Examples of Leading coefficient in the following topics:
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- anxn is called the leading term of f(x), while $a_n \not = 0$ is known as the leading coefficient.
- The properties of the leading term and leading coefficient indicate whether f(x) increases or decreases continually as the x-values approach positive and negative infinity:
- which has −14x4 as its leading term and −141 as its leading coefficient.
- As the degree is even and the leading coefficient is negative, the function declines both to the left and to the right.
- Because the degree is odd and the leading coefficient is positive, the function declines to the left and inclines to the right.
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- If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
- If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
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- However, synthetic division works for every polynomial with leading coefficient equal to 1. The most useful aspects of synthetic division are that it allows one to calculate without writing variables and uses fewer calculations.
- We start with writing down the coefficients from the dividend and the negative second coefficient of the divisor.
- As the leading coefficient of the divisor is 1, the leading coefficient of the quotient is the same as that of the dividend:
- Hence to proceed, we add a 3 to the −12. Note that 3 equals the coefficient we have just written down, multiplied by the coefficient on the left.
- The result of −12+3 is 9, so since the leading coefficient of the divisor is still 1, the second coefficient of the quotient is −9:
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- When given a polynomial with integer coefficients, we can plug in all of these candidates and see whether they are a zero of the given polynomial.
- Since every polynomial with rational coefficients can be multiplied with an integer to become a polynomial with integer coefficients and the same zeroes, the Rational Root Test can also be applied for polynomials with rational coefficients.
- Now we use a little trick: since the constant term of (x−x0)k equals x0k for all positive integers k, we can substitute x by t+x0 to find a polynomial with the same leading coefficient as our original polynomial and a constant term equal to the value of the polynomial at x0.
- In this case we substitue x by t+1 and obtain a polynomial in t with leading coefficient 3 and constant term 1.
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- As a result, the PED coefficient is almost always negative.
- The numerical values for the PED coefficient could range from zero to infinity.
- A PED coefficient equal to one indicates demand that is unit elastic; any change in price leads to an exactly proportional change in demand (i.e. a 1% reduction in demand would lead to a 1% reduction in price).
- A PED coefficient equal to zero indicates perfectly inelastic demand.
- Finally, demand is said to be perfectly elastic when the PED coefficient is equal to infinity.
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- These orbital coefficients also have a sign (plus or minus) reflecting their phase.
- In the case of 1,3-butadiene, shown in the diagram below, the lowest energy pi-orbital (π1) has smaller orbital coefficients at C-1 and C-4, and larger coefficients at C-2 and C-3.
- The remaining three pi-orbitals have similar coefficients (± 0.37 or 0.60), but the location of the higher coefficient shifts to the end carbons in the HOMO and LUMO orbitals (π2 & π3 respectively).
- Calculations of orbital coefficients in such cases leads to an attractive explanation of the regioselectivity that characterizes their Diels-Alder chemistry.
- The dienophile data is reasonably consistent, but the diene LUMO coefficients show variability.
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- Nonparametric methods for testing the independence of samples include Spearman's rank correlation coefficient, the Kendall tau rank correlation coefficient, the Kruskal–Wallis one-way analysis of variance, and the Walk–Wolfowitz runs test.
- If Y tends to increase when X increases, the Spearman correlation coefficient is positive.
- If Y tends to decrease when X increases, the Spearman correlation coefficient is negative.
- The Kendall τ coefficient is defined as:
- When the Kruskal–Wallis test leads to significant results, then at least one of the samples is different from the other samples.
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- The coefficient of determination is actually the square of the correlation coefficient.
- This leads to the alternative approach of looking at the adjusted r2.
- The correlation coefficient is r=0.6631.
- Therefore, the coefficient of determination is r2=0.66312=0.4397.
- Interpret the properties of the coefficient of determination in regard to correlation.
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- We can write the emission and absorption coefficients in terms of the Einstein coefficients that we have just examined.
- The emission coefficient jν has units of energy per unit time per unit volume per unit frequency per unit solid angle!
- The Einstein coefficient A21 gives spontaneous emission rate per atom, so dimensional analysis quickly gives
- We can now write the absorption coefficient and the source function using the relationships between the Einstein coefficients as
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- The discriminant of a polynomial is a function of its coefficients that reveals information about the polynomial's roots.
- The discriminant of a quadratic function is a function of its coefficients that reveals information about its roots.
- Where a, b, and c are the coefficients in f(x)=ax2+bx+c.
- Since adding zero and subtracting zero in the quadratic equation lead to the same outcome, there is only one distinct root of the quadratic function.