Problem 3 - Calculus Early Transcendentals

Question

Assume the derivatives of \(f\) and \(g\) exist. Give a nonzero function that is its own derivative.

Step-by-Step Solution

Answer

Answer: The function f(x) = e^x is a nonzero function that is its own derivative.

Step 1: Define the problem

We are looking for a nonzero function, let's call it \(f(x)\), such that its derivative, \(f'(x)\), is equal to the function itself: \(f'(x) = f(x)\).

Step 2: Guess and check a possible solution

One possible solution for this problem is the exponential function. Specifically, the natural exponential function, denoted by \(e^x\), has the property that its derivative is equal to itself. Let's verify this: Given the function \(f(x) = e^x\), its derivative is \(f'(x) = e^x\). We can see that this derivative is indeed equal to the original function: \(f'(x) = e^x = f(x)\)

Step 3: Confirm that the solution is nonzero

We are looking for a nonzero function. The exponential function \(e^x\) is nonzero, since it is always positive for any real number \(x\). Thus, \(f(x) = e^x\) satisfies the given condition, it is its own derivative and it is nonzero. In conclusion, the nonzero function that is its own derivative is \(f(x) = e^x\).

Key Concepts

Derivative

The concept of a derivative is central to calculus and helps us understand how a function changes. In simple terms, the derivative of a function at a given point measures the rate at which the function's value changes as its input changes. Think of it as the slope of the tangent line to the curve at that point.
For example, for a function like \( f(x) = e^x \), when we calculate the derivative, we want to know how \( e^x \) changes as \( x \) changes.

• The derivative provides insight into the behavior of functions.
• It is often denoted as \( f'(x) \) or sometimes as \( \frac{df}{dx} \).
• In the context of this problem, the derivative of \( e^x \) is \( e^x \) itself, which is quite unique and simplifies problem-solving.

Natural Exponential Function

The natural exponential function, often denoted as \( e^x \), is a very special and important function in mathematics. It is part of the family of exponential functions but uniquely uses the number \( e \), approximately 2.71828, which is an irrational and transcendental number with numerous applications in mathematics.

• Unlike other exponential functions, \( e^x \) has the fascinating property that its rate of growth is proportional to its current value.
• This means that when you take the derivative of \( e^x \), it remains \( e^x \), a quality that makes it incredibly useful in various fields such as economics, ecology, and even physics.
• The natural exponential function is always positive, meaning that for any real number \( x \), \( e^x \) never equals zero, ensuring it is always a nonzero function.

Nonzero Function

When we talk about a nonzero function, we're describing a function that does not take the value zero for any input in its domain. This is crucial in many mathematical contexts where being nonzero implies certain useful properties.

• A function that is nonzero generally implies it is either always positive or always negative; in the case of \( e^x \), it is always positive.
• Why does this matter? If a function is its own derivative, as in our example with \( f(x) = e^x \), it does not "vanish" or become zero over any real input. This allows predictions and calculations that rely on the function remaining distinct and evaluative for any \( x \).
• In practical terms, this property of \( e^x \) means it consistently exhibits growth, which can model real-world phenomena such as population growth or compound interest effectively.