Question

Sketch the graph of the given function over the interval \([a, b] ;\) then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=3 x^{2}+x+1 ; a=-1, b=1, n=10 $$

Short answer

The area of the circumscribed polygon is 4.112.

Step-by-step solution

Sketch the Graph of f(x)

The given function is \(f(x) = 3x^2 + x + 1\), which is a quadratic function with its graph being a parabola. For this specific function, the parabola opens upwards because the coefficient of \(x^2\) is positive. We can sketch the graph by identifying the vertex and the y-intercept. The y-intercept is \(f(0) = 1\). Since \(a = -1\) and \(b = 1\), the graph should be focused within these intervals. The vertex can be calculated by the formula \(x = -\frac{b}{2a}\), where \(a = 3\) and \(b = 1\), yielding \(x = -\frac{1}{6}\). Calculate \(f(-1)\) and \(f(1)\) to find the endpoints of the parabola for this interval.

Divide the Interval [a, b] into n Equal Subintervals

We are given that the interval \([-1, 1]\) needs to be divided into \(n=10\) equal subintervals. The width of each subinterval, \(\Delta x\), is given by \(\Delta x = \frac{b-a}{n} = \frac{1 - (-1)}{10} = \frac{2}{10} = 0.2\). The endpoints of each subinterval are \(-1, -0.8, -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1\).

Calculate the Area of the Circumscribed Polygon

For a circumscribed polygon using right endpoints of each subinterval, calculate the area of rectangles whose heights are given by the function values at the right endpoints. The right endpoints for each interval are: \(-0.8, -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1\). Calculate \(f(x)\) for each right endpoint and sum the areas. The total area is \(\sum_{i=1}^{10} f(x_i) \times \Delta x\). For instance, \(f(-0.8) = 3(-0.8)^2 + (-0.8) + 1 = 2.92\), and similarly calculate for others, then sum all including multiplying by \(\Delta x = 0.2\).

Compute Each Rectangle's Area

Calculate the area for each rectangle by evaluating the function for each right endpoint and multiplying by \(\Delta x\): \(f(-0.8) \times 0.2, f(-0.6) \times 0.2, ..., f(1) \times 0.2\). For example, if \(f(-0.8) = 3.44\), the area of the rectangle is \(3.44 \times 0.2 = 0.688\). Repeat for each right endpoint.

Sum All Rectangle Areas

Add up the areas of all ten rectangles to determine the total area under the curve using the circumscribed polygon approach. Including all the details from above, the total area \(A\) becomes: \(A = 0.584 + 0.512 + 0.456 + 0.416 + 0.384 + 0.368 + 0.368 + 0.384 + 0.416 + 0.544 = 4.112\).

Key concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree of the variable is squared. Its general form is \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. The graph of a quadratic function is a parabola. It can either open upwards, forming a smile shape, or downwards, forming a frown, depending on the sign of the coefficient \( a \).

If \( a \) is positive, the parabola opens upwards, while if \( a \) is negative, it opens downwards. The vertex of the parabola represents either the maximum or minimum point, providing valuable insight into the function's behavior. You can find the vertex using the formula \( x = -\frac{b}{2a} \), and the y-intercept by setting \( x = 0 \) in the function.

Quadratic functions play a crucial role in different fields due to their property of modeling naturally occurring phenomena. Understanding their structure and graphs is essential for tackling more complex math problems, including area calculations involving circumscribed polygons.

Circumscribed Polygon

A circumscribed polygon is a method of approximating the area under a curve on a graph. By using rectangles or other simple geometric shapes, you can estimate the area by circumscribing the polygon around the curve. This approach is beneficial when dealing with integrals or calculating areas that are not easily computed.

In this method, you select certain points—usually at the ends of subdivisions of an interval—and use the function values at those points to determine the height of each rectangle. For the given exercise, the function values at the right endpoints of the subintervals decide the rectangles' heights.

The sum of the area of these rectangles gives an approximation of the area under the curve. By increasing the number of intervals \( n \), and hence the number of rectangles, the approximation becomes more accurate, providing insight into integral concepts and how they relate to real-world applications like physical forces, economics, and engineering.

Subintervals

When analyzing a specific interval \([a, b]\) on the number line, dividing it into equal parts is useful for computations of approximations and integrals. Each of these smaller sections is called a subinterval.

In the context of the original exercise, dividing the interval \([-1, 1]\) into \(n=10\) subintervals results in each subinterval having a width \( \Delta x \). Calculating the width is straightforward: \( \Delta x = \frac{b-a}{n} \), ensuring all sections are of equal length.

The endpoints of these subintervals play a crucial role in evaluating certain sums or approximations, such as in the Riemann sums for integrals. Subdividing helps in minimizing the error in approximation and provides a detailed understanding when dealing with continuous data over a range.

Function Graphing

Function graphing involves plotting a function on a coordinate system to visualize its behavior. This technique is vital in understanding complex mathematical relationships in an intuitive, visual format.

In graphing the quadratic function \( f(x) = 3x^2 + x + 1 \), we plot the curve over the specific interval \([-1, 1]\). To do this effectively, we identify key points such as the vertex—from the formula \( x = -\frac{b}{2a} \) and other intercepts where the function crosses the axis points.

This visual representation allows learners to grasp the interplay between algebraic expressions and their graphical interpretations, which are essential when solving mathematical problems involving functions like calculating the area under a curve using approximation techniques such as circumscribed polygons.