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Given the quadratic function in blue and the line tangent to the curve at A in red, move point A and investigate what happens to the gradient of the tangent line. O O Conic c: y = x²-x-2 Line a: -0.5x + 0.5y = Number b = 1 Point A = (1, -2) .♡ -2 6 5 4 3 2 1 0 -1 -2 -3 a A 1 b=1 3 4 5

Given the quadratic function in blue and the line tangent to the curve at A in red, move point A and investigate what happens to the gradient of the tangent line. O O Conic c: y = x²-x-2 Line a: -0.5x + 0.5y = Number b = 1 Point A = (1, -2) .♡ -2 6 5 4 3 2 1 0 -1 -2 -3 a A 1 b=1 3 4 5

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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**Quadratic Function Exploration** **Objective:** Explore the relationship between a quadratic function and its tangent line by moving point A and observing changes in the gradient of the tangent line. **Description:** 1. **Quadratic Function**: - The graph is shown in blue. - Equation: \( y = x^2 - x - 2 \). 2. **Tangent Line**: - The line tangent to the curve at a specific point A is shown in red. - Equation of the line is represented as \( -0.5x + 0.5y = \). 3. **Controls**: - **Conic**: Toggles the visibility of the quadratic curve. - **Line**: Toggles the visibility of the tangent line. - **Number**: Toggles the display of the constant \( b = 1 \). - **Point**: Toggles the visibility of point A on the graph. 4. **Point A**: - Represents the location where the tangent touches the curve. - Initially located at \( A = (1, -2) \). **Graph Explanation**: - The blue curve represents the quadratic function. - The red line represents the tangent to the curve at point A. - As you move point A along the curve, notice how the slope (gradient) of the red tangent line changes. - The constant \( b = 1 \) is shown as a reference on the tangent. **Instructions**: - Move point A (1, -2) along the quadratic curve. - Observe how both the equation of the tangent line and its slope change. - Analyze the relationship between the position of point A and the gradient of the tangent. By studying this dynamic interplay, learners can develop a deeper understanding of calculus concepts such as differentiation and the geometric interpretation of derivatives.
Transcribed Image Text:**Quadratic Function Exploration** **Objective:** Explore the relationship between a quadratic function and its tangent line by moving point A and observing changes in the gradient of the tangent line. **Description:** 1. **Quadratic Function**: - The graph is shown in blue. - Equation: \( y = x^2 - x - 2 \). 2. **Tangent Line**: - The line tangent to the curve at a specific point A is shown in red. - Equation of the line is represented as \( -0.5x + 0.5y = \). 3. **Controls**: - **Conic**: Toggles the visibility of the quadratic curve. - **Line**: Toggles the visibility of the tangent line. - **Number**: Toggles the display of the constant \( b = 1 \). - **Point**: Toggles the visibility of point A on the graph. 4. **Point A**: - Represents the location where the tangent touches the curve. - Initially located at \( A = (1, -2) \). **Graph Explanation**: - The blue curve represents the quadratic function. - The red line represents the tangent to the curve at point A. - As you move point A along the curve, notice how the slope (gradient) of the red tangent line changes. - The constant \( b = 1 \) is shown as a reference on the tangent. **Instructions**: - Move point A (1, -2) along the quadratic curve. - Observe how both the equation of the tangent line and its slope change. - Analyze the relationship between the position of point A and the gradient of the tangent. By studying this dynamic interplay, learners can develop a deeper understanding of calculus concepts such as differentiation and the geometric interpretation of derivatives.
**Experiment 3**: Slide the blue circle on the parabola to the point where the tangent line is exactly horizontal. **3A.** Does the slope of the tangent line appear to be positive, negative, zero, or undefined? **3B.** State the slope of the tangent line as an integer. **3C.** The equation of the parabola is \( f(x) = x^2 - x - 2 \). Calculate the derivative \( f'(x) \). Evaluate \( f'(1) \) and state its value. **3D.** Compare the slope rise/run to \( f'(1) \). How did they compare? Were they the same or different?
Transcribed Image Text:**Experiment 3**: Slide the blue circle on the parabola to the point where the tangent line is exactly horizontal. **3A.** Does the slope of the tangent line appear to be positive, negative, zero, or undefined? **3B.** State the slope of the tangent line as an integer. **3C.** The equation of the parabola is \( f(x) = x^2 - x - 2 \). Calculate the derivative \( f'(x) \). Evaluate \( f'(1) \) and state its value. **3D.** Compare the slope rise/run to \( f'(1) \). How did they compare? Were they the same or different?
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