Problem 1 - Calculus

Question

Find the derivatives of the following functions \(F(x)\). $$ \int_{1}^{x} \cos ^{2} t d t $$

Step-by-Step Solution

Answer

The derivative is \( F'(x) = \cos^2 x \).

Step 1: Understand the Problem

We are given a function defined as a definite integral with a variable upper limit, \( F(x) = \int_{1}^{x} \cos^2 t \, dt \). We need to find its derivative \( \frac{d}{dx} F(x) \).

Step 2: Recognize the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus part 1, if \( F(x) \) is defined as \( F(x) = \int_{a}^{x} f(t) \, dt \), then its derivative is \( F'(x) = f(x) \). Here, \( f(t) = \cos^2 t \), so according to the theorem, \( F'(x) = \cos^2 x \).

Step 3: Conclusion: Apply the Theorem

Applying the theorem, the derivative of \( F(x) = \int_{1}^{x} \cos^2 t \, dt \) is \( F'(x) = \cos^2 x \). This result follows directly from the Fundamental Theorem of Calculus.

Key Concepts

Derivatives and the Fundamental Theorem of Calculus

Derivatives are the foundation of calculus. They express the rate of change of a function. Essential in physical sciences, engineering, and many fields, derivatives are about understanding how a quantity changes with respect to another. In the context of integrals, specifically, the Fundamental Theorem of Calculus bridges the seemingly distinct areas of differentiation and integration. It comprises two parts. The first part, crucial for our exercise, states that if you define a function as the integral of another function from a constant to a variable limit, the derivative will be the original function being integrated.

Here's how it applies to our problem:

• We have a function defined as an integral: \( F(x) = \int_{1}^{x} \cos^2 t \, dt \).
• By the theorem, the derivative of this function is simply the integrand evaluated at the upper limit: \( F'(x) = \cos^2 x \).

This powerful theorem simplifies the often complex task of finding derivatives of functions defined by integrals.

Understanding Definite Integrals

Definite integrals are a key concept in calculus that calculate the area under a curve within a specific interval. They are distinguished from indefinite integrals, which do not have limits and generally include an arbitrary constant of integration. In our problem, you encountered a definite integral with limits from 1 to \(x\).

Some key points about definite integrals include:

• The integral has both a lower limit (1) and an upper limit (\(x\)).
• The process involves the integrand function, \(\cos^2 t\), being evaluated between these two points.
• This work gives an accumulated sum (area) of values between those points.

Definite integrals play a vital role in many applications, such as calculating distances, areas, and even probabilities. In a calculus problem, solving a definite integral often involves finding the antiderivative and evaluating it at the limits.

The Cosine Function and Its Squares

The cosine function, noted as \( \cos(t) \), generates a wave-like graph, which is periodic and oscillates between -1 and 1. In our exercise, the function \( \cos^2(t) \) was involved.

Here are some important insights about this function:

• \( \cos^2(t) \) is derived from taking the square of the \( \cos(t) \) function.
• Squaring the cosine function results in a curve that is always non-negative, ranging from 0 to 1.
• This function has an important property: it can convert trigonometric functions into easier quadratic forms through identities like \( \cos^2(t) = \frac{1 + \cos(2t)}{2} \).

Understanding the behavior of \( \cos^2(t) \) is critical for solving integrals and derivatives involving this function, as it simplifies calculations and interpretations. Trigonometric identities are useful here, transforming expressions into manageable forms. The periodic nature and symmetry of \( \cos(t) \) make it a frequent subject in calculus and mathematics.