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Smooth function has derivatives of all orders. Polynomials have derivatives of all orders, thus polynomials are smooth functions.
For example: f(x)=2x+3 => f'(x)=2 => f''(x)=0 => f'''(x)=0...
So all derivatives exist. (Derivative being zero is ok.)
Their continuity can be proven using the Weierstrass (epsilon-delta) definition.
What else can I help you with?
How polynomials and non polynomials are alike?
they have variable
What is a straight continuous arrangement of an infinite number of points?
That would be an infinitely long line. Or just called a line if you are talking about graphs
Can polynomials with the same graph have different roots?
False! If the graph is exactly the same, then the x-intercepts will be the same which implies the roots are them same. However, you can have the same roots and different graphs. So while the first statement is true, the converse if not.
What are the operation of polynomials?
Adding and subtracting polynomials is simply the adding and subtracting of their like terms.
Can the sum of three polynomials again be a polynomial?
The sum of two polynomials is always a polynomial. Therefore, it follows that the sum of more than two polynomials is also a polynomial.
Why do all polynomials have graphs that look like the graphs of their leading terms?
Polynomials have graphs that look like graphs of their leading terms because all other changes to polynomial functions only cause transformations of the leading term's graph.
Polynomials with the same roots always have the same graphs?
false
What are the similarities between quadratic function and linear function?
Both are polynomials. They are continuous and are differentiable.
How do you find the points of discontinuity2x2-5x-3?
Polynomials are continuous everywhere, so there are not points of discontinuity.
Is matter smooth and continuous?
No it is not.
Line graphs can be used to illustrate the relationship between continuous quantities?
Yes.
What is the technique of creating a smooth and continuous gliding effect in music called?
The technique of creating a smooth and continuous gliding effect in music is called "legato."
What are the advantages of using broken line graphs?
Easy to compare multiple continuous data sets
Why do scientists use line graphs instead of bar graphs to show population change?
Usually the change is tracked against time and time is a continuous variable. It is customary, when the independent variable is continuous, to use line graphs. However, if the data are populations at five- or ten-year intervals - for example, from censuses - then you might find that they are shown as a bar graph.
What does a data table look like in a science experiment?
Mostly bar graphs unless the experiment is continuous
Polynomials have factors that are?
Other polynomials of the same, or lower, order.
What is piecewise smooth function?
f is a piecewise smooth funtion on [a,b] if f and f ' are piecewise continuous on [a,b]
