Final answer:
The graph of a continuous function f within an interval [s, d] will be smooth, with no interruptions like breaks, jumps, or holes, and the first derivative must also be continuous unless the function approaches infinity.
Step-by-step explanation:
If function f is continuous on an interval [s, d], the graph of f will not show any breaks, jumps, or holes within that interval. This means you should be able to draw the graph from s to d without lifting your pencil off the paper. Some properties we can infer from this are:
- y(x) must be a continuous function over the interval.
- The first derivative of y(x) with respect to space, dy(x)/dx, must also be continuous within that interval, unless the value of the function approaches infinity at some point.
If we consider a function f(x) defined on the interval 0 ≤ x ≤ 20, the graph of such a function will be constrained to this domain. For instance, if f(x) = 20, we have a horizontal line at y = 20, but it would only extend from x = 0 to x = 20.
For probability distributions, these rules slightly change:
- The graph of a continuous probability distribution is a curve, typically known as the probability density function (pdf).
- Probabilities correspond to the area under the pdf curve within a certain interval.
In the context of physics, for an object moving with constant acceleration, the slope of a 2d vs. t graph will represent that acceleration. The position graph of an accelerating object, like a jet-powered car, will show increasing slopes as velocity increases, which reflects in the shape of the graph as a curve that gets steeper over time.