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O The graph of f(x)=2r²+2r-12 is shown alongside. a Find f'(x). b Hence find the point where the tangent has gradient -2. onh and include the information in b. f(x)=2r²+2r ◄ 2

O The graph of f(x)=2r²+2r-12 is shown alongside. a Find f'(x). b Hence find the point where the tangent has gradient -2. onh and include the information in b. f(x)=2r²+2r ◄ 2

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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## Transcription for Educational Website ### Exercise 5. The graph of \( f(x) = 2x^2 + 2x - 12 \) is shown alongside: a. Find \( f'(x) \). b. Hence find the point where the tangent has gradient \(-2\). c. Copy the graph, and include the information in b. ### Graph Explanation The graph provided is a parabola representing the quadratic function \( f(x) = 2x^2 + 2x - 12 \). - The graph is plotted on a Cartesian coordinate system with the x-axis and y-axis labeled. - The parabola opens upwards, indicating a positive leading coefficient (\(2\)). - The vertex of the parabola is located at the bottom point. - The axis of symmetry is shown as a vertical line passing through the vertex. - Some points on the graph have been plotted, including the intercepts: - The x-intercepts are labeled as approximately \(-3\) and \(2\). - The y-intercept is labeled as \(-12\). ### Tasks - **Finding \( f'(x) \):** This requires differentiating the function \( f(x) = 2x^2 + 2x - 12 \). - **Finding the Tangent:** Identify the point on the graph where the derivative (gradient of the tangent) equals \(-2\). - **Graph Copying:** Replicate the graph, ensuring to mark the point identified in part b.
Transcribed Image Text:## Transcription for Educational Website ### Exercise 5. The graph of \( f(x) = 2x^2 + 2x - 12 \) is shown alongside: a. Find \( f'(x) \). b. Hence find the point where the tangent has gradient \(-2\). c. Copy the graph, and include the information in b. ### Graph Explanation The graph provided is a parabola representing the quadratic function \( f(x) = 2x^2 + 2x - 12 \). - The graph is plotted on a Cartesian coordinate system with the x-axis and y-axis labeled. - The parabola opens upwards, indicating a positive leading coefficient (\(2\)). - The vertex of the parabola is located at the bottom point. - The axis of symmetry is shown as a vertical line passing through the vertex. - Some points on the graph have been plotted, including the intercepts: - The x-intercepts are labeled as approximately \(-3\) and \(2\). - The y-intercept is labeled as \(-12\). ### Tasks - **Finding \( f'(x) \):** This requires differentiating the function \( f(x) = 2x^2 + 2x - 12 \). - **Finding the Tangent:** Identify the point on the graph where the derivative (gradient of the tangent) equals \(-2\). - **Graph Copying:** Replicate the graph, ensuring to mark the point identified in part b.
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