Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Understanding the Cost Function in Ice Cream Production
The following problem explores the cost \( C \) (in dollars) associated with the production of \( g \) gallons of ice cream. The cost can be described by the function \( C = f(g) \). We'll use mathematical notation and units to explain the meaning behind specific statements about the cost function.
1. **Given Statements:**
- **\( f(200) = 350 \)**
This statement indicates that the cost of producing 200 gallons of ice cream is 350 dollars. Mathematically, it means when \( g = 200 \), \( C = 350 \).
- **\( f'(200) = 1.4 \)**
This represents the marginal cost at 200 gallons. Specifically, it means that the rate of change of the cost with respect to the amount of ice cream produced (in gallons) at \( g = 200 \) is 1.4 dollars per gallon. In simpler terms, producing one additional gallon of ice cream when already producing 200 gallons costs an additional 1.4 dollars.
2. **Estimation of \( f(202) \) Using the Given Information:**
To estimate \( f(202) \), we'll use the provided marginal cost. We know that:
\[
f(200) = 350
\]
and
\[
f'(200) = 1.4
\]
We can approximate the cost for 202 gallons by adding the marginal cost for 2 additional gallons to the cost for 200 gallons.
\[
f(202) \approx f(200) + 2 \times f'(200)
\]
Substituting the known values:
\[
f(202) \approx 350 + 2 \times 1.4
\]
\[
f(202) \approx 350 + 2.8
\]
\[
f(202) \approx 352.8
\]
Thus, the estimated cost to produce 202 gallons of ice cream is approximately 352.8 dollars.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa55013e1-91aa-48be-9c4b-bf8a0e47de22%2Fde00b15d-dfb4-4ff0-96f2-bedc7959b1e9%2Foauaqr_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Cost Function in Ice Cream Production
The following problem explores the cost \( C \) (in dollars) associated with the production of \( g \) gallons of ice cream. The cost can be described by the function \( C = f(g) \). We'll use mathematical notation and units to explain the meaning behind specific statements about the cost function.
1. **Given Statements:**
- **\( f(200) = 350 \)**
This statement indicates that the cost of producing 200 gallons of ice cream is 350 dollars. Mathematically, it means when \( g = 200 \), \( C = 350 \).
- **\( f'(200) = 1.4 \)**
This represents the marginal cost at 200 gallons. Specifically, it means that the rate of change of the cost with respect to the amount of ice cream produced (in gallons) at \( g = 200 \) is 1.4 dollars per gallon. In simpler terms, producing one additional gallon of ice cream when already producing 200 gallons costs an additional 1.4 dollars.
2. **Estimation of \( f(202) \) Using the Given Information:**
To estimate \( f(202) \), we'll use the provided marginal cost. We know that:
\[
f(200) = 350
\]
and
\[
f'(200) = 1.4
\]
We can approximate the cost for 202 gallons by adding the marginal cost for 2 additional gallons to the cost for 200 gallons.
\[
f(202) \approx f(200) + 2 \times f'(200)
\]
Substituting the known values:
\[
f(202) \approx 350 + 2 \times 1.4
\]
\[
f(202) \approx 350 + 2.8
\]
\[
f(202) \approx 352.8
\]
Thus, the estimated cost to produce 202 gallons of ice cream is approximately 352.8 dollars.
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