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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![### Calculating the Cost Function from the Marginal Cost
This exercise helps you understand how to integrate a given marginal cost function to find the total cost function.
#### Problem Statement
Find the cost function if the marginal cost function is given by:
\[ C'(x) = x^{\frac{2}{5}} + 3 \]
Additionally, it is known that the cost for producing 32 units is $208.
\[ C(32) = 208 \]
#### Solution Approach
1. **Integration of the Marginal Cost Function:**
The given marginal cost function is:
\[ C'(x) = x^{\frac{2}{5}} + 3 \]
To find the total cost function \(C(x)\), integrate \(C'(x)\):
\[ \int C'(x) \, dx = \int \left( x^{\frac{2}{5}} + 3 \right) \, dx \]
2. **Integrate Term by Term:**
- For \( x^{\frac{2}{5}} \):
\[ \int x^{\frac{2}{5}} \, dx = \frac{5}{7} x^{\frac{7}{5}} \]
- For the constant 3:
\[ \int 3 \, dx = 3x \]
Putting it all together:
\[ C(x) = \frac{5}{7} x^{\frac{7}{5}} + 3x + C \]
where \( C \) is the constant of integration.
3. **Determine the Constant of Integration:**
We know that \( C(32) = 208 \).
Substitute \( x = 32 \) and \( C(32) = 208 \):
\[ 208 = \frac{5}{7} (32)^{\frac{7}{5}} + 3(32) + C \]
Calculate \( (32)^{\frac{7}{5}} \):
\[ (32)^{\frac{7}{5}} = 128 \]
So the equation becomes:
\[ 208 = \frac{5}{7} \cdot 128 + 96 + C \]
Simplify:
\[ 208 = \frac{640}{7} + 96 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02b4a1f4-9e35-42db-aa2a-674bb26df94b%2F70792109-ea9f-4307-ab77-5bc1d4c63293%2F6jvka18_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Cost Function from the Marginal Cost
This exercise helps you understand how to integrate a given marginal cost function to find the total cost function.
#### Problem Statement
Find the cost function if the marginal cost function is given by:
\[ C'(x) = x^{\frac{2}{5}} + 3 \]
Additionally, it is known that the cost for producing 32 units is $208.
\[ C(32) = 208 \]
#### Solution Approach
1. **Integration of the Marginal Cost Function:**
The given marginal cost function is:
\[ C'(x) = x^{\frac{2}{5}} + 3 \]
To find the total cost function \(C(x)\), integrate \(C'(x)\):
\[ \int C'(x) \, dx = \int \left( x^{\frac{2}{5}} + 3 \right) \, dx \]
2. **Integrate Term by Term:**
- For \( x^{\frac{2}{5}} \):
\[ \int x^{\frac{2}{5}} \, dx = \frac{5}{7} x^{\frac{7}{5}} \]
- For the constant 3:
\[ \int 3 \, dx = 3x \]
Putting it all together:
\[ C(x) = \frac{5}{7} x^{\frac{7}{5}} + 3x + C \]
where \( C \) is the constant of integration.
3. **Determine the Constant of Integration:**
We know that \( C(32) = 208 \).
Substitute \( x = 32 \) and \( C(32) = 208 \):
\[ 208 = \frac{5}{7} (32)^{\frac{7}{5}} + 3(32) + C \]
Calculate \( (32)^{\frac{7}{5}} \):
\[ (32)^{\frac{7}{5}} = 128 \]
So the equation becomes:
\[ 208 = \frac{5}{7} \cdot 128 + 96 + C \]
Simplify:
\[ 208 = \frac{640}{7} + 96 +
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