Problem 63 - Calculus Early Transcendentals

Question

Suppose \(f\) is positive and its first two derivatives are continuous on \([a, b] .\) If \(f^{\prime \prime}\) is positive on \([a, b],\) then is a Trapezoid Rule estimate of \(\int_{a}^{b} f(x) d x\) an underestimate or overestimate of the integral? Justify your answer using Theorem 7.2 and an illustration.

Step-by-Step Solution

Answer

Answer: Under these conditions, the Trapezoid Rule estimate of the integral is an underestimate.

Step 1: Recall the Trapezoid Rule formula and the error bound in Theorem 7.2

The Trapezoid Rule formula is an approximation for the definite integral and is given by: \[ \int_{a}^{b} f(x) \, dx \approx T_n = \frac{b-a}{2n} \left[ f(a) + 2\sum_{i=1}^{n-1} f(a + i \cdot \frac{b-a}{n}) + f(b) \right] \] Theorem 7.2 provides us with the error bound for the Trapezoid Rule estimate of the integral: \[ \textit{Error in trapezoid rule} = \frac{(b-a)^3}{12n^2} \cdot K \] where \(K\) is an upper bound for \(|f^{\prime\prime}(x)|\) on the interval \([a,b]\) and \(n\) is the number of subintervals.

Step 2: Analyzing the conditions and Theorem 7.2

We are given that \(f^{\prime\prime}(x) > 0\) on the interval \([a,b]\). Since \(f^{\prime\prime}(x)\) is positive, it means that the function \(f(x)\) is concave up on the interval \([a,b]\). This tells us that in the trapezoid rule estimate, the approximation will lie below the actual curve for a concave-up function. Now let's examine the error term from Theorem 7.2, given by \(\frac{(b-a)^3}{12n^2}\cdot K\). Since we have a concave-up function with a positive second derivative, \(K > 0\). Moreover, since \((b-a)\) and \(n\) are both positive, the error term is also positive. This error term being positive implies that the Trapezoid Rule approximation for the integral \(\int_{a}^{b} f(x) \, dx\) is an underestimate.

Step 3: Providing an illustration

To further illustrate this, let's consider a simple concave-up function such as \(f(x) = x^2\). When using the Trapezoid Rule to approximate the integral of \(f(x)\), the area of the trapezoids will underestimate the actual area under the curve of \(f(x) = x^2\), as seen in the image below:  This illustration confirms that the Trapezoid Rule estimate is an underestimate when the given conditions are met. Hence, to answer the question: When \(f^{\prime\prime}(x)\) is positive on \([a, b]\), a Trapezoid Rule estimate for the integral \(\int_{a}^{b} f(x) \, dx\) will be an underestimate. This is justified by Theorem 7.2 and supported by the visual illustration of an concave-up function.

Key Concepts

Concavity

Concavity is an important concept that you encounter when dealing with functions and integrals. In calculus, it refers to the direction in which a function curves. Specifically, if a function is **concave up**, it means that the function is curving upwards like a cup. This is typically characterized by a positive second derivative, \( f''(x) > 0 \).In our original problem, the function \( f(x) \) is concave up, since \( f''(x) \) is positive on the interval \( [a, b] \). When a function is concave up, the tangent lines lie below the curve. Therefore, approximations involving line segments, like those used in the Trapezoid Rule, also tend to lie below the actual curve, leading to an underestimate.To visualize concavity, consider a simple function like \( f(x) = x^2 \). Its graph is a parabola curving upwards. Any tangent or linear approximation to this curve stays beneath the parabola for any segment on the interval, thus reinforcing the rule that a positive second derivative results in a concave-up function.

Error Bound

The Trapezoid Rule is a numerical method for approximating definite integrals, and like all approximations, it comes with an error. The **error bound** in the Trapezoid Rule helps us understand how close our approximation is to the actual integral value.According to Theorem 7.2, the error in the Trapezoid Rule can be estimated using the formula:\[\text{Error} = \frac{(b-a)^3}{12n^2} \cdot K\]where \( (b-a) \) is the length of the interval, \( n \) is the number of subintervals, and \( K \) represents an upper bound for \(|f''(x)|\) on \([a, b]\).In our exercise, since \( f''(x) > 0 \,\) \( K \) is positive, making the error positive. This means the Trapezoid Rule produces an estimate that is a little less than the true integral value. As \( n \) increases, meaning more subintervals, the error decreases, which reflects the importance of using more subintervals for better accuracy.

Definite Integral

A **definite integral** represents the accumulation of quantities, such as area under a curve between two points, \( a \) and \( b \). Mathematically, it is expressed as \( \int_{a}^{b} f(x) dx \).In the context of the original problem, we wish to find the area under the curve \( f(x) \) from \( a \) to \( b \). However, dealing with definite integrals analytically can sometimes be difficult or unsolvable. That's why numerical methods like the Trapezoid Rule are useful because they provide a way to approximate these integrals.The Trapezoid Rule specifically approximates the area under the curve by dividing it into trapezoidal sections, summing up their areas to estimate the whole. While this method does offer a result that might not be perfectly accurate, understanding the properties of the function, like concavity, can give us clues on whether the approximation is an overestimate or an underestimate. In this problem, since the function is concave up, the Trapezoid Rule delivers an underestimated value of the definite integral.