Problem 68 - Calculus

Question

Give an example of:A continuous function which is always increasing and positive.

Step-by-Step Solution

Answer

The function \( f(x) = e^x \) is a continuous, always increasing, and positive function.

Step 1: Recognize the Requirements

To find a continuous function that is always increasing and positive, we need a function that does not decrease as its input increases and remains above zero for every input in its domain.

Step 2: Select a Suitable Function Type

A commonly used function that is always increasing is the exponential function, because as the input increases, the output also increases without limit.

Step 3: Choose a Specific Function

An example of such a function is the exponential function: \( f(x) = e^x \). This function is continuous for all real numbers, always positive, and always increasing.

Step 4: Verify Continuity

The exponential function \( e^x \) is continuous over its entire domain, which is the set of all real numbers, as it can be expressed as an infinite series.

Step 5: Verify Positivity

Since \( e^x > 0 \) for all real numbers \( x \), the function is always positive.

Step 6: Verify Increase

The derivative of \( f(x) = e^x \) is \( f'(x) = e^x \), which is positive for all real numbers, confirming the function is always increasing.

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. This function is written as \( f(x) = a^x \), where \( a \) is a positive constant greater than zero, and \( x \) represents the variable. Among exponential functions, the natural exponential function \( f(x) = e^x \) is particularly significant. Here, \( e \) is an irrational constant approximately equal to 2.718281828. The unique quality of the exponential function, specifically \( e^x \), is that its rate of growth increases as \( x \) grows. This property makes exponential functions well-suited for modeling phenomena where growth accelerates, such as population growth and compound interest.
Exponential functions can transform rapidly due to their intrinsic nature. Unlike linear functions, which change at a constant rate, exponential functions grow progressively faster. This rapid growth is reflected in their graphs, which can start relatively gently and then quickly become steep.
The significance of the exponential function extends beyond mathematics into fields such as biology, finance, and physics, where it is used to describe processes that involve rapid change.

Increasing Function

An increasing function is a type of function where the value of the function rises as the input variable increases. In simple terms, if you move to the right along the horizontal \( x \)-axis, the height of the function rises. Mathematically, a function \( f(x) \) is said to be increasing if for any two numbers \( x_1 \) and \( x_2 \) in its domain, where \( x_1 < x_2 \), the inequality \( f(x_1) \leq f(x_2) \) holds.
When talking about a function being 'always increasing,' it means that this sort of behavior doesn't change over its entire domain. Such functions are pivotal in various real-world applications where predictability of growth or measurement is crucial.
The exponential function \( e^x \) is a classic example of an increasing function. Its derivative \( e^x \) is always positive, indicating a further guarantee that as \( x \) becomes larger, \( f(x) \) also grows. Thus, no matter what two points you choose, moving from left to right on its graph will consistently show an upward trend.

Positive Function

A positive function is one where the output is greater than zero for every input within its domain. In other words, the entire graph of a positive function lies above the \( x \)-axis. This feature makes positive functions useful in contexts where negative values may not make sense or be feasible, such as when modeling length, area, or population, where these quantities must logically remain above zero.
In mathematics, ensuring that a function always remains positive can help stabilize certain models or calculations. A simple and notable example of a positive function is \( f(x) = e^x \), as discussed earlier. It remains positive because the exponential base \( e \), approximately 2.718, raised to any real number power results in a positive value.
The property of being always positive aligns with consistent requirements across various applications, where ensuring non-negative outputs is essential for accuracy and relevance. Whether used in engineering, finance, or natural sciences, maintaining positivity ensures that models align correctly with real-world constraints and interpretability.