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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.
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.
What are the operation of polynomials?
Adding and subtracting polynomials is simply the adding and subtracting of their like terms.
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
Can equations for discrete and continuous graphs be written the same?
Equations for discrete and continuous graphs can share similar forms, but they represent different concepts. Discrete graphs consist of distinct, separate points and are typically represented by functions defined only for specific values, often using sequences or step functions. In contrast, continuous graphs represent smooth curves where the function is defined for all values within an interval. While the mathematical expressions may look alike, their applications and interpretations differ significantly.
What are the similarities between quadratic function and linear function?
Both are polynomials. They are continuous and are differentiable.
What is a graph with no breaks called?
A graph with no breaks is called a "continuous graph." In mathematical terms, this means that the graph can be drawn without lifting the pencil from the paper, indicating that the function it represents is continuous over its domain. Continuous graphs typically exhibit smooth transitions without any jumps, holes, or asymptotes.
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 are graphs that have connected lines or curves?
Graphs that have connected lines or curves are typically referred to as continuous graphs. These graphs represent a function or relationship where the points are connected without any breaks, indicating that for every input within a certain range, there is a corresponding output. Examples include linear functions, polynomial functions, and trigonometric functions. Continuous graphs are important in calculus and mathematical analysis because they allow for the application of concepts such as limits, derivatives, and integrals.
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.
