Problem 124 - 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 Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus relates differentiation and integration. It states that if \( F(x) = \int_{a}^{x} f(t) \, dt \), then for \( f \) to be the derivative of the integral, \( F'(x) = f(x) \).

Step 2: Differentiate Both Sides with Respect to x

Differentiate the given equation \( \int_{0}^{x} f(t) \, dt = 2x^2 \) with respect to \( x \). Using the fundamental theorem, we know \( F'(x) = f(x) \), and differentiating the right side gives \( \frac{d}{dx}(2x^2) = 4x \).

Step 3: Equate the Derivatives

According to Step 1, \( F'(x) = f(x) \). Since differentiating both sides of the initial equation gives us \( f(x) = 4x \), the function \( f(x) \) is found.

Key Concepts

Differentiation and Integration

Differentiation and integration are two of the most fundamental operations in calculus. They are intuitive yet powerful tools used to analyze changes and accumulate quantities, respectively.
Understanding differentiation means recognizing it as the process of finding the derivative of a function. A derivative represents how a function changes at any given point, essentially giving the rate of change or slope of a curve. For example, if we have a position function of an object with respect to time, its derivative will give us the velocity.
Integration, on the other hand, can be viewed as the reverse process of differentiation. It accumulates quantities to find the total, such as area under a curve, total distance traveled, or accumulated growth. In simple words, integration adds up infinitely small pieces to find the whole or accumulated value.

• Differentiation = Finding how fast something changes.
• Integration = Aggregating small changes to find the total.

The Fundamental Theorem of Calculus connects these two operations, showing that they are, in a sense, inverse processes. This theorem is a cornerstone in calculus, linking the concept of differentiating a function with integrating it.

Derivative of an Integral

The idea of the derivative of an integral is a direct result of the Fundamental Theorem of Calculus. This theorem states that if you have an integral of a function from a fixed point to a variable endpoint, the derivative of that integral is the original function evaluated at the endpoint.
Mathematically, if we have a function defined as \ F(x) = \int_{a}^{x} f(t) \, dt \, then its derivative \ F'(x) = f(x) \. This is because as you take the derivative of the integral, you're essentially undoing the accumulation process, and what remains is the function evaluated at the upper limit.
This concept is crucial when solving problems like the one provided. In the solution, the integration from 0 to x of an unknown function \( f(t) \) is given as \( 2x^2 \). Differentiating both sides of this equation with respect to \( x \) gives us \( f(x) \), using the theorem to establish that after differentiation, \( f(x) = 4x \).
Understanding this property allows us to move fluidly between integration and differentiation, solving for unknown functions with ease.

Calculus for Biology and Medicine

Calculus plays a pivotal role in biology and medicine, providing tools to model and understand complex biological processes. It allows researchers and practitioners to describe how populations grow, how drugs disperse through the body, and how fast diseases spread.
In biomedical applications, derivatives can describe rates such as:

• The rate of heartbeats over time.
• The speed at which a drug is metabolized.

Integration helps in determining overall quantities, such as:

• Total exposure to a drug over a specific period.
• The number of cells counted in a sample area.

It's the application of differentiation and integration that enables meaningful interpretation of biological data, turning raw numbers into insights that can lead to better understanding and new medical treatments. By understanding calculus, medical professionals can predict outcomes, optimize treatments, and innovate in the field of healthcare, ultimately improving patient care and health outcomes.