Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f^{\prime}\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0,\) then \(\int_{0}^{\infty} f^{\prime}(x) d x=-f(0)\)

Short answer

The statement is true. Because the Fundamental Theorem of Calculus, Part I states that the integral of a function's derivative over its domain equals the function evaluated at the upper limit of the interval of integration minus the function evaluated at the lower limit. As such, we can apply this to the given function, considering that \( \lim _{x \rightarrow \infty} f(x)=0.\)

Step-by-step solution

Use the Fundamental Theorem of Calculus

The first thing we need to do is use the Fundamental Theorem of Calculus. According to this theorem, if F is an antiderivative of \(f\), then \(\int_{a}^{b} f(x) d x=F(b)-F(a).\). Let's apply this theorem to the given integral to give us \(\int_{0}^{\infty} f^{\prime}(x) d x=f(\infty)-f(0)\)

Substitute the limit

Next, we substitute \(\lim _{x \rightarrow \infty} f(x)=0\) into the equation from Step 1, so we get: \(f(\infty)-f(0) = 0 - f(0) = -f(0)\)

Compare the results

Comparing the results from Step 1 and Step 2, we can see they are the same. Thus the given statement is true.

Key concepts

Continuous Functions

When we discuss continuous functions, we're looking at functions that are smooth and unbroken. This means you can draw the graph of the function without lifting your pencil from the paper. Formally, a function is continuous at a point if the following three conditions are met: the function is defined at that point, the limit exists at that point, and the function's value at that point equals the limit value. For example, the function defined by f(x) = x^2 is continuous everywhere.

Continuity is crucial in calculus because it ensures that functions behave predictably and that certain properties hold, such as the ability to find antiderivatives. The Fundamental Theorem of Calculus, for instance, requires the function to be continuous over the interval of integration to apply. When we integrate a continuous derivative like f'(x) over an interval, it tells us the net change in the original function f(x) over that interval.

Improper Integrals

An improper integral involves the integration over an unbounded interval or an integrand that approaches infinity within the interval. To manage these infinities, we use limits to define the integral. Consider \(\int_{a}^{\infty} f(x) \, dx\), we can't just plug infinity into the upper limit of integration, so we replace it with a variable bound, like \(\int_{a}^{b} f(x) \, dx\), and then take the limit as \(b\) approaches infinity.

For the given exercise, we're dealing with \(\int_{0}^{\infty} f'(x) \, dx\) which turns into an improper integral because the upper limit is infinity. Here, by assuming \(f'\) is continuous, we are allowed to interpret the integral in terms of the limit of the antiderivative of \(f(x)\) as \(x\) approaches infinity. Doing this carefully is important to avoid misinterpreting the behavior of the function at infinity, which could lead to incorrect solutions.

Limits of Functions

The concept of limits is foundational in calculus. It describes the behavior of a function as the input approaches a certain value. For example, \(\lim_{x \to c} f(x) = L\) means that as \(x\) gets closer and closer to \(c\), the function's values get arbitrarily close to \(L\). This does not necessarily mean that \(f(c) = L\), especially if \(f\) is not defined at \(c\).

In the context of the exercise, we examine the limit as \(x\) tends towards infinity, which informs us about the end behavior of the function. As \(\lim_{x \to \infty} f(x) = 0\), this means that as \(x\) grows larger and larger, the values of \(f(x)\) approach 0. In evaluating the improper integral, this limit plays a part in determining the value of the integral as it gives us the behavior of \(f(x)\) 'at infinity'—a pivotal concept when integrating with respect to unbounded domains.