Examples of piecewise function in the following topics:
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- Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated intervals.
- Example 1: Consider the piecewise definition of the absolute value function:
- Substitute those values into the first part of the piecewise function f(x)=x2:
- Since there is an closed AND open dot at x=1 the function is piecewise continuous there.
- When x=2, the function is also piecewise continuous.
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- One has to be a little careful about saying that a particular function is equal to its Fourier series since there exist piecewise continuous functions whose Fourier series diverge everywhere!
- If f is piecewise continuous and has left and right derivatives at a point c (A right derivative would be: limt→0(f(c+t)−f(c))/t , t>0 .
- If f is continuous with period 2π and f′ is piecewise continuous, then the Fourier series for f converges uniformly to f .
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- A line integral is an integral where the function to be integrated is evaluated along a curve.
- A line integral (sometimes called a path integral, contour integral, or curve integral) is an integral where the function to be integrated is evaluated along a curve.
- The function to be integrated may be a scalar field or a vector field.
- For some scalar field f:U⊆Rn→R, the line integral along a piecewise smooth curve C⊂U is defined as:
- For a vector field F:U⊆Rn→Rn, the line integral along a piecewise smooth curve C⊂U, in the direction of r, is defined as:
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- In order to minimize risk of losing material in a failed reaction and to use time efficiently, a piecewise synthesis scheme is often used.
- Piecewise synthesis involves breaking target material B into several pieces that can be synthesized separately, and then combining them.
- For example, a molecule containing an imine, carboxylic acid and ketone will have all three of those functional groups reduced by lithium aluminum hydride.
- But what if a chemist's intention is to leave the carboxylic acid and ketone functionalities, while reducing the imine?
- Once the ester is reduced, the acetal can be removed, thus yielding the original ketone functionality.
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- Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C.
- If L and M are functions of (x,y) defined on an open region containing D and have continuous partial derivatives there, then:
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- A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function.
- An example of such an integrand is f(x)=exp(x2), the antiderivative of which (the error function, times a constant) cannot be written in elementary form.
- A 'brute force' kind of numerical integration can be done, if the integrand is reasonably well-behaved (i.e. piecewise continuous and of bounded variation), by evaluating the integrand with very small increments.
- A definite integral of a function can be represented as the signed area of the region bounded by its graph.
- Solve for the definite integral of a continuous function over a closed interval
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- To find total capacitance of the circuit, simply break it into segments and solve piecewise .
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- Suppose V is a subset of Rn (in the case of n=3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S (also indicated with ∂V=S).
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- Functional groups are atoms or small groups of atoms (two to four) that exhibit a characteristic reactivity when treated with certain reagents.
- A particular functional group will almost always display its characteristic chemical behavior when it is present in a compound.
- Because of their importance in understanding organic chemistry, functional groups have characteristic names that often carry over in the naming of individual compounds incorporating specific groups.
- In the following table the atoms of each functional group are colored red and the characteristic IUPAC nomenclature suffix that denotes some (but not all) functional groups is also colored.
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- A rational function is one such that f(x)=Q(x)P(x), where Q(x)≠0; the domain of a rational function can be calculated.
- A rational function is any function which can be written as the ratio of two polynomial functions.
- Any function of one variable, x, is called a rational function if, and only if, it can be written in the form:
- Note that every polynomial function is a rational function with Q(x)=1.
- A constant function such as f(x)=π is a rational function since constants are polynomials.