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

4. Applications of Derivatives

Related Rates

Calculus

4. Applications of Derivatives

Related Rates

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2:35 minutes

Problem 21a

Textbook Question

Economics

Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².

a. Find the average cost per machine of producing the first 100 washing machines.

Verified step by step guidance

First, understand the given cost function: c(x) = 2000 + 100x - 0.1x², where c(x) represents the total cost of producing x washing machines.

To find the average cost per machine for the first 100 machines, we need to calculate the total cost for producing 100 machines, which is c(100).

Substitute x = 100 into the cost function: c(100) = 2000 + 100(100) - 0.1(100)².

Calculate c(100) to find the total cost of producing 100 machines.

Finally, divide the total cost c(100) by 100 to find the average cost per machine: Average cost = c(100) / 100.

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Key Concepts

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

Average Cost

Average cost is calculated by dividing the total cost by the number of units produced. It provides insight into the cost efficiency of production. For the given function c(x), the average cost per machine for producing x washing machines is c(x)/x, which helps determine the cost per unit for a specific production level.

Cost Function

A cost function, such as c(x) = 2000 + 100x − 0.1x², represents the total cost of producing x units. It includes fixed costs, variable costs, and sometimes quadratic terms that account for changes in cost efficiency. Understanding this function is crucial for calculating total and average costs at different production levels.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically in the form ax² + bx + c. In economics, they can model cost functions where the cost changes non-linearly with production levels. The term −0.1x² in the cost function indicates diminishing returns or increasing inefficiencies as production increases, affecting the average cost calculation.

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Economics

Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².

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Textbook Question

Economics

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