Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Argue that \(e^{x}\) is an increasing function.

Step-by-Step Solution

Answer

The function \( e^{x} \) is increasing because its derivative \( e^{x} \) is always positive.

Step 1: Understanding the problem

We want to show that the function \( f(x) = e^{x} \) is an increasing function. A function \( f(x) \) is said to be increasing if, for any two numbers \( a \) and \( b \) such that \( a < b \), we have \( f(a) < f(b) \).

Step 2: Use the definition of the derivative

To determine if a function is increasing, we can use the derivative test. If the derivative \( f'(x) \) is positive for all \( x \) in the interval, then the function is increasing on that interval.

Step 3: Differentiate the function

Compute the derivative of \( f(x) = e^{x} \). The derivative is \( f'(x) = \, e^{x} \).

Step 4: Analyze the derivative

Evaluate the sign of the derivative \( f'(x) = e^{x} \). Since the exponential function \( e^{x} \) is always positive for all real numbers \( x \) (i.e., \( e^{x} > 0 \)), the derivative is always positive.

Step 5: Conclusion from the derivative test

Since the derivative \( f'(x) = e^{x} \) is positive for all \( x \), the function \( f(x) = e^{x} \) is an increasing function.

Key Concepts

Derivative Test

To determine whether a function is increasing or decreasing, mathematicians utilize a helpful tool called the derivative test. The core idea is pretty straightforward: if the derivative of a function, denoted as \( f'(x) \), is greater than zero across an interval, the function is said to be increasing over that interval. Conversely, if \( f'(x) \) is negative, the function is decreasing.

In the case of \( f(x) = e^x \), we calculate its derivative to find \( f'(x) = e^x \). The exponential function \( e^x \) is uniquely interesting because it is always positive for all \( x \). Thus, \( e^x > 0 \) for any real number \( x \). This fact directly implies that the function \( f(x) = e^x \) is continuously increasing as the derivative remains positive.

Understanding the derivative test empowers you to analyze various functions effectively and determine their behavior over given intervals.

Exponential Functions

Exponential functions are a category of mathematical functions where an independent variable appears in the exponent. Their general form is \( f(x) = a^{x} \), with \( a \) being a constant and \( x \) the variable. A defining characteristic of exponential functions is their rapid growth (or decay) rate, which is determined by the base \( a \). Specifically, when \( a > 1 \), the function grows as \( x \) increases.

For the function \( e^x \), the base is the mathematical constant \( e \), approximately equal to 2.71828. The function exhibits continuous growth and is often used in mathematical modeling of real-world scenarios involving growth processes, such as population dynamics or compound interest.

With their continuous, smooth curves and inherent growth characteristics, exponential functions like \( e^x \) serve as essential tools across various fields, including economics, biology, and physics.

Function Derivatives

The derivative of a function is a core concept in calculus, providing valuable information about how a function changes. By calculating a function's derivative, one can understand its rate of change at any given point. Derivatives are fundamentally the slope of the tangent to the curve at a location \( x \).

In the context of \( f(x) = e^x \), the derivative \( f'(x) = e^x \) suggests that the rate at which \( e^x \) changes is precisely the same as its current value. This unique property means any increase in \( x \) results in an increase in \( e^x \).

• A derivative provides insight into the behavior of a function, whether it's increasing or decreasing.
• For increasing functions, the derivative is positive over their domain.

Function derivatives are not just a mathematical abstraction; they have profound applications, including optimized design, equilibrium states in physics, and economic predictions.