Problem 6 - Calculus for Biology and Medicine

Question

Find \(\frac{d y}{d x} .\) $$ y=\int_{0}^{x} \sqrt{1+u^{2}} d u, x>0 $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \sqrt{1+x^2} \).

Step 1: Identify the Problem Type

The problem asks us to find the derivative \( \frac{dy}{dx} \) of a function defined as an integral, which suggests using the Fundamental Theorem of Calculus.

Step 2: Recognize the Application of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \( F(x) = \int_{a}^{x} f(u) \, du \), then \( F'(x) = f(x) \). Here, \( f(u) = \sqrt{1+u^2} \) and \( a = 0 \).

Step 3: Differentiate the Given Function

Based on the Fundamental Theorem of Calculus, differentiate \( y = \int_{0}^{x} \sqrt{1+u^2} \, du \), giving \( \frac{dy}{dx} = \sqrt{1+x^2} \).

Key Concepts

Derivative

A derivative is a fundamental concept in calculus that represents the rate at which a function changes at any given point. Specifically, it measures how the output (like position, cost, or temperature) changes in reaction to a change in input (like time, quantity, or distance). When you see a problem asking for \( \frac{dy}{dx} \), it's essentially asking for the derivative of the function \( y \) with respect to \( x \). This derivative tells us the slope of the tangent line to the curve at any point \( x \). There are different interpretations of derivatives:
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Velocity, when dealing with space and time.

Instantaneous rate of change, such as how quickly a temperature is changing.

In the given exercise, we find \( \frac{dy}{dx} \) by differentiating an integral function. This will reveal the function's behavior at each point \( x \).

Integral Calculus

Integral Calculus is one of the two main branches of calculus, with its focus on accumulation of quantities and the areas under and between curves. While differential calculus (involving derivatives) deals with rates of change, integral calculus helps in evaluating the total change from a rate of change. In mathematical terms, an integral aggregates parts to find the whole.
There are two main types:

• Definite Integrals, which evaluate to a specific numerical value and represent the exact accumulation over a given interval.
• Indefinite Integrals, which represent a family of functions and include an arbitrary constant \( C \).

In our exercise, we are given a definite integral \( \int_{0}^{x} \sqrt{1+u^2} \, du \). This represents the accumulation or the area under the curve from 0 to \( x \). The Fundamental Theorem of Calculus helps us find the derivative of such an integral, linking both integral and differential calculus together.

Differentiation

Differentiation is the mathematical process used to find a derivative. It is the act of calculating how a function changes as its input changes. Let's delve into the steps of differentiation using the Fundamental Theorem of Calculus, which directly applies here.
The Fundamental Theorem of Calculus posits that differentiation and integration are inverse processes. When dealing with integrals, differentiation helps us "undo" the accumulation to find a rate of change function. For a function \( y = \int_{0}^{x} f(u) \, du \), the derivative \( \frac{dy}{dx} \) is simply \( f(x) \). This is a powerful tool as it allows us to easily move from the realm of integrals to the realm of derivatives.
This process is crucial in many applications, such as in physics for determining velocity and acceleration from a given displacement-time function, or in economics for understanding cost functions.