Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.4.107

Chapter 4, Problem 4.4.107

107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

Verified step by step guidance

Step 1: Understand the concept of marginal cost. Marginal cost is the derivative of the cost function c=f(x) with respect to the number of items produced, x. It represents the rate of change of cost as production increases.

Step 2: Analyze the graph provided. The graph shows the cost function c=f(x) as a curve. To determine where the marginal cost changes from decreasing to increasing, we need to identify the point where the slope of the tangent to the curve transitions from decreasing to increasing.

Step 3: Look for the inflection point on the graph. An inflection point is where the curvature of the graph changes, which corresponds to the second derivative of the cost function changing sign. This is the point where the marginal cost transitions from decreasing to increasing.

Step 4: Estimate the production level at the inflection point. From the graph, observe the curve's behavior and approximate the x-value (production level) where the slope of the tangent stops decreasing and starts increasing. This appears to be around x=60 thousand units.

Step 5: Conclude that the marginal cost changes from decreasing to increasing at approximately x=60 thousand units. This is based on the visual analysis of the graph and the concept of inflection points.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Marginal Cost

Marginal cost refers to the additional cost incurred when producing one more unit of a good or service. It is derived from the cost function, c = f(x), by calculating the derivative, f'(x). Understanding where marginal cost increases or decreases helps businesses make informed production decisions.

Cost Function

The cost function, represented as c = f(x), describes the total cost of producing x units of a product. It typically reflects fixed and variable costs and can be analyzed to determine how costs change with varying production levels. The shape of this function is crucial for identifying points of marginal cost change.

Inflection Point

An inflection point on a graph is where the curvature changes, indicating a transition in the behavior of the function. In the context of marginal cost, it marks the production level where the marginal cost shifts from decreasing to increasing, which is essential for optimizing production efficiency and cost management.

Recommended video:

04:50

Critical Points

Related Practice

Textbook Question

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

f(x) = x − 6√(x − 1)

Textbook Question

Identifying Extrema

In Exercises 15–18:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local and absolute extreme values, if any, saying where they occur.

Textbook Question

26. Constructing cylinders Compare the answers to the following two construction problems.

a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?

" style="" width="350">

Textbook Question

Theory and Examples

In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.

y = x¹¹ + x³ + x − 5

Textbook Question

Finding Functions from Derivatives

In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.

f'(x) = 2x − 1, P(0,0)

Textbook Question

10. Catching rainwater A 1125 ft^3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy.

a. If the total cost is c=5(x^2+4xy) + 10xy, what values of x and y will minimize it?

b. Give a possible scenario for the cost function in part (a).