Mathematics, 14.06.2020 02:57, rapryce

When a company produces and sells x thousand units per week, its total weekly profit is P thousand dollars, where Upper P equals StartFraction 1000 x Over 100 plus x squared EndFraction . The production level at t weeks from the present is x equals 4 plus 2 t. Find the marginal profit, StartFraction dY Over dx EndFraction and the time rate of change of profit, StartFraction dP Over dt EndFraction . How fast (with respect of

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answered: tmarie03

The marginal profit is $\frac{1000(100-x^{2})}{(x^{2}+100)^{2}}$ .

Step-by-step explanation:

The profit function p (x) is the difference between the revenue and cost function.

The profit function is given as follows:

$p (x) = \frac{1000x}{100+x^{2}}$

Determine the marginal profit function as follows:

$\text{Marginal Profit}=\frac{\text{d}}{\text{dx}} p (x)$

$={\tfrac{\mathrm{d}}{\mathrm{d}x}\left[\dfrac{1000x}{x^2+100}\right]}}\\\\={1000\cdot{\tfrac{\mathrm{d}}{\mathrm{d}x}\left[\dfrac{x}{x^2+100}\right]}}}}\\\\$

$=1000\cdot\frac{\frac{\text{d}}{\text{dx}}(x)\cdot (x^{2}+100)+x\cdot \frac{\text{d}}{\text{dx}} (x^{2}+100)}{ (x^{2}+100)^{2}}\\\\=\frac{1000(x^{2}-2x^{2}+100)}{(x^{2}+100)^{2}}\\\\=\frac{1000(100-x^{2})}{(x^{2}+100)^{2}}$

Thus, the marginal profit is $\frac{1000(100-x^{2})}{(x^{2}+100)^{2}}$ .

answered: Guest

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range:

answered: Guest

hello from mrbilldoesmath!

answer: (7x + 6y) ( 49x^2 - 42xy + 36y^2)

discussion:

in general, the factorization of a^3 + b^3 is

a^3 + b^ 3 = (a + b) ( a^2 - ab + b^2)

in our case note that 343 = 7^3 and 216 = 6^3 so the equation becomes:

343x^3 + 216y^3 = (7x)^3 + (6y)^3

if you think of a (above) as 7x and b as 6y the factorization becomes