Question

Prove the mean value theorem for integrals, stated below. Assume that \(g \in C[a, b]\) and that \(\psi\) is an integrable function that is either nonnegative or nonpositive throughout the interval \([a, b]\). Then there is a point \(\xi \in[a, b]\) such that $$ \int_{a}^{b} g(x) \psi(x) d x=g(\xi) \int_{a}^{b} \psi(x) d x $$ [Hint: Bound \(g\) below and above by its minimum and maximum values on the interval, respectively, then bound the desired integral value likewise and use the Intermediate Value Theorem given on page \(10 .]\)

Short answer

Question: Prove the Mean Value Theorem for integrals by following the given plan: 1. Bound the function g by its minimum and maximum values on the interval [a, b]. 2. Bound the integral value similarly. 3. Apply the Intermediate Value Theorem to find a point ξ in [a, b] that satisfies the given equation. Answer: To prove the Mean Value Theorem for integrals, we follow these steps: Step 1: Since the function g is continuous on the interval [a, b], it attains its minimum, m, and maximum, M, on that interval. Therefore, we have m ≤ g(x) ≤ M for all x in [a, b]. Step 2: Multiply the inequality found in Step 1 by ψ(x) and integrate each term with respect to x from a to b. We obtain the following inequality: m ∫ψ(x) dx ≤ ∫g(x)ψ(x) dx ≤ M ∫ψ(x) dx Step 3: Now, we apply the Intermediate Value Theorem for continuous functions. Since the function G(x) = g(x)ψ(x) is continuous on [a,b], there exists a ξ in [a, b] such that G(ξ) = ∫g(x)ψ(x) dx. This gives us: g(ξ)ψ(ξ) = ∫g(x)ψ(x) dx Dividing both sides by ψ(ξ) (provided ψ(ξ) ≠ 0, otherwise the integral on both sides is 0 and the theorem holds trivially) yields the desired result: g(ξ) = ∫g(x)ψ(x) dx / ∫ψ(x) dx

Step-by-step solution

Find the minimum and maximum of \(g\) on [a, b]

Since \(g\) is continuous on the interval \([a, b]\), by the Extreme Value Theorem, it attains its minimum, \(m\), and maximum, \(M\), on that interval. Therefore, we have \(m \leq g(x) \leq M\) for all \(x\in[a, b]\).

Bound the integral value

Multiply the inequality found in Step 1 by \(\psi(x)\): $$ m \psi(x) \leq g(x)\psi(x) \leq M \psi(x) $$ Since \(\psi(x)\) is either nonnegative or nonpositive, we can now integrate each term with respect to \(x\) from \(a\) to \(b\): $$ m\int_{a}^{b} \psi(x) d x \leq \int_{a}^{b} g(x) \psi(x) d x \leq M \int_{a}^{b} \psi(x) d x $$

Apply the Intermediate Value Theorem

Now, we apply the Intermediate Value Theorem for continuous functions. Since the function \(G(x) = g(x)\psi(x)\) is continuous on \([a,b]\), there exists a \(\xi \in [a, b]\) such that $$ G(\xi) = \int_{a}^{b} g(x) \psi(x) d x $$ which gives us $$ g(\xi) \psi(\xi) = \int_{a}^{b} g(x) \psi(x) d x $$ Dividing both sides by \(\psi(\xi)\) (provided \(\psi(\xi)\neq0\), otherwise the integral on both sides is \(0\) and the theorem holds trivially) gives the desired result: $$ g(\xi) = \frac{\int_{a}^{b} g(x) \psi(x) d x}{\int_{a}^{b} \psi(x) d x} $$

Key concepts

Extreme Value Theorem

The Extreme Value Theorem is a fundamental result in calculus that assures us that a continuous function defined on a closed interval has both a minimum and a maximum. In simpler terms, if you have a smooth curve with no breaks on a piece of paper that starts and ends at distinct points, you will definitely be able to point at the highest and lowest points on this curve.

For a function, say f(x), that is continuous on an interval [a, b], there exist points c and d in the interval such that f(c) is the minimum value and f(d) is the maximum value. It doesn't matter how the curve wiggles in between, those extreme values must exist. This fact is crucial in calculus, especially when dealing with integrals and functions' behaviors, as it lays the groundwork for finding bounded areas and understanding the behavior of functions.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is great for filling in the gaps. It states that for a function f that is continuous on an interval [a, b], and for any number N between f(a) and f(b), there exists a c in [a, b] such that f(c) = N.

Imagine you're driving from the bottom to the top of a hill continuously—no teleporting allowed! The IVT tells us that at some point you must have been at every elevation between the bottom and top. The theorem is instrumental when we work with integrals, as it helps us guarantee the existence of certain averages or values without necessarily knowing their exact location.

Continuous Functions

Continuous functions are the bread and butter of calculus—they're the ones that draw a complete, unbroken line without any jumps or gaps. More formally, a function is continuous at a point if its limit as it approaches that point is equal to the function's value at that point. This means no surprises; what you expect is exactly what you get.

To check if a function is continuous over an interval, we look to see if it fulfills this 'no surprise' condition at every point within the interval. This not only includes the points within the range but also the ends—assuming the interval is closed. Continuous functions are nice to us because they are predictable and we can apply the Intermediate Value Theorem to them, as well as use them reliably in the Mean Value Theorem for Integrals.

Integrable Functions

Integrable functions may not always play as nice as continuous functions, but they still follow rules that allow us to calculate the area under their curves. In crude terms, if you can draw a function's graph and envision calculating the area under it without pulling your hair out, it's probably integrable. More formally, if a function can be approximated by sums of areas of rectangles (in a process called Riemann sum) as closely as we like, it's integrable.

The beauty of integrable functions is that they don't have to be continuous everywhere, they might have some points of discontinuity (think a sudden jump), but as long as certain conditions are met (like being bounded and having only a finite number of discontinuities on a closed interval), they can still play the integral game. This flexibility allows us to work with a wider range of functions mathematically when analyzing real-world scenarios.