A total revenue function is given by R(x) = 1000(x^2 - 0.1x)^1/2 , where R(x) is the total revenue, in thousands of dollars, from the sale of x items. Find the rate at which total revenue is changing when 20 items have been sold.
Calculate the amount of interest on a loan of $3,200 at 6% interest for 60 days using
To find the rate at which total revenue is changing when 20 items have been sold, we need to find the derivative of the total revenue function with respect to x and then evaluate it at x = 20.
First, let's find the derivative of the total revenue function R(x):
R(x) = 1000(x^2 - 0.1x)^1/2
To find the derivative, we can use the chain rule. Let's simplify the expression inside the parentheses:
(x^2 - 0.1x)^1/2 = (x(x - 0.1))^1/2 = (x^2 - 0.1x)^(1/2) = (x(x - 0.1))^1/2 = (x^2 - 0.1x)^(1/2)
Now, let's differentiate R(x) using the chain rule. Let u = x^2 - 0.1x:
dR/dx = dR/du * du/dx
To find dR/du, we differentiate R(u) with respect to u:
dR/du = 1000 * (u)^(-1/2) = 1000 * (x^2 - 0.1x)^(-1/2)
To find du/dx, we differentiate u with respect to x:
du/dx = d(x^2 - 0.1x)/dx = 2x - 0.1
Now, let's combine the results to find dR/dx:
dR/dx = dR/du * du/dx
= 1000 * (x^2 - 0.1x)^(-1/2) * (2x - 0.1)
Now, we can evaluate dR/dx at x = 20:
dR/dx at x = 20 = 1000 * (20^2 - 0.1*20)^(-1/2) * (2*20 - 0.1)
= 1000 * (400 - 2)^(1/2) * (40 - 0.1)
= 1000 * (398)^(1/2) * (39.9)
≈ 792,011.19
Therefore, the rate at which the total revenue is changing when 20 items have been sold is approximately $792,011.19.
To find the rate at which total revenue is changing when 20 items have been sold, we need to find the derivative of the total revenue function R(x) and then substitute x = 20 to evaluate the derivative at that point.
Here's how to do it step by step:
Step 1: Take the derivative of the total revenue function R(x) with respect to x. The exponent of (x^2 - 0.1x) is 1/2, so we can use the power rule for differentiation.
R'(x) = d/dx [1000(x^2 - 0.1x)^1/2]
Using the power rule, the derivative is:
R'(x) = 1000 * 1/2 * (x^2 - 0.1x)^-1/2 * (2x - 0.1)
Simplifying this expression, we get:
R'(x) = 500 * (x^2 - 0.1x)^-1/2 * (2x - 0.1)
Step 2: Substitute x = 20 into the derived expression to evaluate the derivative at that point.
R'(20) = 500 * (20^2 - 0.1 * 20)^-1/2 * (2 * 20 - 0.1)
First, calculate the expressions in the parentheses:
(20^2 - 0.1 * 20) = (400 - 2) = 398
(2 * 20 - 0.1) = (40 - 0.1) = 39.9
Now substitute these values into the expression:
R'(20) = 500 * 398^-1/2 * 39.9
To simplify this, calculate the square root of 398 and substitute the value:
√398 ≈ 19.9499
R'(20) ≈ 500 * 1/19.9499 * 39.9
Now calculate the expression:
R'(20) ≈ 500 * 0.0500984 * 39.9
R'(20) ≈ 1001.976
Therefore, the rate at which total revenue is changing when 20 items have been sold is approximately $1001.976 (in thousands of dollars per item).
that would be dR/dx when x=20
R = 1000(x^2-.1x)^2
R' = 2000(x^2-.1x)(2x-.1)
R'(20) = 31,760,400