Stefan Hendricks
2021-12-20
Answered

A manufacturer of lighting fixtures has daily production costs of $C=800-10x+0.25{x}^{2}.$ where C is the total cost (in dollars) and X is the number of units produced. How many fixtures should be produced each day to yield a minimum cost?

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asked 2021-02-25

True or False. The graph of a rational function may intersect a horizontal asymptote.

asked 2022-06-21

I know how to use this algorithm when I am integrating rational functions, but my textbook has omitted the actual proof for why it works. If someone could please help me with this question:

Prove that given a and b, there are constants ${c}_{1}$ and ${c}_{2}$ so that

$\frac{ax+b}{(x-1{)}^{2}}=\frac{{c}_{1}}{x-1}+\frac{{c}_{2}}{(x-1{)}^{2}}$

Prove that given a and b, there are constants ${c}_{1}$ and ${c}_{2}$ so that

$\frac{ax+b}{(x-1{)}^{2}}=\frac{{c}_{1}}{x-1}+\frac{{c}_{2}}{(x-1{)}^{2}}$

asked 2021-06-28

One method of graphing rational functions that are reciprocals of polynomial functions is to sketch the polynomial function and then plot the reciprocals of the y-coordinates of key ordered pairs. Use this technique to sketch

b)f(x)=

d)

asked 2022-05-21

I am trying to prove that

${\int}_{C}\frac{P(z)}{Q(z)}dz=0$

If polynomial order of $Q$ is 2 or more than that of $P$, using the theorem stating that if a function has a finite number of singular points all interior to a contour $C$, then

${\int}_{C}f(z)dz=2\pi i\text{Res}{\textstyle [}\frac{1}{{z}^{2}}f(\frac{1}{z}){\textstyle ]}$

I received the hint that, under the conditions described above, the rational function in the integrand can be written as a McLauran Series with no negative powers of z. This would imply that the residue is zero...

My problem is, I can't seem to wrap my head around how the rational function given to me can be written into the form of a power series with only positive exponents...

So, as the title says, When can a rational function be represented as a power series? I'm not looking for a full proof, but a few details would be nice, just so I feel more comfortable running with the hint.

${\int}_{C}\frac{P(z)}{Q(z)}dz=0$

If polynomial order of $Q$ is 2 or more than that of $P$, using the theorem stating that if a function has a finite number of singular points all interior to a contour $C$, then

${\int}_{C}f(z)dz=2\pi i\text{Res}{\textstyle [}\frac{1}{{z}^{2}}f(\frac{1}{z}){\textstyle ]}$

I received the hint that, under the conditions described above, the rational function in the integrand can be written as a McLauran Series with no negative powers of z. This would imply that the residue is zero...

My problem is, I can't seem to wrap my head around how the rational function given to me can be written into the form of a power series with only positive exponents...

So, as the title says, When can a rational function be represented as a power series? I'm not looking for a full proof, but a few details would be nice, just so I feel more comfortable running with the hint.

asked 2022-09-04

Is $\frac{y}{6}=x$ a direct or inverse variation?

asked 2021-05-02

Graph the rational functions. Include the graphs and equations of the asymptotes and dominant terms. $\frac{x+3}{x+2}$

asked 2022-02-15

Is the section df associated to a rational function f on a curve X a global section of the canonical sheaf $\omega}_{X$ ? I know its zeroes are the ramification points, but does it have poles?