Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Concavity

Calculus

5. Graphical Applications of Derivatives

Concavity

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2:14 minutes

Problem 4.4.107

Textbook Question

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.

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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.

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Textbook Question

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

Textbook Question

Graph f(x) = x cos x and its second derivative together for 0 ≤ x ≤ 2pi. Comment on the behavior of the graph of f in relation to the signs and values of f".

Textbook Question

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

Textbook Question

103. A function f(x) has domain (-2, 2). The graph below is a plot of the derivative of f, not a plot of f itself. In other words, this is a graph of y = f'(x). Either use this graph to determine on which intervals the graph of f is concave up and on which intervals the graph of f is concave down, or explain why this information cannot be determined from the graph.

Textbook Question

93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \y'' as positive, negative, or zero.

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Textbook Question

99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.

Multiple Choice

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

Multiple Choice

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

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