Problem 12 - Calculus for Biology and Medicine

Question

Find \(\frac{d y}{d x}\) \(y=\int_{-1}^{x} \frac{2}{t^{2}+t} d t\)

Step-by-Step Solution

Answer

\( \frac{d y}{d x} = \frac{2}{x^2 + x} \).

Step 1: Understand the Problem

We are given a function defined as an integral: \[ y = \int_{-1}^{x} \frac{2}{t^2 + t} \, dt \]. We need to find \( \frac{d y}{d x} \), the derivative of \( y \) with respect to \( x \). This involves using the Fundamental Theorem of Calculus.

Step 2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus tells us that if \( F \) is an antiderivative of \( f \), then:\[ \frac{d}{d x} \int_{a}^{x} f(t) \, dt = f(x). \]For our function, \( f(t) = \frac{2}{t^2 + t} \).

Step 3: Differentiate Using the Fundamental Theorem of Calculus

By applying the theorem, we differentiate the integral directly:\[ \frac{d y}{d x} = \frac{2}{x^2 + x}. \] This result comes from simply replacing the variable \( t \) with \( x \) in the integrand.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a crucial bridge between differential calculus and integral calculus. It states that differentiation and integration are essentially inverse operations. This theorem consists of two parts:

• The first part tells us that the integral of a function can be undone by differentiation.
• The second part allows us to evaluate definite integrals using antiderivatives.

In simpler terms, if you have a function that you integrate and then differentiate, you return to your original function (granted certain conditions are met). This theorem is crucial for solving problems where you're asked to find the rate of change or derivative of a function defined as an integral, just like in our original exercise. The key takeaway here is that, by applying this theorem, you can directly differentiate an integral with respect to its upper limit, given the lower limit is constant, simplifying the process immensely.

Integral Calculus

Integral Calculus deals with the concept of integrals, which are used to find areas under curves, among other things. There are two main types of integrals:

• Definite integrals, which provide a numerical value representing the area under a curve within a certain range.
• Indefinite integrals, which represent families of functions and include a constant of integration.

In the original exercise, the problem involves a definite integral from \(-1\) to \(x\) of the function \(\frac{2}{t^2 + t}\). Solving such problems often involves utilizing the rules and properties of integrals such as linearity and the substitution method. Understanding integrals as the reverse of derivatives is important, as it lays the foundation for applying the Fundamental Theorem of Calculus in later steps.

Derivative

In calculus, derivatives represent the rate at which a function is changing at any given point, essentially giving us the slope of a function's graph. Monitored through different notations like \(\frac{dy}{dx}\), derivatives help us understand how one quantity changes with respect to another.In the given exercise, the task is to determine \(\frac{d y}{d x}\), the derivative of \(y\). The function \(y\) is presented as an integral with respect to \(x\). Using the Fundamental Theorem of Calculus, the solution involves differentiating this integral. Here, it gives us the value of the original integrand, \(\frac{2}{x^2 + x}\), evaluated at \(x\). This ability to seamlessly move between derivative and integral calculus is what makes these concepts so powerful in mathematics, enabling us to solve complex problems in physics, engineering, and beyond.