Problem 78 - Calculus: Early Transcendentals

Question

If \( f \) is continuous and \( g \) and \( h \) are differentiable functions, find a formula for $$ \displaystyle \frac{d}{dx} \int^{h(x)}_{g(x)} f(t) \, dt $$

Step-by-Step Solution

Answer

The derivative is \( f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x) \).

Step 1: Identify the Given Expression

We need to differentiate the definite integral \( \int^{h(x)}_{g(x)} f(t) \, dt \) with respect to \( x \). The integral has variable limits \( h(x) \) and \( g(x) \).

Step 2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \( f \) is continuous on \([a, b]\), then \( \frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x) \). We use this theorem to differentiate the integral, but here, both limits are functions of \( x \).

Step 3: Use the Leibniz Rule for Differentiating Under the Integral Sign

The Leibniz Rule states that if both the upper and lower limits are functions of \( x \), then \[\frac{d}{dx} \int_{g(x)}^{h(x)} f(t) \, dt = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x)\]This formula accounts for changes in function values at both upper and lower limits due to their dependence on \( x \).

Step 4: Write the Final Formula

Incorporating the previous step, the formula to differentiate the given expression is:\[\frac{d}{dx} \int_{g(x)}^{h(x)} f(t) \, dt = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x)\]

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration, two central operations in calculus. It consists of two parts. The first part provides an anti-derivative for a continuous function. It states that if a function \( f \) is continuous over an interval \( [a, b] \), then the function \( F(x) = \int_{a}^{x} f(t) \, dt \) is differentiable over \( (a, b) \) and \( F'(x) = f(x) \).
This means that integration can be reversed by differentiation. The theorem gives us a powerful tool to evaluate definite integrals and compute areas under curves.

• Integration: The process of finding the area under a curve.
• Differentiation: The process of calculating the rate at which a function changes.

The second part of the theorem provides a method to find definite integrals. If \( F \) is an antiderivative of \( f \) on an interval \( [a, b] \), then \( \int_{a}^{b} f(t) \, dt = F(b) - F(a) \). This formula is the reason we can evaluate integrals so efficiently.

Leibniz Rule

The Leibniz Rule is essential for dealing with integrals with variable limits that depend on another variable. This is especially common in problems involving parameters or conditions that change dynamically. For an expression like \( \int_{g(x)}^{h(x)} f(t) \, dt \), the Leibniz Rule gives us the derivative of this integral with respect to \( x \):

• \( f(h(x)) \cdot h'(x) \) is the contribution from the upper limit \( h(x) \).
• \( f(g(x)) \cdot g'(x) \) is the contribution from the lower limit \( g(x) \).

The rule essentially breaks down the complex differentiation process into manageable parts:
1. Differentiate the upper limit's effect while considering the function's value there.
2. Subtract the effect from the lower limit's derivative.
This is a straightforward extension of the Fundamental Theorem of Calculus to integrals with function limits, allowing us to account for the rate of change at both ends.

Differentiation

Differentiation is a core concept in calculus. It is the calculation of a derivative, which represents the rate at which a function is changing at any given point. For example, when you want to understand how a function behaves as you adjust an input, differentiation provides this insight by describing how the output value rapidly increases or decreases.

• Notation: Commonly denoted as \( f'(x) \) or \( \frac{df}{dx} \).
• Chain Rule: A key rule used for differentiating composite functions.

Differentiation allows us to:
- Analyze trends and predict behavior in data or models.
- Solve complex real-world problems like finding the velocity of an object.
Example: If \( y = x^2 \), the derivative \( \frac{dy}{dx} = 2x \) tells us that the slope of the curve \( y = x^2 \) at any point \( x \) is \( 2x \).
This foundational process enables deeper exploration of mathematical and physical phenomena.

Continuous Functions

A continuous function is a function that, intuitively, can be drawn without lifting your pen from the paper. Mathematically, a function \( f(x) \) is continuous at a point \( a \) if the following conditions are met:

• \( f(a) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) equals \( f(a) \).

This means there are no breaks, no holes, and no jumps in the function's graph around that point. Continuity is crucial because many fundamental calculus theorems, like the Intermediate Value Theorem and, importantly, the Fundamental Theorem of Calculus, rely on it.

• No gaps: Continuous functions do not have interruptions.
• Smooth paths: They ensure smooth transitions in their respective graphs.

Understanding continuity allows us to grasp more complex ideas in calculus, like integration and the behavior of functions under differentiation, guaranteeing predictable outcomes.