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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![### Transcription of Mathematical Problems
**Problem 3:**
If \( f \) is the function given by \( f(x) = x^3 + 11x^2 + 3x - 2 \), name the open interval where the graph of \( f \) is increasing.
- The derivative of the function \( f \) is given as \( f'(x) = 3x^2 + 22x + 3 \).
- The problem attempts to solve for intervals where the graph is increasing by setting up the inequality \( 3(x+1)(x+3) > 0 \).
**Problem 4:**
Find the slope of the tangent line to the graph of \( 3x^3 + 2y^3 = -10 \) at \( y = -2 \).
- Differentiate the equation implicitly: \( 9x^2 dx + 6y^2 dy = 0 \).
- Solve for \( \frac{dy}{dx} \) using the derived relation \( 9x^2 + 6y^2 \frac{dy}{dx} = 0 \).
**Problem 5:**
If \( f(x) = \frac{1}{3}x^3 - \frac{1}{2}x^2 - 25 \), then what is the minimum value of the function on the closed interval \([-2, 5]\)?
- Calculate function values at the endpoints:
- \( f(-2) = 3(-2)^2 - 2(-2) - \frac{25}{3} \),
- \( f(5) = 3(5)^2 - 2(5) - \frac{25}{3} \).
- Compare these values to find the minimum value on the interval.
### Additional Notes
- The problems involve calculations of derivatives and understanding the behavior of polynomial functions in various contexts.
- Students will need to apply skills in differentiation, inequality solving, and evaluating functions at specific points to tackle these problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb492f7ae-3a50-4a0d-856d-c60c2ea37af5%2Fb574d233-1214-4aa1-b544-74fd403ddb17%2Fj9992d5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription of Mathematical Problems
**Problem 3:**
If \( f \) is the function given by \( f(x) = x^3 + 11x^2 + 3x - 2 \), name the open interval where the graph of \( f \) is increasing.
- The derivative of the function \( f \) is given as \( f'(x) = 3x^2 + 22x + 3 \).
- The problem attempts to solve for intervals where the graph is increasing by setting up the inequality \( 3(x+1)(x+3) > 0 \).
**Problem 4:**
Find the slope of the tangent line to the graph of \( 3x^3 + 2y^3 = -10 \) at \( y = -2 \).
- Differentiate the equation implicitly: \( 9x^2 dx + 6y^2 dy = 0 \).
- Solve for \( \frac{dy}{dx} \) using the derived relation \( 9x^2 + 6y^2 \frac{dy}{dx} = 0 \).
**Problem 5:**
If \( f(x) = \frac{1}{3}x^3 - \frac{1}{2}x^2 - 25 \), then what is the minimum value of the function on the closed interval \([-2, 5]\)?
- Calculate function values at the endpoints:
- \( f(-2) = 3(-2)^2 - 2(-2) - \frac{25}{3} \),
- \( f(5) = 3(5)^2 - 2(5) - \frac{25}{3} \).
- Compare these values to find the minimum value on the interval.
### Additional Notes
- The problems involve calculations of derivatives and understanding the behavior of polynomial functions in various contexts.
- Students will need to apply skills in differentiation, inequality solving, and evaluating functions at specific points to tackle these problems.
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