Question

The demand for a certain commodity is D(x) = 5000e−.02x units per month when the market price is x dollars per unit.

(a) At what rate is the consumer expenditure E(x) = xD(x) changing with respect to price x when the price is equal to $110 dollars?
(b) At what price does consumer expenditure stop increasing and begin to decrease?
(c) At what price does the rate of consumer expenditure begin to increase?

Answer 1

Answer:

Step-by-step explanation:

We have two functions:

$D(x) = 5000e^{0.02x} \text{commodity}$

$E(x) = xD(x) \text{expenditure}$

firstly, we can express D(x) in the above equation.

$E(x) =  5000xe^{0.02x}$

Thinking of the questions mathematically,

part a) is only asking what is the rate of change of the expenditure(E) with respect to x(price) when x is 110 dollars? that is,

$\left\dfrac{dE}{dx}\right|_{x=110}}$ evaluated at x = 110.

$\dfrac{dE}{dx} = \dfrac{d}{dx}\left(5000xe^{-0.02x}\right)$

using the product rule for differentiation.

$\dfrac{dE}{dx} = 5000(-0.02xe^{-0.02x} + e^{-0.02x})$

for simplification we can take e^0.02x common

$\dfrac{dE}{dx} = 5000e^{-0.02x}(-0.02x + 1)$

now put x= 110, and we'll get the rate of change of the expenditure at x = 110. (answer for part a):

$\dfrac{dE}{dx} = 5000e^{-0.02x}(-0.02x + 1)$

$\left\dfrac{dE}{dx}\right|_{x=110}} = 5000e^{-0.02(110)}(-0.02(110) + 1)$

$\left\dfrac{dE}{dx}\right|_{x=110}} = 5000e^{-0.02(110)}(-0.02(110) + 1)$

$\left\dfrac{dE}{dx}\right|_{x=110}} = -664.82\,\text{dollars}$

part(b) at what value of x does E'(x) = 0? i.e, at what price(x) does the expenditure E changes from increasing to decreasing?

$\dfrac{dE}{dx} = 0$

$5000e^{-0.02x}(-0.02x + 1) = 0$

we can split the equation and solve them separately.

$5000e^{-0.02x}=0\,\,,\,\,(-0.02x + 1) = 0$

$-0.02x=\ln{(0)}\,\,,\,\,(-0.02x + 1) = 0$

$\text{ln(0) cannot be solved}\,\,,\,\,x = 50$

at x =50 dollars, the price changes from increasing to decreasing.

part(c) is only asking: from what value of x is the price increasing.

Well, judging from the answer of E'(x) = 0, the only value from where the price is changing from increasing to decreasing is x =50. so, the price has been increasing from x < 50. BUT since price cannot be negative (depending on the context), we can't say $x = -\infty$, instead our answer should be: that the rate of consumer expenditure (dE/dx) begin increasing from x = 0.  

Answer 2

The consumer expenditure rate at 110 dollars is -664.82. The consumer rate stop at x = 50. And the rate of cosumer expenditure (dE/dx) begin increasing from x = 0.

What is an exponent?

Exponential notation is the form of mathematical shorthand which allows us to write complicated expressions more succinctly. An exponent is a number or letter is called the base. It indicates that the base is to raise to a certain power. X is the base and n is the power.

Given

The demand for a certain commodity is $\rm D(x) = 5000e^{-0.02x}$ units per month when the market price is x dollars per unit.

(a)  The consumer expenditure E(x) = xD(x) changing with respect to price x when the price is equal to $110 dollars.

$\rm E(x) = xD(x) \\\\ E(x) = 5000xe^{-0.02}$

Then

$\rm \dfrac{dE(x)}{dx}|_{x=110}$ evaluated at x = 110.

$\rm \dfrac{dE}{dx} = \dfrac{d}{dx} (5000xe^{-0.02x})$

Using product rules for differentiation.

$\rm \dfrac{dE}{dx} = 5000(-0.02xe^{-0.02x} + e^{-0.02x ) \\\\\dfrac{dE}{dx} = 5000e^{-0.02x}(-0.02x + 1)  $

At x = 110, then

$\rm \dfrac{dE(x)}{dx}|_{x=110} = 5000e^{-0.02*110} (-0.02*110 + 1) \\\\\dfrac{dE(x)}{dx}|_{x=110} = -664.82$

(b) The consumer expenditure stop increasing and begin to decrease.

At E'(x) = 0 that is

$\rm \dfrac{dE}{dx} = 0$

Then

$\rm 5000e^{-0.02*110} (-0.02*110 + 1) = 0$

On simplifying, we have

$\rm x = 50$

At x = 50 dollars, the price changes from increasing or decreasing.

(c) The rate of consumer expenditure begins to increase.

Well, judging from the answer of E'(x) = 0, the only value from where the price is changing from increasing to decreasing is x = 50. So, the price has been increasing from x < 50. But, since the price cannot be negative, we cannot say $\rm x = -\infty$, instead our solution should be that

The rate of cosumer expenditure (dE/dx) begin increasing from x = 0.

More about the exponent link is given below.

https://brainly.com/question/5497425