Problem 81 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(f\) is continuous on \([a, b]\). (a) Let \(G(x)=\int_{a}^{x} f(t) d t\). Show that \(G\) is continuous on \([a, b]\). (b) Let \(F(x)\) be any antiderivative of \(f\) on \([a, b]\). Show that \(F\) is continuous on \([a, b]\).

Step-by-Step Solution

Answer

Both \( G(x) \) and \( F(x) \) are continuous on \([a, b]\) because \( f \) is continuous on this interval.

Step 1: Understanding the Given Function

We are given that the function \( f \) is continuous on the interval \([a, b]\). We need to show that the function \( G(x) = \int_{a}^{x} f(t) \, dt \) is continuous on the same interval.

Step 2: Apply the Fundamental Theorem of Calculus (FTC)

According to the Fundamental Theorem of Calculus, if \( f \) is continuous on \([a, b]\), then the function \( G(x) = \int_{a}^{x} f(t) \, dt \) is continuous on \([a, b]\). This is because the integral of a continuous function over a closed interval yields another continuous function.

Step 3: Show Continuity of \( G \) Using Limits

For any point \( c \in [a, b] \), we need to show that the limit of \( G(x) \) as \( x \to c \) equals \( G(c) \). Since \( f \) is continuous, by properties of definite integrals and the continuity of \( f \), \( \lim_{x\to c} \int_{a}^{x} f(t) \, dt = \int_{a}^{c} f(t) \, dt = G(c) \), proving \( G \) is continuous.

Step 4: Understanding \( F(x) \) as an Antiderivative

An antiderivative \( F(x) \) of \( f(x) \) satisfies the condition \( F'(x) = f(x) \). Our task is to show that \( F(x) \) is continuous over \([a, b]\).

Step 5: Show Continuity of Antiderivative \( F \)

Since \( f \) is continuous on \([a, b]\) and thus \( F'(x) = f(x) \) is well-defined and continuous on \([a, b]\), \( F \) must be continuous everywhere that its derivative exists. Thus, \( F \) is continuous on \([a, b]\).

Step 6: Concluding the Proofs

We have shown that both \( G(x) \) and \( F(x) \) are continuous on \([a, b]\) by applying the Fundamental Theorem of Calculus and the definition of an antiderivative, combined with the given that \( f \) is continuous.

Key Concepts

Continuity of Functions

Understanding the continuity of a function is crucial in calculus. A function is said to be continuous on an interval if, roughly speaking, you can draw it without lifting your pencil from start to finish. More formally, a function \( f(x) \) is continuous at a point in its domain if the limit of \( f(x) \) as \( x \) approaches any point \( c \) is equal to \( f(c) \).
This concept ensures that there are no sudden jumps or breaks in the graph of the function.For a function defined over a closed interval, such as \([a, b]\), continuity implies a smooth transition across that interval. Here are a few highlights:

• A continuous function on \([a, b]\) will have defined limits as \( x \) approaches any point within this interval.
• Continuity is crucial for the application of the Fundamental Theorem of Calculus, as it guarantees the existence of a smooth antiderivative and the evaluation of definite integrals.

The continuous nature of \( f \) on \([a, b]\) reassures us of constructing integrals and antiderivatives that are themselves continuous. This forms the basis of many problem-solving techniques in calculus.

Definite Integrals

Definite integrals serve as a powerful tool in calculus for finding the total accumulation of a quantity over an interval. When calculating a definite integral \( \int_{a}^{b} f(t) \, dt \), where \( f(t) \) is continuous on \([a, b]\), this process involves finding the signed area between the function \( f(t) \) and the \( t \)-axis from \( a \) to \( b \).The Fundamental Theorem of Calculus tells us that if \( f \) is continuous, then the integral \( \int_{a}^{x} f(t) \, dt \) is continuous. This means the integral, resulting in the function \( G(x) \), smoothly transitions without jumps as \( x \) moves across different points in \([a, b]\).

• This smooth transition reflects the natural idea that the accumulation of values does not suddenly leap or drop.
• The evaluation of definite integrals helps in finding the total change or total quantity over a specified interval.

Definite integrals provide more than just numerical results; they also represent the continuity and connectivity of changes over an interval, making them fundamental to understanding calculus' deeper insights.

Antiderivatives

Antiderivatives, or indefinite integrals, are crucial in reversing the process of differentiation. Given a continuous function \( f(x) \), an antiderivative \( F(x) \) is anything whose derivative gives back the original function, meaning \( F'(x) = f(x) \).In essence, finding an antiderivative involves discovering a function that represents the accumulated change of \( f(x) \). Now, since \( f(x) \) is given as continuous over the interval \([a, b]\), its antiderivative will also exhibit continuity across this interval.

• The continuity of \( F(x) \) is rooted in the fact that its derivative, \( f(x) \), is defined and smooth across \([a, b]\).
• This continuity assures that expectations such as smooth graphs and well-behaved limits at boundaries are maintained.

Antiderivatives not only place functions within a broader spectrum of continuity but also serve as a bridge to evaluating definite integrals and comprehending accumulative progressions on intervals.