What is the difference between the actual increase in profit as production increases from 5 to 6 units, and the marginal profit at a production level of 5 units where the profit function is given by
p(x)= 3x^2 - 5x + 2
First I plugged 5 into the equation and got 52. Then I took the derivative which is 6x -5 and plugged in 5 and got 25. 52 - 25 = 27. Is this correct?
No, you are not correct. But, you almost had it!
To compute the actual increase in profit, first plug in 5, then plug in 6. The difference between f(5) and f(6) is the actual.
p(x)= 3x^2 - 5x + 2
p(5) = 3(5)^2 - 5(5) + 2 = 52
p(6) = 3(6)^2 - 5(6) + 2 = 80
p(6) - p(5) = 80 - 52 = 28
To compute the marginal profit
dp/dx = 6x - 5
6(5) - 5 = 25
Actual - marginal = 28 - 25 = 3
Check my math
To determine the difference between the actual increase in profit and the marginal profit at a production level of 5 units, you need to follow the steps correctly.
First, find the profit at a production level of 5 units by substituting x=5 into the profit function:
p(x) = 3x^2 - 5x +2
p(5) = 3(5^2) - 5(5) + 2
= 3(25) - 25 + 2
= 75 - 25 + 2
= 52
So the profit at a production level of 5 units is 52.
Next, find the marginal profit at a production level of 5 units by taking the derivative of the profit function and substituting x=5 into the derivative:
p'(x) = 6x - 5
p'(5) = 6(5) - 5
= 30 - 5
= 25
So the marginal profit at a production level of 5 units is 25.
Finally, to find the difference between the actual increase in profit and the marginal profit at a production level of 5 units, subtract the marginal profit from the actual increase in profit:
Difference = Actual Increase in Profit - Marginal Profit
= 52 - 25
= 27
Therefore, the difference between the actual increase in profit and the marginal profit at a production level of 5 units is 27, which confirms your calculation.
To determine the difference between the actual increase in profit as production increases from 5 to 6 units and the marginal profit at a production level of 5 units, you followed the correct steps. However, there seems to be a slight calculation error in your solution.
Let's break it down again:
1. Start with the profit function: p(x) = 3x^2 - 5x + 2.
2. To find the actual increase in profit, evaluate the profit function at x = 6 and x = 5, then subtract the two values.
p(6) = 3(6)^2 - 5(6) + 2 = 108 - 30 + 2 = 80.
p(5) = 3(5)^2 - 5(5) + 2 = 75 - 25 + 2 = 52.
The actual increase in profit is: 80 - 52 = 28.
3. To calculate the marginal profit at x = 5, take the derivative of the profit function and evaluate it at x = 5.
p'(x) = 6x - 5.
p'(5) = 6(5) - 5 = 30 - 5 = 25.
4. Finally, subtract the marginal profit from the actual increase in profit to find the difference.
Difference = Actual increase in profit - Marginal profit = 28 - 25 = 3.
Therefore, the correct difference is 3, not 27 as you mentioned.