Problem 13 - Calculus for Biology and Medicine

Question

Find \(\frac{d y}{d x}\) $$ y=\int_{\pi / 2}^{x} \sin \left(t^{2}+1\right) d t $$

Step-by-Step Solution

Answer

\(\frac{d y}{d x} = \sin(x^2 + 1)\).

Step 1: Understand the Problem

We need to find the derivative of the given function with respect to \(x\). The function \(y\) is defined as an integral with a variable upper limit \(x\). This is a problem suitable for the Fundamental Theorem of Calculus.

Step 2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus, specifically the Leibniz rule, states that if \(F(x) = \int_{a}^{x} f(t) \, dt\), then the derivative \(\frac{dF}{dx} = f(x)\). In our exercise, \(f(t) = \sin(t^2 + 1)\), so the solution to the derivative \(\frac{d y}{d x} = \sin(x^2 + 1)\).

Key Concepts

Understanding Derivatives of Integral Functions

In calculus, a derivative represents the rate of change or the slope of a function at any given point. When dealing with integral functions, especially those with variable upper limits, understanding how to differentiate them becomes crucial. The Fundamental Theorem of Calculus makes this process straightforward. For a function defined as an integral with a variable upper limit, finding its derivative is a structured process:

• Identify the function within the integral's upper limit as a variable you're differentiating with respect to.
• Recognize that the derivative of the integral function is, in essence, the original function's value at this upper limit.

This simplifies the derivative process considerably and provides access to evaluating the change rate of accumulated functions, as we see in the given example with \(rac{d y}{d x} = \sin(x^2 + 1)\). Understanding these concepts allows for efficient handling of more complex calculus challenges.

Leibniz Rule Simplifies Differentiation

Leibniz rule, part of the Fundamental Theorem of Calculus, helps evaluate the derivatives of integral functions. It is particularly useful when the integral has a variable upper limit, as seen in our exercise. Here's how it applies:

• The rule states that for a function \( F(x) = \int_a^x f(t) \, dt \), the derivative \( \frac{dF}{dx} \) is simply \( f(x) \), the integrand evaluated at the upper limit of integration \( x \).
• This can be understood as the process of undoing the integration, meaning the derivative returns the function \( f(x) \) involved in the original integral.

By simplifying the approach to finding derivatives of functions defined as integrals, the Leibniz rule makes complex calculus problems more digestible for students and professionals alike, ensuring efficiency and clarity in mathematical analysis.

Variable Upper Limit in Integrals

An integral with a variable upper limit, such as \( \int_{b}^{x} f(t) \, dt \), introduces an interesting dynamic to differentiation. The upper limit \( x \) changes with respect to which we're differentiating, making it a focal point in the derivation process. Here's how it works:

• When the upper limit is variable, it means the area under the curve of \( f(t) \) is calculated up to \( x \).
• The change in this area is what the derivative represents, aligning perfectly with the value of the function \( f(x) \) evaluated at this new boundary, or upper limit.

This concept isn't just theoretical but also allows for practical interpretation and application of integrals where boundaries shift, a common occurrence in real-world scenarios where dynamic functions are analyzed. Understanding variable upper limits are crucial in applying the Fundamental Theorem of Calculus effectively.