Problem 126 - Calculus for Biology and Medicine

Question

Suppose that $$\int_{0}^{x} f(t) d t=\frac{1}{2} \tan (2 x)$$ Find \(f(x)\).

Step-by-Step Solution

Answer

The function \( f(x) = \sec^2(2x) \).

Step 1: Use the Fundamental Theorem of Calculus

To find the function \( f(x) \) from the given integral equation, we apply the Fundamental Theorem of Calculus. This theorem states that if \( F(x) \) is an antiderivative of \( f(x) \), then \( \frac{d}{dx} \left( \int_{0}^{x} f(t) \, dt \right) = f(x) \). Given \( \int_{0}^{x} f(t) \, dt = \frac{1}{2} \tan(2x) \), we differentiate both sides with respect to \( x \).

Step 2: Differentiate Both Sides

Differentiate both sides of the equation \( \int_{0}^{x} f(t) \, dt = \frac{1}{2} \tan(2x) \). The left side is simply \( f(x) \), by the Fundamental Theorem of Calculus. The right side is differentiated using the chain rule. This gives us:\[ f(x) = \frac{d}{dx} \left( \frac{1}{2} \tan(2x) \right) = \frac{1}{2} \cdot 2 \cdot \sec^2(2x) = \sec^2(2x). \]

Step 3: Verify the Solution

To ensure the derived \( f(x) = \sec^2(2x) \) is correct, we can reverse the process by integrating this function over \( [0, x] \) and confirming the original equation. The integral \( \int_{0}^{x} \sec^2(2t) \, dt \) results in \( \frac{1}{2} \tan(2x) \), consistent with the original expression \( \int_{0}^{x} f(t) \, dt = \frac{1}{2} \tan(2x) \). Thus, the function is verified.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is an important, bridging concept between differential calculus and integral calculus. It establishes the connection between the derivative and the integral.
This theorem is divided into two main parts.

• Part 1: It states that if you have a continuous function and integrate it over an interval, you obtain a function that can be differentiated back to the original function. In simpler terms, integration and differentiation are inverse processes.
• Part 2: It tells us that if you take the derivative of an integral, you regain the original function being integrated. This is particularly useful for evaluating definite integrals.

In our problem, we use Part 2 of the theorem. Here, the integral from 0 to x of a function is given, and we find the original function by differentiating the result. This is how we can discover the function from its integral representation.

Integration

Integration is a core concept in calculus that involves finding the area under a curve represented by a function. It has applications in various fields such as physics, engineering, and economics.
The process of integration can be thought of as the reverse of differentiation.

• Definite Integration: This type of integration involves calculating the integral over a specific interval, such as from 0 to x. The result is a number that represents the area under the curve in that interval.
• Indefinite Integration: This is where we integrate a function without specific limits, resulting in a general formula plus an arbitrary constant.

In the original problem, we applied definite integration and used the result to work backwards, using differentiation to find the unknown function. This showcases the inverse nature of integration and differentiation.

Differentiation

Differentiation is the process of finding the derivative of a function. It measures how a function changes as its input changes. This is crucial in understanding the rate at which quantities change.
Derivatives can be found for all sorts of functions, and the basic rules of differentiation simplify this process.

• Power Rule: For a function, the power rule states that the derivative of \(x^n\) is \(nx^{n-1}\).
• Product Rule: Used when differentiating a product of two functions.
• Chain Rule: Crucial for finding the derivative of a composite function. In our exercise, differentiating \(\frac{1}{2} \tan(2x)\) involved using the chain rule, as it's a function within a function.

Thus, differentiation helped us to "undo" the integration from the given expression, leading us to the function \(f(x) = \sec^2(2x)\), thus illustrating how integral and differential calculus reflect each other.