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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![**Problem Statement:**
Find the area of the shaded region.
**Diagram Description:**
The image contains a graph of a line with the equation \( y = 2x + 1 \). The line passes through two points marked on the graph: \( (1, 3) \) and \( (3, 7) \). The shaded region forms a right triangle with the x-axis, bounded by vertical lines \( x = 1 \) and \( x = 3 \).
- The base of the triangle lies on the x-axis from \( x = 1 \) to \( x = 3 \).
- The height of the triangle is along the y-axis from the y-values of the line at these x-coordinates.
**Multiple Choice Options:**
- 5
- 7.5
- 12.5
- 10
**Calculation of Area:**
1. **Base of the Triangle:** The length along the x-axis from 1 to 3, which is \(3 - 1 = 2\).
2. **Height of the Triangle:** Difference in y-values from \( (1, 3) \) to \( (3, 7) \), which is \(7 - 3 = 4\).
3. **Area Formula for a Triangle:** \(\frac{1}{2} \times \text{base} \times \text{height}\).
Calculate the area:
\[
\frac{1}{2} \times 2 \times 4 = 4
\]
However, the minimum y-value at \( x = 1 \) was disregarded (point starts at \( y = 3 \) not zero), implying recalculation or graphic mismatch verification.
If distinct y-axis consideration or graph mismatch calibration provides:
\[
Area = \frac{1}{2} \times 2 \times 6 \quad (\text{using vertical span 3 at \(x=3\) to 1 at \(x=1\)})
\]
Please consult on graphical verification for calculated area calibration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34715e37-b64b-42a1-b0ce-fdbcb06d67b0%2F7882e791-ad4f-4eeb-a8ee-d8fa777b5c9b%2Fxdn8po_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the area of the shaded region.
**Diagram Description:**
The image contains a graph of a line with the equation \( y = 2x + 1 \). The line passes through two points marked on the graph: \( (1, 3) \) and \( (3, 7) \). The shaded region forms a right triangle with the x-axis, bounded by vertical lines \( x = 1 \) and \( x = 3 \).
- The base of the triangle lies on the x-axis from \( x = 1 \) to \( x = 3 \).
- The height of the triangle is along the y-axis from the y-values of the line at these x-coordinates.
**Multiple Choice Options:**
- 5
- 7.5
- 12.5
- 10
**Calculation of Area:**
1. **Base of the Triangle:** The length along the x-axis from 1 to 3, which is \(3 - 1 = 2\).
2. **Height of the Triangle:** Difference in y-values from \( (1, 3) \) to \( (3, 7) \), which is \(7 - 3 = 4\).
3. **Area Formula for a Triangle:** \(\frac{1}{2} \times \text{base} \times \text{height}\).
Calculate the area:
\[
\frac{1}{2} \times 2 \times 4 = 4
\]
However, the minimum y-value at \( x = 1 \) was disregarded (point starts at \( y = 3 \) not zero), implying recalculation or graphic mismatch verification.
If distinct y-axis consideration or graph mismatch calibration provides:
\[
Area = \frac{1}{2} \times 2 \times 6 \quad (\text{using vertical span 3 at \(x=3\) to 1 at \(x=1\)})
\]
Please consult on graphical verification for calculated area calibration.
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