Final answer:
To sketch the region between y = 3 cos x and y = 8x² - 2, plot both curves on the same graph, find their points of intersection, and shade the enclosed region that lies between them.
Step-by-step explanation:
To sketch the region enclosed by the curves y = 3 cos x and y = 8x² - 2, you need to follow these steps:
- Determine the interval for x within which the curves might intersect. For trigonometric functions like cosine, considering one or two periods is typically sufficient. For y = 3 cos x, since cosine fluctuates between -1 and 1, y will range from -3 to 3.
- Plot the function y = 3 cos x which will be a cosine wave with an amplitude of 3 on a graph.
- Plot the function y = 8x² - 2 which is a parabola opening upwards with its vertex at (0,-2).
- Identify the points of intersection by setting the two equations equal to each other and solving for x: 3 cos x = 8x² - 2. This may require numerical methods or graphing to approximate since it's not solvable with elementary algebra.
- Shade the region on the graph that is enclosed between the two curves, this is the area where the curve y = 8x² - 2 lies above the curve y = 3 cos x.
These intersections will determine the limits of the region you're interested in. The enclosed region can be shaded to ensure clarity.