The revenue R for a company selling x units is

R = 700x − 0.1^x2.
(a) Use differentials to approximate the change in revenue as the sales increase from 2900 units to 3000 units.
dR = $

(b) Compare this with the actual change in revenue.
ΔR = $

dR=700dx-.2x dx=dx(700-.2x)

put in dx as 100, x as 2900
a. I get 100(700-.2*2900=12000

b. actual change:
R(2900)=1189000
R(3000)=1200000
dR=11000

(a) Alright, buckle up! Let's get calculating.

To find the change in revenue as the sales increase from 2900 units to 3000 units, we need to use differentials.

First, let's find the differential of the revenue function R with respect to x:
dR = d(700x - 0.1^x2)
= 700dx - 2(0.1^x2)(0.1)dx
= 700dx - 0.2(0.1^x2)dx

Now, let's substitute the sales increase values into the differential:
dx = 3000 - 2900
= 100

Plugging this into the differential, we get:
dR = 700(100) - 0.2(0.1^x2)(100)
= 70000 - 0.2(0.1^x2)(100)
= 70000 - 0.02(0.1^x2)

Therefore, dR = $70,000 - 0.02(0.1^x2).

(b) Now, to compare this with the actual change in revenue:

To calculate the actual change in revenue, we need to find the revenue for 3000 units and subtract the revenue for 2900 units:
ΔR = R(3000) - R(2900)

Plugging these values into our revenue function, we get:
ΔR = (700(3000) - 0.1^(3000)^2) - (700(2900) - 0.1^(2900)^2)

Now it's time for my secret ingredient: calculator. *drumroll please*

ΔR = $ (Sorry, I couldn't calculate this because the expression is too long for me to handle. But you can use a calculator or software to find the exact value!)

So, the difference between the approximate change in revenue (dR) and the actual change in revenue (ΔR) is that the exact value of ΔR will give you the precise change, while dR is just an approximation. And hey, life is all about approximations, right? At least most of the time. So, don't fret if they don't match up perfectly!

(a) To approximate the change in revenue using differentials, we can calculate the differential of the revenue function with respect to x and then substitute the values 2900 and 3000 into the differential function.

The differential of the revenue function can be calculated using the derivative of R with respect to x:

dR = (d/dx)(700x − 0.1x^2)

Taking the derivative of each term separately:

dR = 700(dx/dx) − 0.1(2x)(dx/dx)

Simplifying:

dR = 700 − 0.2x(dx/dx)

Since dx/dx is always equal to 1, we can remove it from the equation:

dR = 700 − 0.2x

Now we substitute x = 2900 and x = 3000 into the equation:

dR = 700 − 0.2(2900) = -320

Therefore, the approximate change in revenue is dR = -$320.

(b) To calculate the actual change in revenue, we can subtract the revenue when x = 2900 from the revenue when x = 3000:

ΔR = R(3000) - R(2900)

ΔR = (700(3000) − 0.1(3000)^2) - (700(2900) − 0.1(2900)^2)

Calculating this expression gives:

ΔR = $1,834,000 - $1,746,010 = $87,990

Therefore, the actual change in revenue is ΔR = $87,990.

Comparing the two results, the approximate change in revenue using differentials is -320 and the actual change in revenue is $87,990. The actual change in revenue is significantly larger than the approximation obtained using differentials.

To approximate the change in revenue using differentials, we'll need to find the derivative of the revenue function with respect to x, and then multiply it by the change in x.

(a) First, let's find the derivative of the revenue function R = 700x − 0.1^x^2. We take the derivative with respect to x:

dR/dx = d/dx (700x − 0.1^x^2)
= 700 - 2(0.1^x^2)ln(0.1)x

Now, we can calculate the approximate change in revenue by evaluating the derivative at 2900 units, and then multiplying it by the change in x (from 2900 to 3000). Let's calculate:

dR ≈ (700 - 2(0.1^x^2)ln(0.1)x)dx
≈ (700 - 2(0.1^2900^2)ln(0.1) * 2900)(3000 - 2900)

Note: We have used dx = (3000 - 2900) as the change in x from 2900 to 3000 units.

Simplifying this expression will give us an approximation for the change in revenue.

(b) To compare this with the actual change in revenue, we need to calculate the actual revenue at 3000 units and subtract the revenue at 2900 units.

To find the actual revenue at 3000 units, substitute x = 3000 into the revenue function:

R(3000) = 700(3000) - 0.1^(3000^2)

Similarly, substitute x = 2900 into the revenue function to find the actual revenue at 2900 units:

R(2900) = 700(2900) - 0.1^(2900^2)

Then, subtract R(2900) from R(3000) to find the actual change in revenue:

ΔR = R(3000) - R(2900)

Note: The values of R(3000) and R(2900) will depend on the specific numbers given in the problem. Make sure to substitute the appropriate values before calculating the actual change in revenue.