Problem 21 - Calculus

Question

Find \(G^{\prime}(x).\) $$ G(x)=\int_{x}^{\pi / 4}(s-2) \cot 2 s d s ; 0

Step-by-Step Solution

Answer

\(G'(x) = -(x-2) \cot 2x\).

Step 1: Identify the Fundamental Theorem of Calculus Part 1

The problem provides a function defined as a definite integral with variable limits of integration. The Fundamental Theorem of Calculus Part 1 applies here:\[ \frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x). \]Thus, if you have \(G(x) = \int_{x}^{\pi / 4} g(s) ds\), we need to find \(-g(x)\) because the upper limit \(x\) is in the lower bound of the integral.

Step 2: Apply the Fundamental Theorem of Calculus

Given the function \(G(x) = \int_{x}^{\pi / 4} (s-2) \cot 2s \, ds\), apply the theorem:\[ G'(x) = -[(x-2) \cot 2x]. \]Notice the negative sign due to \(x\) being the lower limit of integration.

Step 3: Simplify the Expression

Substitute \(x\) into the expression for \(g(s)\) to find:\[ G'(x) = -(x-2) \cot 2x. \]This expression represents the derivative of \(G(x)\).

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus used to find the total accumulation of quantities, such as areas under curves or total growth. In simple terms, a definite integral of a function over an interval gives us the net area between the function and the x-axis within that interval. This area can be above or below the x-axis. A standard notation for a definite integral is:

• \( \int_{a}^{b} f(x) \, dx \)

Here, \( a \) and \( b \) denote the boundaries (or limits) of the integration.
One key property of definite integrals is that they can be evaluated using the Fundamental Theorem of Calculus, which links differentiation and integration. This theorem states that if \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then:

• \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \)

This means we compute the antiderivative \( F \), evaluate it at the upper limit \( b \), and subtract the evaluation at the lower limit \( a \) to find the integral's value.

Derivatives

Derivatives represent the rate at which a function changes as its input changes. In other words, the derivative of a function at a particular point tells how the function value changes as you move along its curve. The most common notation for a derivative is \( f'(x) \) or \( \frac{df}{dx} \).
Consider a simple function \( f(x) \). Its derivative, \( f'(x) \), gives us the slope of the tangent line to the curve at any point. This is crucial in many fields, including physics and engineering, where we often need to understand how things change over time.

• The derivative of a constant is 0.
• The power rule for differentiation: \( \frac{d}{dx} \left[ x^n \right] = n x^{n-1} \).
• The product rule: \( \frac{d}{dx} [u v] = u'v + uv' \).
• The chain rule: Used for composite functions: \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \).

With derivatives and integrals both central to calculus, they are two sides of the same coin. The Fundamental Theorem of Calculus bridges them by showing that integrating a derivative brings us back to the original function, up to a constant.

Variable Limits of Integration

Variable limits of integration occur when the limits of an integral are functions of a variable rather than constants. This matches scenarios where the range of integration can shrink or grow, impacting how the integral represents an accumulation of quantities. An integral with variable limits often takes a form like:

• \( \int_{g(x)}^{h(x)} f(t) \, dt \)

Here the limits \( g(x) \) and \( h(x) \) are functions of \( x \), making it necessary to apply the Fundamental Theorem of Calculus with care. When differentiating an integral with variable limits

• the derivative of \( \int_{a}^{x} f(t) \, dt \) with respect to \( x \) gives \( f(x) \),
• if the lower limit is \( x \), like in \( \int_{x}^{b} f(t) \, dt \), the theorem results in \(-f(x)\).

Understanding this concept is vital for problems involving dynamic systems and rates of change, where the limits themselves change with respect to another variable like time or position. This aspect of calculus is widely used in physics and other applied sciences to model real-world problems.