Question

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

Short answer

Question: Describe and sketch the region bounded by the left boundary f(x) = x and the right boundary g(x) = x^2, with limits x = 0 and x = 1. Answer: The region is a shaded area located between a linear function, f(x) = x, and a quadratic function, g(x) = x^2, within the limits x = 0 and x = 1. It is bounded on the left by the line f(x) = x, on the right by the curve g(x) = x^2, and enclosed by the points of intersection (0,0) and (1,1). The sketch of this region will show the area sandwiched between these two functions, clearly illustrating the boundaries and limits.

Step-by-step solution

Choose functions as left and right boundaries

Let's choose two functions, one to represent the left boundary and the other for the right boundary. We'll use functions that can be easily visualized: a linear function for the left boundary and a quadratic function for the right boundary. Let the left boundary be represented by the linear function f(x) = x and the right boundary be represented by the quadratic function g(x) = x^2.

Determine the region between the two functions

To determine the region between the two functions f(x) = x and g(x) = x^2, we should find the points of intersection, that is the values of x for which f(x) = g(x). This will give us the limits of the region we want to sketch. Solve the equation f(x) = g(x): x = x^2 Subtract x from both sides: 0 = x^2 - x Factor x from the expression: 0 = x(x - 1) The equation is satisfied when either x = 0 or x = 1. So, the limits of the region where the two functions intersect are x = 0 and x = 1.

Sketch the region

Now that we have the left and right boundaries (f(x) = x and g(x) = x^2) and the limits of the region (x = 0 and x = 1), we can sketch the region. 1. Draw the axes (x and y-axis) on a coordinate plane. 2. Plot the linear function f(x) = x (a straight line with slope 1 and passing through the origin). 3. Plot the quadratic function g(x) = x^2 (a parabolic curve with the vertex at the origin). 4. Mark the points of intersection between the two functions, (0, 0) and (1, 1). 5. Shade the region between the two functions and within the limits x = 0 and x = 1. The final sketch will show a shaded region bounded on the left by the line f(x) = x, on the right by the curve g(x) = x^2, and between the limits x = 0 and x = 1.