The demand for the video game is modeled by the logistic curve, where q(t) is the total number of units sold t months after its introduction.
q(t)= 10000/(1+0.5 e**(-0.4t))
(a) Use technology to estimate q'(4) to the nearest integer.
mark units per month
(b) Assume that the manufacturers of the video game sell each unit for $810. What is the company's marginal revenue dR/dq?
$ mark /unit
(c) Use the chain rule to estimate the rate at which revenue is growing 4 months after the introduction of the video game. (Round your answer to the nearest thousand.)
$ mark per month.
12
To answer these questions, we need to find the derivative of the demand function q(t) with respect to time t.
(a) To estimate q'(4) using technology, we can use a graphing calculator or a software like Wolfram Alpha. Here's how you can do it on Wolfram Alpha:
1. Open Wolfram Alpha in your web browser.
2. Type in "D[10000/(1+0.5*e^(-0.4*x)), x]" and hit enter.
3. Wolfram Alpha will provide you with the derivative of the function.
4. Substitute x=4 into the derivative you obtained in step 3.
5. Round the result to the nearest integer to get q'(4).
(b) The marginal revenue can be found by taking the derivative of the revenue function R(q) with respect to quantity q.
Given that each unit is sold for $810, the revenue function R(q) is given by R(q) = 810 * q.
To find dR/dq, we need to take the derivative of R(q) with respect to q. The derivative of R(q) with respect to q is constant and equal to the selling price per unit. Therefore, dR/dq = $810/unit.
(c) To estimate the rate at which revenue is growing 4 months after the introduction of the video game using the chain rule, we need to take the derivative of the revenue function R(q(t)) with respect to time t.
Given that q(t) = 10000/(1+0.5*e^(-0.4*t)), the revenue function R(q(t)) can be written as R(q(t)) = 810 * q(t).
To find the rate at which revenue is growing 4 months after the introduction, we need to take the derivative of R(q(t)) with respect to t using the chain rule.
1. Calculate dR(q(t))/dq by taking the derivative of R(q(t)) with respect to q, which we found in part (b) as $810/unit.
2. Calculate dq(t)/dt by taking the derivative of q(t) with respect to t. Use the chain rule to simplify this derivative.
3. Multiply dR(q(t))/dq and dq(t)/dt to calculate the rate at which revenue is growing.
4. Substitute t=4 into the expression you obtained in step 3.
5. Round the result to the nearest thousand to get the rate at which revenue is growing 4 months after the introduction.