Examples of Leading coefficient in the following topics:
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- $a_nx^n$ 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 $-\frac {x^4}{14}$ as its leading term and $- \frac{1}{14}$ 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 $(x-x_0)^k$ equals $x_0^k$ for all positive integers $k$, we can substitute $x$ by $t+x_0$ 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 $x_0$.
- 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 $\tau$ 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 $r^2$.
- The correlation coefficient is $r=0.6631$.
- Therefore, the coefficient of determination is $r^2 = 0.6631^2 = 0.4397$.
- 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 $j_\nu$ has units of energy per unit time per unit volume per unit frequency per unit solid angle!
- The Einstein coefficient $A_{21}$ 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 $f(x) = ax^2 + bx + c$.
- 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.