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Chapter 3: Problem 78

What do we mean when we describe the graph of a polynomial function as smooth and continuous?

Short Answer

Expert verified
When the graph of a polynomial function is described as 'smooth' and 'continuous', it means that the graph has no sharp corners or cusps (smooth) and no jumps, breaks, holes, or asymptotes (continuous).

Step by step solution

01

Define 'smooth'

When we describe a graph as 'smooth', we mean that the graph has no sharp corners or cusps, and it is differentiable everywhere. In other words, you should be able to draw the line of the graph without having to lift your pen.
02

Define 'continuous'

When we say that a graph is 'continuous', we are saying that the function has no jumps, breaks, holes, or asymptotes. In more technical language, a function is continuous if at every point in its domain, the function's limit is equal to the value of the function at that point.
03

Relate 'smooth' and 'continuous' to Polynomials

Now, regarding polynomial functions, they are both smooth and continuous everywhere. This is because polynomial functions can be defined and differentiated for all real numbers.

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