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#11 can you show me how to do this?

#11 can you show me how to do this?

Calculus: Early Transcendentals
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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#11 can you show me how to do this?
**Graph Explanation:** Two graphs are shown, each with a curve on the coordinate plane. They illustrate the concept of the slope of a secant line on a function \( f(x) \). 1. **Graph Details:** - Both graphs feature a blue curve that corresponds to a function \( f(x) \). - The \( x \)-axis ranges from 0 to 8, and the \( y \)-axis ranges from -1 to 3. 2. **Key Points:** - At \( x = 4 \), the point on the curve is labeled \( f(4) \). - At \( x = 4 + h \), another point on the curve is labeled \( f(4 + h) \). - The vertical lines from these points on the function down to the \( x \)-axis create a right triangle. 3. **Slope Calculation:** - The change in \( y \) (vertical side of triangle) is represented as \( f(4 + h) - f(4) \). - The change in \( x \) (horizontal side) is shown as \( h \). - The slope of the secant line connecting these two points is given as \( \frac{f(4 + h) - f(4)}{h} \). **Question:** What line has slope \(\frac{f(4 + h) - f(4)}{h}\)? - The options provided are: - ○ the line from \((f(4), 4)\) to \((f(4) + h, 4 + h)\) - ○ the line from \((f(4), 4)\) to \((f(4 + h), 4 + h)\) - ○ the line from \((4, f(4))\) to \((4 + h, f(4) + h)\) - ○ the line from \((4, f(4))\) to \((4 + h, f(4 + h))\) **Additional Materials:** - eBook The correct choice for the slope representation is the line from \((4, f(4))\) to \((4 + h, f(4 + h))\), which directly corresponds to the definition of a secant line slope using the h-notation.
Transcribed Image Text:**Graph Explanation:** Two graphs are shown, each with a curve on the coordinate plane. They illustrate the concept of the slope of a secant line on a function \( f(x) \). 1. **Graph Details:** - Both graphs feature a blue curve that corresponds to a function \( f(x) \). - The \( x \)-axis ranges from 0 to 8, and the \( y \)-axis ranges from -1 to 3. 2. **Key Points:** - At \( x = 4 \), the point on the curve is labeled \( f(4) \). - At \( x = 4 + h \), another point on the curve is labeled \( f(4 + h) \). - The vertical lines from these points on the function down to the \( x \)-axis create a right triangle. 3. **Slope Calculation:** - The change in \( y \) (vertical side of triangle) is represented as \( f(4 + h) - f(4) \). - The change in \( x \) (horizontal side) is shown as \( h \). - The slope of the secant line connecting these two points is given as \( \frac{f(4 + h) - f(4)}{h} \). **Question:** What line has slope \(\frac{f(4 + h) - f(4)}{h}\)? - The options provided are: - ○ the line from \((f(4), 4)\) to \((f(4) + h, 4 + h)\) - ○ the line from \((f(4), 4)\) to \((f(4 + h), 4 + h)\) - ○ the line from \((4, f(4))\) to \((4 + h, f(4) + h)\) - ○ the line from \((4, f(4))\) to \((4 + h, f(4 + h))\) **Additional Materials:** - eBook The correct choice for the slope representation is the line from \((4, f(4))\) to \((4 + h, f(4 + h))\), which directly corresponds to the definition of a secant line slope using the h-notation.
The image contains the graphs of a function \( f(x) \) displayed across three separate plots, designed to illustrate the concept of increments and differences at a given point on the curve. ### Top Graph: - **Graph Description**: This plot depicts the function \( y = f(x) \) as a blue curve on a standard Cartesian plane. The x-axis ranges from \(-1\) to \(6\), and the y-axis ranges from \(-1\) to \(3\). - **Purpose**: It serves to provide a general view of the function \( y = f(x) \) over the specified domain and range. ### Bottom Left Graph: - **Graph Description**: Focuses on the behavior of the function around \( x = 4 \). The graph highlights the point \( f(4) \) and a nearby point \( f(4+h) \), with \( h > 0 \). - **Markings**: - \( f(4) \): The vertical distance from the x-axis to the curve at \( x = 4 \). - \( f(4+h) \): The vertical distance from the x-axis to the curve at the point \( x = 4 + h \). - \( h \): The horizontal segment on the x-axis between \( x = 4 \) and \( x = 4 + h \). - \( f(4+h) - f(4) \): The vertical segment between the two points on the y-axis, representing the change in \( y \). ### Bottom Right Graph: - **Graph Description**: Similar to the left graph but with a more detailed depiction of the change in \( y \) values and increment \( h \). - **Markings**: - Same elements as in the left graph, labeled clearly for a focused view on the differences and increments calculated between \( x = 4 \) and \( x = 4+h \). These diagrams are instrumental in understanding the concept of the derivative and how changes in the input \( x \) affect the output \( y \) in small increments, reinforcing the foundation for calculus topics such as limits and slopes of tangent lines.
Transcribed Image Text:The image contains the graphs of a function \( f(x) \) displayed across three separate plots, designed to illustrate the concept of increments and differences at a given point on the curve. ### Top Graph: - **Graph Description**: This plot depicts the function \( y = f(x) \) as a blue curve on a standard Cartesian plane. The x-axis ranges from \(-1\) to \(6\), and the y-axis ranges from \(-1\) to \(3\). - **Purpose**: It serves to provide a general view of the function \( y = f(x) \) over the specified domain and range. ### Bottom Left Graph: - **Graph Description**: Focuses on the behavior of the function around \( x = 4 \). The graph highlights the point \( f(4) \) and a nearby point \( f(4+h) \), with \( h > 0 \). - **Markings**: - \( f(4) \): The vertical distance from the x-axis to the curve at \( x = 4 \). - \( f(4+h) \): The vertical distance from the x-axis to the curve at the point \( x = 4 + h \). - \( h \): The horizontal segment on the x-axis between \( x = 4 \) and \( x = 4 + h \). - \( f(4+h) - f(4) \): The vertical segment between the two points on the y-axis, representing the change in \( y \). ### Bottom Right Graph: - **Graph Description**: Similar to the left graph but with a more detailed depiction of the change in \( y \) values and increment \( h \). - **Markings**: - Same elements as in the left graph, labeled clearly for a focused view on the differences and increments calculated between \( x = 4 \) and \( x = 4+h \). These diagrams are instrumental in understanding the concept of the derivative and how changes in the input \( x \) affect the output \( y \) in small increments, reinforcing the foundation for calculus topics such as limits and slopes of tangent lines.
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