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Chapter 3: Problem 57

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

Short Answer

Expert verified
When we describe the graph of a polynomial function as smooth and continuous, we mean that the graph has no breaks, holes or sharp corners (implying it is smooth) and you can draw it without lifting your pen from the paper (implying it is continuous). This means polynomial functions have derivatives that exist at every point in their domain.

Step by step solution

01

Defining a Polynomial Function

A polynomial function is a function that can be expressed in the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(a_i\) are constants for \(i = 0,1, ..., n\) and \(n\) is a nonnegative integer.
02

Defining Smoothness

In mathematics, a function is described as smooth if it has derivatives of all orders everywhere in its domain. In other words, the function is smooth if it does not have any jumps or kinks, or any sudden changes in direction. All polynomial functions are smooth functions.
03

Defining Continuity

A function is said to be continuous if it does not have any holes, jumps or vertical asymptotes. Put simply, if a function is continuous, you could draw it without lifting your pen from the paper. All polynomial functions are continuous everywhere. This is because for each value of \(x\), there exists a corresponding value of \(y=f(x)\).
04

Conclusion

So, when we describe the graph of a polynomial function as smooth and continuous, we mean that its graph has no breaks, holes, jumps, or kinks and that it can be drawn without lifting the pen from the paper, signifying that it has derivatives of all orders everywhere in its domain.

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Most popular questions from this chapter

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=7 x^{2}+9 x^{4}$$

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=3 x^{2}-x^{3}$$

What are the zeros of a polynomial function and how are they found?

Can the graph of a polynomial function have no \(y\) -intercept? Explain,

Use the four-step procedure for solving variation problems given on page 356 to solve. The electrical resistance of a wire varies directly as its length and inversely as the square of its diameter. A wire of 720 feet with \(\frac{1}{4}\)-inch diameter has a resistance of \(1 \frac{1}{2}\) ohms. Find the resistance for 960 feet of the same kind of wire if its diameter is doubled.
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