Question

) Identify the name of the following below: f(x)dx Each part of the f(x)dx can also be identified by name. As a result, B) What does the represent and provide a definition using your own words. C) What is the f(x) called and provide a definition using your own words. D) What is the symbol dx called and provide a definition using your own words. E) The f(x)dx

has a real-world application, so let’s say the monthly marginal cost for a product is = x 20. If fixed costs are $50, find the total cost function C(x) for the month and C(0).

Answer 1

Explanation:

A) The expression f(x)dx represents a mathematical integral.

B) The integral symbol (∫) represents the process of integration, which is a fundamental concept in calculus. Integration is used to find the area under a curve, among other applications. In this context, the integral symbol (∫) indicates that we are finding the antiderivative of the function f(x) with respect to x.

C) The expression f(x) is called the integrand. It represents the function being integrated.

D) The symbol dx is called the differential of x. It represents an infinitesimally small change in the variable x.

E) The f(x)dx expression has various real-world applications, particularly in physics, engineering, and economics. For example, it can be used to calculate the total cost of production or the total amount of work done.

To illustrate this, let's consider the given scenario where the monthly marginal cost for a product is equal to 20x and fixed costs are $50. To find the total cost function C(x) for the month, we integrate the marginal cost function with respect to x:

C(x) = ∫(20x)dx

Integrating 20x with respect to x gives us (1/2)x^2. Therefore:

C(x) = (1/2)x^2 + C

Since fixed costs are $50, we can determine the value of C by evaluating C(0):

C(0) = (1/2)(0)^2 + C

C(0) = 0 + C

C(0) = C = $50

Therefore, the total cost function for the month is:

C(x) = (1/2)x^2 + $50