Problem 3 - Calculus for Biology and Medicine

Question

Find \(\frac{d y}{d x}\) \(y=\int_{0}^{x}(4 t-3) d t\)

Step-by-Step Solution

Answer

The derivative is \(\frac{dy}{dx} = 4x - 3\).

Step 1: Set up the problem

We are asked to find the derivative of the integral function with respect to \(x\). The function is given as \(y=\int_{0}^{x}(4t-3) dt\).

Step 2: Recognize the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus helps us relate differentiation to integration. According to the theorem, if \(F(x)\) is an antiderivative of \(f(t)\), then \(\frac{d}{dx}\int_{a}^{x} f(t) dt = f(x)\). Apply this theorem here.

Step 3: Apply the Fundamental Theorem of Calculus

Using the theorem, we identify \((4t-3)\) as the function \(f(t)\) inside the integral. Differentiate \(y=\int_{0}^{x}(4t-3) dt\); thus \(\frac{dy}{dx} = (4x-3)\).

Step 4: Simplify the derivative

The derivative \(\frac{dy}{dx} = 4x-3\) is already simplified, meaning this is our final derivative.

Key Concepts

Derivative of Integral Function

When dealing with the derivative of an integral function, we are often applying the Fundamental Theorem of Calculus. This theorem bridges the world of differentiation and integration, two core operations in calculus. In the exercise, we are tasked with differentiating an integral function: \( y = \int_{0}^{x}(4t-3) \, dt \). Here, the goal is to find \( \frac{dy}{dx} \), the rate at which \( y \) changes as \( x \) changes.To achieve this, the Fundamental Theorem of Calculus provides a vital shortcut. It states that if you have a definite integral with a variable upper limit of integration, its derivative is just the integrand evaluated at the upper limit. In simpler terms, this means:

• Set the integrand of the integral function as \( f(t) = 4t - 3 \).
• According to the theorem, \( \frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x) \).

So, here the derivative is simply \( 4x - 3 \), reflecting how straightforward this theorem makes what could otherwise be a complex calculation.

Antiderivative

An antiderivative is a function that essentially reverses differentiation. In other words, if you have a function \( f(x) \), its antiderivative, often noted as \( F(x) \), is a function such that when you take its derivative, you get back \( f(x) \). In the context of the integral we are working with, which is \( \int_{0}^{x}(4t-3) \, dt \), we can say that the integrals are finding antiderivatives. However, thanks to the way lower and upper limits work in a definite integral, you often don't need to explicitly find the antiderivative to solve for its derivative.Consider that if you know the antiderivative, differentiating it would ideally return the original integrand. This entire process is elegantly encapsulated by the Fundamental Theorem of Calculus. Hence, when applying this theorem, the term \( 4t - 3 \) which serves as \( f(t) \) is integral to understanding the immediate application of this concept.

Differentiation and Integration

The relationship between differentiation and integration is pivotal in calculus. They are inverse processes, where differentiation gives us the rate of change and integration finds the accumulated function based on this change rate. This relationship is precisely what the Fundamental Theorem of Calculus captures. Essentially, it clarifies how these two processes are intertwined.For instance, in the integral \( \int_{0}^{x}(4t-3) \, dt \), integration is first used to create a function \( y(x) \) representing accumulated values. By applying differentiation to this integral according to the theorem, we smoothly revert the process, extracting \( f(x) \), the simple integrand function.Here’s how this process unfolds:

• Integration: Accumulates the values described by a function over an interval.
• Differentiation: Using the theorem, we differentiate to retrieve the instantaneous "rate" function \( (4x-3) \).

In summary, while integration provides a cumulative aspect, differentiation allows us to tap back into specifics, like zooming in on the exact snapshot of change.