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Chapter 16: Problem 2

Describe and sketch a region that is bounded on the left and on the right by two curves.

Short Answer

Expert verified
Question: Sketch the region bounded by the quadratic function g(x) = x^2 and the linear function f(x) = x, and describe the shape of the region. Answer: The region bounded by the quadratic function g(x) = x^2 and the linear function f(x) = x is a triangular area enclosed by the points (0,0), (1,0) and (1,1).

Step by step solution

01

Choose two functions to bound the region

To keep things simple, let's choose a linear function and a quadratic function. We'll use the functions: - f(x) = x - g(x) = x^2 The region will be bounded on the left by the quadratic function (g) and on the right by the linear function (f).
02

Find the intersection points of the curves

To find the intersection points, we need to solve the equation f(x) = g(x), i.e., x = x^2. Here's how to solve it: x = x^2 x - x^2 = 0 x(1 - x) = 0 So x = 0 or x = 1. So the intersection points of the curves are (0,0) and (1,1).
03

Sketch the curves and the bounded region

To sketch the region, draw the curves y=x and y=x^2 on the same coordinate plane. Then, plot the points (0,0) and (1,1) on the graph and connect them with a straight line. Now, shade the region that lies between the x-axis and the two curves, which will create a region bounded on the left by the quadratic function (x^2) and on the right by the linear function (x). To summarize: 1. Draw the curves y=x and y=x^2. 2. Plot the points (0,0) and (1,1). 3. Connect the two points with a straight line. 4. Shade the region between the x-axis and the two curves. The bounded region is a triangular area enclosed by the points (0,0), (1,0), and (1,1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intersection Points
Intersection points are the coordinates where two or more functions meet. In our problem, we are dealing with a linear function, \( f(x) = x \), and a quadratic function, \( g(x) = x^2 \). To find where these two curves intersect, we equate them: \( f(x) = g(x) \). This gives us the equation \( x = x^2 \). By rearranging and simplifying, we have:\[ x - x^2 = 0 \]Factoring gives us \( x(1 - x) = 0 \). Solving this, we find two intersection points, \( x = 0 \) and \( x = 1 \). Therefore, the intersection points are at the coordinates \((0,0)\) and \((1,1)\). These points are vital because they define the boundaries of the region we are interested in sketching.
Linear Function
A linear function is one of the simplest types of functions, characterized by a constant rate of change. The standard form of a linear function is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In our exercise, the linear function is \( y = x \). This function has a slope of 1, indicating a 45-degree angle with the x-axis.
  • The graph of \( y = x \) passes through the origin \((0,0)\).
  • Every point on this line has equal x and y coordinates.
  • It is a straight line that plays a role in bounding our region on the right.
Linear functions are known for their simplicity and direct proportional relationship between variables.
Quadratic Function
A quadratic function is a polynomial of degree 2, generally represented as \( y = ax^2 + bx + c \). In our problem, the quadratic function is \( y = x^2 \), which means:
  • The function opens upwards, as indicated by the positive coefficient of \( x^2 \).
  • Its vertex is at the origin \((0,0)\).
This function forms a parabolic shape on the graph. Quadratic functions are crucial in many areas of study because they can model complex systems in a simple parabolic form. They help create boundaries such as the left side in this particular problem.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we graphically represent algebraic equations. It consists of a horizontal x-axis and a vertical y-axis. In this exercise:
  • We plot our linear and quadratic functions on the coordinate plane.
  • The intersection points \((0,0)\) and \((1,1)\) are clearly marked.
  • The plane allows us to visually determine the bounded region between the curves.
Understanding the coordinate plane is crucial for visualizing how functions interact and the regions they form. It helps us locate and clearly define areas of intersection and boundary.
Curve Sketching
Curve sketching is the process of drawing graphs of functions to understand their shapes and relationships. It involves:
  • Plotting key points such as intersections, turning points, and boundaries.
  • Understanding the behavior of functions, like slope and curvature.
  • Visually representing solutions to equations and inequalities.
In this problem, we sketched \( y = x \) and \( y = x^2 \) on the coordinate plane, marked the intersection points, and drew the bounded region. Curve sketching guides us in analyzing complex relationships and solutions through simple visuals.

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Most popular questions from this chapter

Charge distribution A spherical cloud of electric charge has a known charge density \(Q(\rho),\) where \(\rho\) is the spherical coordinate. Find the total charge in the cloud in the following cases. a. \(Q(\rho)=\frac{2 \times 10^{-4}}{\rho^{4}}, 1 \leq \rho<\infty\). b. \(Q(\rho)=\left(2 \times 10^{-4}\right) e^{-0.01 p^{3}}, 0 \leq \rho<\infty\).

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