Problem 125 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The function \( f(x) \) is \( 4x \).

Step 1: Understand the Relationship

We are given the integral \( \int_{0}^{x} f(t) \, dt = 2x^2 \). This is a form of a function described by the Fundamental Theorem of Calculus. Our goal is to find the function \( f(x) \).

Step 2: Use the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \( F(x) = \int_{0}^{x} f(t) \, dt \), then \( f(x) = F'(x) \).

Step 3: Differentiate the Given Function

Differentiate the expression \( 2x^2 \) with respect to \( x \) to find \( f(x) \). The derivative is \( \frac{d}{dx}(2x^2) = 4x \).

Step 4: Conclude the Calculation

The integration relationship given allows us to conclude, based on differentiation, that \( f(x) = 4x \).

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function. In essence, it is the process of adding up tiny quantities to determine the whole. It is often used to compute areas under curves or to find the accumulated quantity, like distance from velocity.

There are two main types of integrals:

• Definite Integrals: Represented as \( \int_{a}^{b} f(x) \, dx \), it calculates the accumulated quantity between the limits \( a \) and \( b \).
• Indefinite Integrals: Represented simply as \( \int f(x) \, dx \), it is the antiderivative of the function, reflecting a family of functions rather than a single value.

The Fundamental Theorem of Calculus bridges the connection between differentiation and integration. It tells us that if we can take a derivative of a function, we can integrate it, and vice versa. In the given exercise, the definite integral \( \int_{0}^{x} f(t) \, d t \) results in \( 2x^2 \). This represents a scenario where you are trying to reverse the process done by differentiation using integration.

By understanding integration this way, you can see it is vital in numerous fields, providing powerful tools for analyzing changes and accumulated values.

Differentiation

Differentiation is another core concept in calculus and refers to the process of finding the derivative of a function. It is essentially the reverse of integration. The derivative represents the rate of change or the slope of the function at any given point.

In our problem, the differentiation step involved taking the derivative of the expression \( 2x^2 \) to find the form of \( f(x) \). Differentiation tells us how a function is changing at any point, turning the integral into a more manageable algebraic expression.

Key features of differentiation include:

• Derivatives can tell us where the function increases or decreases.
• They can be used to find maximum or minimum points by setting the derivative to zero.
• They provide insights into the function's behavior, how steep it is at any given point.

In summary, learning how to apply differentiation allows you to find the rate of change, which is crucial in physics for velocity, economics for cost mediations, and engineering to determine material stress levels. In short, differentiation reveals the dynamics of functions, providing a deeper insight into their behavior over time or space.

Derivative Calculation

The calculation of derivatives is central to understanding how functions behave. It involves applying different rules and techniques to find the rate at which a function changes.

To solve for \( f(x) \) in the exercise, we differentiate \( 2x^2 \) with respect to \( x \), resulting in \( f(x) = 4x \). This process involves using basic differentiation rules:

• Power Rule: For any \( x^n \), the derivative is \( nx^{n-1} \).
• Constant Multiplication Rule: If a constant is multiplied by a function, differentiate the function normally and multiply it by the constant.

These rules help simplify what initially might appear complex.

Derivative calculations are essential in predicting how quantities change and are extensively used in various fields. For example, in real-life applications, they're used to solve problems related to motion, where the derivative of position gives velocity; thus, helping in trajectory predictions or optimizing processes for better efficiency.

By mastering derivative calculations, you gain a powerful toolset for not only solving calculus problems but also applying it practically across many scientific and engineering disciplines.