Problem 12 - Precalculus: Mathematics for Calculus

Question

In these exercises we estimate the area under the graph of a function by using rectangles. (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. (b) Improve your estimates in part (a) by using eight rectangles.

Step-by-Step Solution

Answer

Area estimates are more accurate with eight rectangles than four.

Step 1: Understand the problem

We need to estimate the area under the curve of the function \(f(x) = e^{-x}\) from \(x = 0\) to \(x = 4\). We will use approximating rectangles to estimate the area, considering both right and left endpoints as sample points. Additionally, we'll improve the estimate by using eight rectangles instead of four.

Step 2: Divide the interval for four rectangles

First, divide the interval between 0 and 4 into four equal parts, each with a width \(\Delta x = \frac{4 - 0}{4} = 1\). The subintervals are: [0,1], [1,2], [2,3], and [3,4].

Step 3: Estimate using right endpoints with four rectangles

For right endpoints, evaluate \(f(x)\) at the right endpoint of each subinterval: - \(x=1\), \(f(1)=e^{-1}\)- \(x=2\), \(f(2)=e^{-2}\)- \(x=3\), \(f(3)=e^{-3}\)- \(x=4\), \(f(4)=e^{-4}\)The estimated area is: \[A_{right} = \Delta x \sum_{i=1}^{4} f(x_i) = 1(e^{-1} + e^{-2} + e^{-3} + e^{-4})\]

Step 4: Estimate using left endpoints with four rectangles

For left endpoints, evaluate \(f(x)\) at the left endpoint of each subinterval:- \(x=0\), \(f(0)=e^{0}\)- \(x=1\), \(f(1)=e^{-1}\)- \(x=2\), \(f(2)=e^{-2}\)- \(x=3\), \(f(3)=e^{-3}\)The estimated area is:\[A_{left} = \Delta x \sum_{i=0}^{3} f(x_i) = 1(e^{0} + e^{-1} + e^{-2} + e^{-3})\]

Step 5: Divide the interval for eight rectangles

Now divide the interval between 0 and 4 into eight equal parts, each with a width \(\Delta x = \frac{4 - 0}{8} = 0.5\). The subintervals are: [0,0.5], [0.5,1], [1,1.5], [1.5,2], [2,2.5], [2.5,3], [3,3.5], and [3.5,4].

Step 6: Estimate using right endpoints with eight rectangles

For right endpoints, evaluate \(f(x)\) at the right end of each subinterval:- \(f(0.5), f(1), f(1.5), \ldots, f(4)\) which equals to \(f(0.5)=e^{-0.5}, f(1)=e^{-1}, f(1.5)=e^{-1.5}, \ldots, f(4)=e^{-4}\)The estimated area is:\[A_{right} = \Delta x \sum_{i=1}^{8} f(x_i)= 0.5(e^{-0.5} + e^{-1} + e^{-1.5} + \ldots + e^{-4})\]

Step 7: Estimate using left endpoints with eight rectangles

For left endpoints, evaluate \(f(x)\) at the left end of each subinterval:- \(f(0), f(0.5), f(1), f(1.5), \ldots, f(3.5)\) which equals to \(f(0)=e^{0}, f(0.5)=e^{-0.5}, f(1)=e^{-1}, \ldots, f(3.5)=e^{-3.5}\)The estimated area is:\[A_{left} = \Delta x \sum_{i=0}^{7} f(x_i) = 0.5(e^{0} + e^{-0.5} + e^{-1} + \ldots + e^{-3.5})\]

Step 8: Compare the estimates

Calculate each expression from Steps 3 to 7 to get numerical values for each estimate. By increasing the number of rectangles from four to eight, the approximation should become more accurate and closer to the actual area under the curve.

Key Concepts

Function Graph

A function graph is a visual representation of a mathematical function. It shows how the output of a function depends on different input values. For the exercise at hand, the function we are dealing with is the exponential function: \( f(x) = e^{-x} \). Here, the graph is a smooth, decaying curve that starts at \((0, 1)\) and approaches the x-axis as x increases. Observing a function graph can help us estimate areas under the curve, using methods like rectangular approximation. This method involves drawing rectangles under the curve to approximate its area. The width and height of these rectangles depend on chosen sample points, such as left or right endpoints. The more rectangles you use, the better the approximation of the actual area under the graph. This graphical approach not only makes the understanding of integration more intuitive but also adds a new dimension to the study of functions visually.

Rectangular Approximation

Rectangular approximation is a technique used in calculus to estimate the area under a curve. We do this by dividing the area into rectangles and summing up their areas. The basic idea is to approximate the region under the curve with shapes whose areas we can calculate easily. Here's the straightforward process:

• Choose the number of rectangles.
• Divide the interval into smaller subintervals, each representing the width of a rectangle.
• Select a sample point in each subinterval to determine the height of the rectangle.

The total area is then the sum of the areas of these rectangles. The more rectangles you use, the closer the approximation will be to the true area. This method is handy especially when dealing with functions that do not have an easy-to-find antiderivative. It's a stepping stone towards understanding definite integrals and the fundamental theorem of calculus.

Left and Right Endpoints

In rectangular approximation, choosing the sample points can be key to obtaining a good estimate. The two common choices are left and right endpoints of each subinterval.**Left Endpoints:**- When using left endpoints, the height of each rectangle is determined by the function value at the start of each subinterval. - This method can sometimes underestimate or overestimate areas depending on whether the function is increasing or decreasing. - In our exercise, using left endpoints for \(f(x) = e^{-x}\) from \( x=0 \) to \( x=4 \) leads to underestimation since the function is decreasing.**Right Endpoints:**- The right endpoint method uses the function value at the end of each subinterval for the height.- It generally provides a better estimate than the left endpoint for decreasing functions.- For the exercise, using right endpoints results in an overestimate for the same interval. Both methods are used to offer different perspectives and insights, which aid in understanding the behavior of the function.

Exponential Function

The exponential function, particularly \(f(x) = e^{-x}\), is crucial in mathematics for modeling decay processes, such as radioactive decay or depreciation. It is characterized by a constant relative rate of change, meaning the rate at which the function decreases is proportional to its current value.**Key characteristics include:**

• The base, \(e\), is an irrational constant approximately equal to 2.71828.
• The function is decreasing as \(x\) increases, leading the graph to slope downwards to the right.
• At \(x=0\), the function equals 1, denoted as \(f(0) = e^0 = 1\).
• As \(x\) approaches infinity, \(f(x)\) tends towards zero.

In practice, understanding this function helps in analyzing growth and decay scenarios. It's essential for calculus students to grasp how change in inputs affects outputs and how these functions can be represented graphically.