Problem 24 - Calculus Early Transcendentals

Question

Complete the following steps for the given function, interval, and value of \(n\). a. Sketch the graph of the function on the given interval. b. Calculate \(\Delta x\) and the grid points \(x_{0}, x_{1}, \ldots, x_{n}\) c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve. d. Calculate the left and right Riemann sums. $$f(x)=2 x^{2} \text { on }[1,6] ; n=5$$.

Step-by-Step Solution

Answer

a) The left Riemann sum overestimates the area under the curve, while the right Riemann sum underestimates the area. b) The left Riemann sum underestimates the area under the curve, while the right Riemann sum overestimates the area. c) Both the left and right Riemann sums underestimate the area under the curve. d) Both the left and right Riemann sums overestimate the area under the curve.

Step 1: Sketch the graph of the function on the given interval [1, 6]

To sketch the graph of the function \(f(x) = 2x^2\) on the interval [1,6], plot points of the function by substituting different x-values and then connect them by a smooth curve.

Step 2: Calculate \(\Delta x\) and grid points.

Divide the interval [1, 6] into 5 equal subintervals (as n=5). Calculate \(\Delta x\) which is the width of each subinterval by using the formula \(\Delta x = \frac{b-a}{n} = \frac{6-1}{5} = 1.\) Now, find the grid points \(x_0, x_1, \ldots, x_n\) as follows: \(x_0=1, x_1=2, x_2=3, x_3=4, x_4=5,\) and \(x_5=6\).

Step 3: Illustrate the left and right Riemann sums.

For the left Riemann sum, use the left endpoint of each subinterval to evaluate the function. For the right Riemann sum, use the right endpoint of each subinterval to evaluate the function. Plot rectangles whose heights are given by the function values at the left and right endpoints, and whose width is \(\Delta x\). The left Riemann sum will underestimate the area, and the right Riemann sum will overestimate the area under the curve because \(f(x)\) is increasing on [1, 6], so the function's value in the subintervals increases from left to right.

Step 4: Calculate the left and right Riemann sums.

For the left Riemann sum, the sum is given by: $$L_n= \Delta x \sum_{i=0}^{n-1}f(x_i) = 1[f(1) + f(2) + f(3) + f(4) + f(5)] = 1[2 + 8 + 18 + 32 + 50] = 110$$ For the right Riemann sum, the sum is given by: $$R_n= \Delta x \sum_{i=1}^{n}f(x_i) = 1[f(2) + f(3) + f(4) + f(5) + f(6)] = 1[8 + 18 + 32 + 50 + 72] = 180$$ Since the function \(f(x)=2x^2\) is increasing on the interval [1, 6], the left Riemann sum (110) underestimates the area under the curve, while the right Riemann sum (180) overestimates the area under the curve.

Key Concepts

Increasing Function

An increasing function is a fundamental concept in calculus, where the function's value grows as the input increases. In simpler terms, as you move from left to right on the graph of an increasing function, the height of the graph does not go down. This means that for any two points, say \(x_1\) and \(x_2\), if \(x_1 < x_2\), then \(f(x_1) \leq f(x_2)\). Increasing functions have several practical applications in mathematics and other fields.

• Predicting market trends in economics.
• Understanding growth patterns in biology.
• Analyzing algorithms in computer science.

In the example we worked through, the function \(f(x) = 2x^2\) is increasing on the interval \([1, 6]\). As x increases, the value of \(f(x)\) grows larger. This characteristic plays a critical role in determining the behavior of different types of Riemann sums that we encounter.

Definite Integrals

Definite integrals are a part of integral calculus used to calculate the area under a curve between two points on a graph. They provide a precise numerical value for that area, unlike indefinite integrals, which represent a family of functions. The notation for a definite integral from point \(a\) to point \(b\) is \(\int_a^b f(x) \, dx\).

The steps to solve for a definite integral typically include:

• Identifying the function \(f(x)\) whose area you want to calculate.
• Determining the limits of integration \(a\) and \(b\).
• Calculating the antiderivative of \(f(x)\).
• Evaluating the antiderivative at \(b\) and \(a\), and finding their difference.

In our example with \(f(x)=2x^2\) on the interval \([1, 6]\), determining the area under the curve is precisely what definite integrals aim to achieve. Riemann sums provide a way of approximating this integral when calculating it directly might not be feasible.

Numerical Approximation

Numerical approximation is a technique used to find approximate results when an exact solution is difficult to obtain. In calculus, numerical approximation often deals with estimating the value of integrals using methods like Riemann sums. These sums provide an approximation of the area under a curve by summing the areas of multiple rectangles.

The accuracy of a numerical approximation depends on several factors, including:

• The size of the subintervals – smaller subintervals generally lead to more accurate approximations.
• The type of Riemann sum used – left, right, or midpoint.

In the exercise involving \(f(x)=2x^2\), both left and right Riemann sums are used to approximate the area. While neither gives an exact value, they do provide insight into the definite integral by showing how the approximation varies depending on the method used.

Calculus

Calculus is a branch of mathematics that focuses on change and motion, and is divided into two main areas: differential and integral calculus. Differential calculus concerns rates of change and slopes of curves, whereas integral calculus deals with the accumulation of quantities and the areas under and between curves.

Important concepts include:

• Limits – fundamental to defining derivatives and integrals.
• Derivatives – representing the rate of change of a function.
• Integration – finding the accumulated area under a curve.

In our example, calculus allows us to explore how the area under the curve, represented by \(f(x)=2x^2\), can be interpreted using definite integrals and Riemann sums, thereby tying together these key concepts of change and accumulation in mathematics.