural phenomena.

One of the unique characteristics of \( e \) is that it serves as the base for the natural exponential function \( e^{t} \), whose rate of change at \( t = 0 \) is exactly 1. This makes \( e \) the only base where the derivative of the function at 0 equals 1, giving it an appealing simplicity in calculus.

Euler's number emerges in many areas such as compound interest calculations, where it helps model continuous growth or decay, and in defining logarithms, with the natural logarithm having \( e \) as its base. Understanding \( e \) and its applications can enrich one's knowledge in calculus and beyond, broadening comprehension of mathematical behaviors and functions.