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

Let $C(q)$ represent the cost and $R(q)$ represent the revenue, in dollars, of producing $q$ items. 5 (a) If $C(50)=4300$ and $C^{\prime}(50)=24,$ estimate $C(52)$ (b) If $C^{\prime}(50)=24$ and $R^{\prime}(50)=35,$ approximately how much profit is earned by the $51^{\text {st }}$ item? (c) If $C^{\prime}(100)=38$ and $R^{\prime}(100)=35,$ should the company produce the $101^{\text {st }}$ item? Why or why not?

Let $C(q)$ represent the cost and $R(q)$ represent the revenue, in dollars, of producing $q$ items.
5
(a) If $C(50)=4300$ and $C^{\prime}(50)=24,$ estimate $C(52)$
(b) If $C^{\prime}(50)=24$ and $R^{\prime}(50)=35,$ approximately how much profit is earned by the $51^{\text {st }}$ item?
(c) If $C^{\prime}(100)=38$ and $R^{\prime}(100)=35,$ should the company produce the $101^{\text {st }}$ item? Why or why not?

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Key Concepts

Marginal Analysis

Marginal analysis refers to the study of the additional or incremental costs and revenues associated with producing one more unit of a product. It uses the derivatives of the cost and revenue functions to quantify these changes, providing insight into the effective impact of small variations in production levels.

Profit Optimization

Profit optimization involves comparing the marginal revenue and marginal cost to determine whether the production of an additional unit is profitable. If the marginal revenue exceeds the marginal cost, producing one more unit will increase profit; conversely, if the marginal cost is higher, it may lead to a decrease in profit.

Linear Approximation

This concept involves using the value and derivative of a function at a known point to estimate the function's value at a nearby point. By using the tangent line to the function, one can approximate small changes effectively, which is especially useful when calculating function values without having the explicit formula.

Transcript

0:00 Hi there.

00:01 In this problem, we're given different information in each of the three parts.

00:05 So for part a, we're given that the cost of making 50 items is $4 ,300.

00:11 We're given c prime, which, remember, is just the marginal cost at 50 items, is 24.

00:19 So we're asked to estimate, now what's the cost of making 52 items? let's remember, we know the cost of making 50 items, so it should be at least that much.

00:31 We also know, at least an approximation, how much does it cost to add one additional item to production? and that's what the marginal cost is.

00:41 So it adds $24 per item that we're adding.

00:46 So now we're adding two items.

00:50 And so we'll want to multiply that by the marginal cost.

00:55 And remember, the marginal cost, let's cross off the m and put a prime here.

00:58 Marginal cost is c prime.

01:01 It's the derivative of cost.

01:02 So just make sure we're adding two of those, since we're adding two items to our production.

01:09 The cost of making 50 was 4 ,300 plus two times the cost of adding an item is 24.

01:18 So we get 4300 plus, well, two times 24 is 48.

01:24 So we end up with 4348.

01:31 It's 4 ,348.

01:32 That's our best estimate for the cost of making 52 items.

01:36 For part b, we're told c prime at 50 items is 24, which again remember, this is just marginal costs...

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