Problem 10 - Precalculus: Mathematical for Calculus

Question

(a) Estimate the area under the graph of \(f(x)=e^{-x}\) \(0 \leq x \leq 4,\) using four approximating rectangles and taking the sample points to be (i) right endpoints (ii) left endpoints In each case, sketch the curve and the rectangles. In each case, sketch the curve and the rectangles. (b) Improve your estimates in part (a) by using eight rectangles.

Step-by-Step Solution

Answer

Using four rectangles: right endpoints = 0.5713, left endpoints = 1.5530. Using eight rectangles: right endpoints = 0.6150, left endpoints = 0.8771.

Step 1: Understanding the Problem

We need to estimate the area under the curve of the function, \( f(x) = e^{-x} \), over the interval \([0, 4]\) using two different methods: right endpoints and left endpoints, with four and then eight rectangles.

Step 2: Calculate Width of Rectangles

For four rectangles, the width \( \Delta x \) is calculated as \( \Delta x = \frac{4 - 0}{4} = 1 \). For eight rectangles, \( \Delta x = \frac{4 - 0}{8} = 0.5 \).

Step 3: Right Endpoint Approximation (4 Rectangles)

Using the right endpoint method with four rectangles, the sample points are \( x = 1, 2, 3, 4 \). The area is approximated as \( \Delta x [f(1) + f(2) + f(3) + f(4)] \). Calculate: \[1 \times (e^{-1} + e^{-2} + e^{-3} + e^{-4})\approx 1 \times (0.3679 + 0.1353 + 0.0498 + 0.0183)\approx 0.5713.\]

Step 4: Left Endpoint Approximation (4 Rectangles)

Using the left endpoint method with four rectangles, the sample points are \( x = 0, 1, 2, 3 \). The area is approximated as \( \Delta x [f(0) + f(1) + f(2) + f(3)] \). Calculate: \[1 \times (1 + e^{-1} + e^{-2} + e^{-3}) \approx 1 \times (1 + 0.3679 + 0.1353 + 0.0498) \approx 1.5530.\]

Step 5: Right Endpoint Approximation (8 Rectangles)

With eight rectangles and right endpoints, the sample points are \( x = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4 \). Calculate the area: \[0.5 \times (f(0.5) + f(1) + f(1.5) + f(2) + f(2.5) + f(3) + f(3.5) + f(4))\]\[\approx 0.5 \times (0.6065 + 0.3679 + 0.2231 + 0.1353 + 0.0821 + 0.0498 + 0.0302 + 0.0183) \approx 0.6150.\]

Step 6: Left Endpoint Approximation (8 Rectangles)

Using eight rectangles with left endpoints, the sample points are \( x = 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5 \). Calculate the area: \[0.5 \times (f(0) + f(0.5) + f(1) + f(1.5) + f(2) + f(2.5) + f(3) + f(3.5))\]\[\approx 0.5 \times (1 + 0.6065 + 0.3679 + 0.2231 + 0.1353 + 0.0821 + 0.0498 + 0.0302) \approx 0.8771.\]

Key Concepts

Area Under a Curve

Calculating the area under a curve is a crucial aspect of understanding how functions behave over a particular interval. This area is often calculated using methods from calculus to provide a precise measurement of the space between the curve and the x-axis.
For practical purposes, especially when dealing with integrals that can be complex to solve analytically, approximations like the Riemann Sum are used.

• The Riemann Sum divides the area into multiple rectangles, making it easier to mentally picture the area.
• These rectangles can vary in number, and increasing the number usually leads to a more accurate approximation of the area.

In the context of the exercise, the function given, specifically, is an exponential function over the interval \([0, 4]\), and our goal is to determine the area under this curve. The more rectangles used, the closer our approximation comes to representing the true area under the curve.

Right Endpoint Approximation

The right endpoint approximation is a method within the Riemann Sum framework used to estimate the area under a curve. This approach involves using the rightmost point of each subinterval to determine the height of the rectangles.
This means that, for a function over a certain interval that has been divided into several segments, you use the value of the function at the right endpoint of each segment to calculate the height of your rectangles.

• In our exercise, for 4 rectangles, the right endpoints are at x-values 1, 2, 3, and 4.
• For 8 rectangles, they are at 0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4.

The choice of using the right endpoint makes this method susceptible to either underestimation or overestimation depending on whether the function is increasing or decreasing. In the case of the function \(f(x)=e^{-x}\), which is a decreasing function, using the right endpoints results in an underestimation of the actual area under the curve. This happens because each rectangle formed is shorter than if we would have used the taller left endpoint rectangle.

Left Endpoint Approximation

Similarly to the right endpoint approximation, the left endpoint approximation is another technique used to estimate the area under a curve. The difference lies in the choice of points to measure each rectangle's height.
With the left endpoint approach, you use the value at the leftmost point of each subinterval. This decision impacts the result, especially in estimating the area and in terms of accuracy.

• For the function in our problem, using 4 rectangles entails sample points at x = 0, 1, 2, and 3.
• If using 8 rectangles, the sample points become x = 0, 0.5, 1, 1.5, 2, 2.5, 3, and 3.5.

Due to the decreasing nature of the exponential function \(f(x)=e^{-x}\), using the left endpoint results in overestimating the area. The left endpoint rectangles are taller than the actual values within each interval, covering more area than the curve itself.

Exponential Function

An exponential function like \(f(x) = e^{-x}\) is fundamental in mathematics, often involved in growth and decay models. In general, an exponential function can be expressed in the form \(f(x) = a \, e^{bx}\), where "e" is Euler's number (approximately 2.71828).
This type of function is interesting because it consistently grows or decays at an exponential rate across the x-values.

• For \(f(x)=e^{-x}\), the function decreases as x increases, drastically affecting the shape of the curve and thus the calculations of the area under it.
• The function approaches zero as x approaches infinity, indicating the curve never quite touches the x-axis but gets arbitrarily close.

Understanding the behavior of exponential functions is crucial when applying area approximation techniques such as the Riemann sums. The exponential decay leads to distinct approximation outcomes like those visible in right and left endpoint methods—the key visual here being the smooth, continuously decreasing nature of the graph.