Understanding Marginal Cost and Integration

The video tutorial explains how to calculate the total cost of producing the first 49 units using the company's marginal cost function, C'(x) = 8/√x. It begins by introducing the problem and graphically representing the marginal cost function. The tutorial then demonstrates how to use the definite integral of the marginal cost function from 0 to 49 to find the total cost. The integrand is transformed for easier integration, and the antiderivative is calculated. Finally, the integral is evaluated at the limits to determine the total cost, which is $112.

What is the marginal cost function given in the problem?

• C'(x) = 8x^2

• C'(x) = 8 / sqrt(x)

• C'(x) = 8 + x

• C'(x) = 8x

What does the horizontal axis represent in the graph of the marginal cost function?

• Cost in dollars per unit

• Profit

• Total cost

• Number of units sold

How can the total cost for the first 49 units be determined?

• By finding the area under the marginal cost function from 0 to 49

• By multiplying cost per unit by 49

• By adding the marginal costs of each unit

• By calculating the average cost

What is the process of finding the total cost using integration called?

• Differentiation

• Summation

• Integration

• Multiplication

How is the integrand 8 / sqrt(x) transformed for integration?

• 8x^2

• 8x^(-1/2)

• 8x^(1/2)

• 8x

What happens to the exponent when 8 / sqrt(x) is moved to the numerator?

• It halves

• It becomes negative

• It doubles

• It becomes positive

What is the antiderivative of 8x^(-1/2)?

• 16x^(1/2)

• 4x^(1/2)

• 2x^(1/2)

• 8x^(1/2)

What is the value of the integral at the upper limit of 49?

• 8 x sqrt(49)

• 16 x sqrt(49)

• 4 x sqrt(49)

• 2 x sqrt(49)

What is the square root of 49?

• 8

• 7

• 6

• 5

What is the total cost for the first 49 units?

• $130

• $120

• $112

• $100