Problem 43 - Calculus Early Transcendentals

Question

Determine whether the following statements are true and give an explanation or counterexample. a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function. b. If the number of subintervals used in the Midpoint Rule is increased by a factor of \(3,\) the error is expected to decrease by a factor of 8. c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of \(4,\) the error is expected to decrease by a factor of 16.

Step-by-Step Solution

Answer

Provide an explanation or counterexample. a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function. b. If the number of subintervals used in the Midpoint Rule is increased by a factor of 3, the error is expected to decrease by a factor of 8. c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of 4, the error is expected to decrease by a factor of 16. Answer: a. True. The Trapezoid Rule will generate the exact area under the curve as a trapezoid for linear functions. b. False. The decrease in error will be by a factor of 9, not 8. c. True. The error will decrease by a factor of 16 when increasing the number of subintervals by a factor of 4 in the Trapezoid Rule.

Step 1: a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function.

The Trapezoid Rule is a method for approximating the definite integral of a function by calculating the area of trapezoids beneath the curve. When a linear function is used, the area under the curve will form a trapezoid, which will be the exact area. Therefore, the statement is true. To understand this, consider any linear function \(f(x)= m * x + c\). When applying the Trapezoid Rule with a single interval, it will generate the exact area under the curve as a trapezoid, and thus the approximation will be equal to the exact value of the definite integral.

Step 2: b. If the number of subintervals used in the Midpoint Rule is increased by a factor of 3, the error is expected to decrease by a factor of 8.

The error in the Midpoint Rule is proportional to the fourth derivative of the function, which is integrated, and the square of the width of the subintervals. If the number of subintervals is increased by a factor of 3, then the width of the subintervals is reduced by a factor of 3. Therefore, the square of the width will be reduced by a factor of 9. Therefore, the statement is false. As a counterexample, consider any smooth function, such as \(f(x) = x^3\). If we increase the number of subintervals by a factor of 3, the decrease in error will be a factor of 9, as opposed to the claimed factor of 8.

Step 3: c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of 4, the error is expected to decrease by a factor of 16.

The error in the Trapezoid Rule is proportional to the maximal value of the second derivative of the function, which is integrated, and the square of the width of the subintervals. If the number of subintervals is increased by a factor of 4, then the width of the subintervals is reduced by a factor of 4. Therefore, the square of the width will be reduced by a factor of 16. This statement is true. For example, consider any smooth function, such as \(f(x)=x^4\). If we increase the number of subintervals by a factor of 4 in this case, the error will decrease by a factor of 16, as stated in the problem.

Key Concepts

Definite Integral

Understanding the definite integral is critical in calculus, as it represents the accumulation of quantities, like area under a curve, and bridges the conceptual gap between antiderivatives and area. In formal terms, the definite integral of a function between two limits represents the signed area under the curve of the function graphed on a coordinate plane, between the vertical lines corresponding to these limits.

For instance, when considering a function graphed on the xy-plane, the definite integral from a point a to b corresponds to the total 'net' area sandwiched between the function curve, the x-axis, and the vertical lines x=a and x=b. In calculus terminology, this is written as \( \[ \int_a^b f(x) \, dx \] \). This integral is a numerical value, which can sometimes be evaluated exactly using anti-differentiation. However, when functions are complex or do not have elementary antiderivatives, numerical methods like the Trapezoid Rule and the Midpoint Rule come into play.

Midpoint Rule

The Midpoint Rule is a numerical method used to approximate the definite integral of a function over a certain interval. Unlike other methods which might use the endpoints or trapezoids to approximate the area, the Midpoint Rule considers the value of the function at the midpoint of each subinterval to approximate the area under the curve.

To apply the Midpoint Rule, you first divide the entire interval into smaller subintervals of equal width, and then calculate the function value at the midpoint of each subinterval. These values are multiplied by the width of the subintervals and summed up to give the total approximate area. Mathematically expressed, if there are n subintervals of width \( \Delta x \) in an interval \( [a, b] \), the Midpoint Rule approximation is given by \( M_n = \Delta x \left[ f(m_1) + f(m_2) + ... + f(m_n) \right] \), where \( m_i \) is the midpoint of the i-th subinterval.

Subinterval Number and Error

As highlighted in the exercise solution, the precision of the Midpoint Rule is sensitive to the number of subintervals used. Increasing the number of subintervals makes the approximation more accurate, effectively reducing the error. The error decreases in proportion to the fourth power of the interval width. This is why if the number of subintervals is tripled, the expectation that the error will decrease by a factor of 8 is erroneous; instead, it will decrease by a factor of 9.

Numerical Integration

Numerical integration encompasses a range of algorithms for calculating the numerical value of a definite integral. This is especially useful in cases where an analytic solution is difficult or impossible to obtain. It can be due to the integrand being complex, lacking a closed-form expression, or too irregular. Numerical integration techniques approximate the integral by summing the values of the function at selected points within the integration interval and multiplying by a factor related to the width of the intervals.

The Trapezoid Rule and the Midpoint Rule are two prime examples of numerical integration methods. These methods substitute the actual area under the curve with shapes like trapezoids and rectangles whose areas are easily calculable. The number of shapes (or subintervals) and their respective accuracy largely determine the precision of the approximate integral.

Role of Subintervals

In addition to understanding the approximation methods themselves, knowing the relationship between the number of subintervals and the estimation error is critical. The rule of thumb is: the finer the subdivision (more subintervals), the lesser the error and the better the approximation. This relation is highlighted in the exercise solution where increasing the number of subintervals in the Trapezoid Rule by a factor of 4 results in the error decreasing by a factor of 16, showing how error diminishes sharply with finer subdivisions.