Let C(q) represent the cost, R(q) the revenue, and π(q) the total profit, in dollars, of produci

step 1 Hello, in this problem we are given the following. With CFQ representing the cost function we ar

Let C(q) represent the cost, R(q) the revenue, and π(q) the...

Let $C(q)$ represent the cost, $R(q)$ the revenue, and $\pi(q)$ the total profit, in dollars, of producing $q$ items. (a) If $C^{\prime}(50)=75$ and $R^{\prime}(50)=84,$ approximately how much profit is earned by the $51^{\text {st }}$ item? (b) If $C^{\prime}(90)=71$ and $R^{\prime}(90)=68,$ approximately how much profit is earned by the $91^{\text {st }}$ item? (c) If $\pi(q)$ is a maximum when $q=78,$ how do you think $C^{\prime}(78)$ and $R^{\prime}(78)$ compare? Explain.

Let $C(q)$ represent the cost, $R(q)$ the revenue, and $\pi(q)$ the total profit, in dollars, of producing $q$ items.
(a) If $C^{\prime}(50)=75$ and $R^{\prime}(50)=84,$ approximately how much profit is earned by the $51^{\text {st }}$ item?
(b) If $C^{\prime}(90)=71$ and $R^{\prime}(90)=68,$ approximately how much profit is earned by the $91^{\text {st }}$ item?
(c) If $\pi(q)$ is a maximum when $q=78,$ how do you think $C^{\prime}(78)$ and $R^{\prime}(78)$ compare? Explain.

Key Concepts

Cost Function

The cost function represents the total expense incurred in producing a given number of items, including both fixed and variable costs. Its derivative, often called the marginal cost, indicates the additional cost incurred by producing one more unit.

Revenue Function

The revenue function displays the total income from selling a specific number of items. Its derivative, known as the marginal revenue, measures the extra income earned from the sale of one additional item.

Profit Function

The profit function is defined as the difference between the revenue and cost functions, typically shown as ?(q) = R(q) - C(q). Its derivative, or marginal profit, approximates the change in profit for producing one more unit and is crucial in evaluating the profitability of incremental changes.

Marginal Analysis

Marginal analysis examines the impact of producing one more unit by using the derivatives of the cost and revenue functions. By comparing the marginal revenue and marginal cost, one can estimate the additional profit or loss from a small change in production levels.

Optimization in Profit Maximization

When the profit function reaches a maximum, its derivative equals zero. This condition implies that the marginal revenue is equal to the marginal cost at the optimal production level, meaning any further increase or decrease in production will not lead to a higher profit.

Transcript

00:01 Hello, in this problem we are given the following.

00:07 With cfq representing the cost function we are given in part a.

00:14 We are given that the marginal cost of producing 75 and the marginal revenue of producing 50 is...

00:32 The question is how much profit does producing the 51st we are? we want to know how much increasing production from 50 to 51, how much does it earn.

00:57 Okay, so let's see one more time.

01:01 Pye of q is the profit function and the profit function equals the revenue earned by selling q units minus the cost of producing q.

01:15 Differentiating, we have marginal profit of q equal the marginal revenue.

01:22 Of q minus the marginal cost of q.

01:28 What does, what does, the fact that the marginal cost of, the marginal cost of 50 units 75? it means that the slope of the curve of the cost function at 50 is 75.

01:53 So we can approximate, we can approximate that the cost of producing one more unit than 50 is $75.

02:11 And the marginal revenue function tells us that selling one more extra unit from of 50, which means 50 first units, we will have a change in revenue of 84 dollars.

02:27 So basically, the marginal profit at 50, we obtained by inserting.

02:36 These two values here.

02:38 Eighty -four minus 75, nine dollars.

02:45 Once again, interpret this.

02:49 The marginal profit, graphically looking, say here at 50, the marginal profit curve is something like this.

02:59 So the slope of the tangent to the graph, the slope is nine.

03:08 So we can approximate, we can approximate, we can approximate, that the profit function at 51, the profit function at producing 51 will be equal to the profit of producing 50 units plus 9.

03:31 So to answer the question, how much profit do we gain or lose by producing an extra unit of 50 first unit? it gains 9 dollar profit...

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