Understanding Marginal Cost and Derivatives

The video tutorial explains the cost function for producing wood stain and derives the marginal cost function C' of x. It demonstrates how to calculate the marginal cost of producing the 101st gallon using the derivative and compares this approximation to the exact cost. The tutorial highlights the difference between linear approximation and exact values, emphasizing the concept of instantaneous rate of change.

What is the initial cost in dollars for producing x gallons of wood stain?

• 3,200 + 0.1x - 0.001x^2 + 0.0004x^3

• 3,200 + 0.1x

• 0.1x - 0.001x^2

• 0.0004x^3

What is the derivative of the constant term 3,200 in the cost function?

• 3,200

• 0.0012

• 0

• 0.1

How is the derivative of the term -0.001x^2 calculated?

• -0.002x

• 0.002x

• -0.001x

• 0.001x

What does the marginal cost function C' of x represent?

• The average cost per gallon

• The fixed cost of production

• The change in cost with respect to a change in quantity

• The total cost of producing x gallons

How do you approximate the cost of producing the 101st gallon using the marginal cost function?

• By calculating C(100) + C'(100)

• By calculating C'(101) and multiplying by 2

• By calculating C'(100) and multiplying by 1

• By calculating C(101) - C(100)

What is the calculated marginal cost of producing the 101st gallon using the derivative?

• $12.02

• $13.50

• $10.00

• $11.90

What is the exact cost of producing the 101st gallon?

• $11.90

• $12.02

• $10.50

• $13.00

Why is there a discrepancy between the marginal cost approximation and the exact cost?

• The cost function is linear

• The cost function is non-linear

• The cost function is constant

• The derivative was calculated incorrectly

What does the slope of the tangent line at a point on the cost function represent?

• The total cost at that point

• The fixed cost at that point

• The average cost at that point

• The marginal cost at that point

What is the significance of using a linear approximation in this context?

• It provides the exact cost

• It is not useful

• It simplifies the calculation of marginal cost

• It increases the cost