What do marginal cost, revenue, and profit approximate? If the cost function is C (x) = 4x² + 15, what is the marginal cost? What is C'(10)? Find the exact cost of making the 11th unit and compare that to the marginal cost. Are they close?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![**Understanding Marginal Cost, Revenue, and Profit**
To delve into these fundamental economic concepts, let’s consider the cost function \( C(x) = 4x^2 + 15 \).
1. **Marginal Cost:**
- Marginal cost represents the cost of producing one more unit of a good. Mathematically, it is the derivative of the cost function, denoted as \( C'(x) \).
- For the given cost function \( C(x) = 4x^2 + 15 \), the marginal cost function is determined by finding the derivative:
\[
C'(x) = \frac{d}{dx}(4x^2 + 15) = 8x.
\]
- The marginal cost when \( x = 10 \) is \( C'(10) = 8(10) = 80 \).
2. **Exact Cost of Making the 11th Unit:**
- To find the exact cost of producing the 11th unit, calculate the difference \( C(11) - C(10) \):
\[
C(11) = 4(11)^2 + 15 = 4(121) + 15 = 484 + 15 = 499,
\]
\[
C(10) = 4(10)^2 + 15 = 4(100) + 15 = 400 + 15 = 415.
\]
- The exact cost of making the 11th unit is \( 499 - 415 = 84 \).
3. **Comparison:**
- Compare the exact cost (84) with the marginal cost (80).
- These values are quite close, demonstrating how marginal cost provides an approximation of the cost for producing an additional unit, especially useful at large scales.
This exercise illustrates how marginal cost provides an efficient way to approximate changes in total cost with changes in production.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37f6a860-f974-494c-9785-179491d557d0%2F6eb8506f-fda4-4e43-89ab-7f46d2e0f707%2F159ctw_processed.png&w=3840&q=75)
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