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6. 4 9- 2 4 Let x(t) = t3 – 4t2 + 1 and y(t) = t³ – 5t2 + 2t + 8 At t = = 2, æ(2) y(2) dx \t=2 dt dy = =2 dt dy tangent slope (give your answer as a whole number or a reduced fraction) %3D t=2 = dx speed(2) (give your answer rounded to 2 decimal places) 2.

6. 4 9- 2 4 Let x(t) = t3 – 4t2 + 1 and y(t) = t³ – 5t2 + 2t + 8 At t = = 2, æ(2) y(2) dx \t=2 dt dy = =2 dt dy tangent slope (give your answer as a whole number or a reduced fraction) %3D t=2 = dx speed(2) (give your answer rounded to 2 decimal places) 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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The image depicts a parametric curve along with equations and questions for analysis at a specific value of the parameter \( t \). The graph shows a looped path plotted in a coordinate system, showcasing the movement of a point determined by the given equations for \( x(t) \) and \( y(t) \). ### Equations Provided: - \( x(t) = t^3 - 4t^2 + 1 \) - \( y(t) = t^3 - 5t^2 + 2t + 8 \) ### Tasks To Perform at \( t = 2 \): 1. Calculate \( x(2) \). 2. Calculate \( y(2) \). 3. Determine \( \frac{dx}{dt} \big|_{t=2} \). 4. Determine \( \frac{dy}{dt} \big|_{t=2} \). 5. Compute the tangent slope \( \frac{dy}{dx} \big|_{t=2} \) and express it as a whole number or a reduced fraction. 6. Calculate the speed at \( t = 2 \) and present it rounded to two decimal places. ### Diagram Explanation: The curve is plotted in the x-y coordinate system. It represents the trajectory formed by the parametric equations, showing a distinct loop shape that reflects the changing values of \( x(t) \) and \( y(t) \). It highlights where calculations need to be made, especially at \( t = 2 \), forming a basis for understanding changes in position and speed. This visual illustration aids in relating the mathematical aspects to geometric representation.
Transcribed Image Text:The image depicts a parametric curve along with equations and questions for analysis at a specific value of the parameter \( t \). The graph shows a looped path plotted in a coordinate system, showcasing the movement of a point determined by the given equations for \( x(t) \) and \( y(t) \). ### Equations Provided: - \( x(t) = t^3 - 4t^2 + 1 \) - \( y(t) = t^3 - 5t^2 + 2t + 8 \) ### Tasks To Perform at \( t = 2 \): 1. Calculate \( x(2) \). 2. Calculate \( y(2) \). 3. Determine \( \frac{dx}{dt} \big|_{t=2} \). 4. Determine \( \frac{dy}{dt} \big|_{t=2} \). 5. Compute the tangent slope \( \frac{dy}{dx} \big|_{t=2} \) and express it as a whole number or a reduced fraction. 6. Calculate the speed at \( t = 2 \) and present it rounded to two decimal places. ### Diagram Explanation: The curve is plotted in the x-y coordinate system. It represents the trajectory formed by the parametric equations, showing a distinct loop shape that reflects the changing values of \( x(t) \) and \( y(t) \). It highlights where calculations need to be made, especially at \( t = 2 \), forming a basis for understanding changes in position and speed. This visual illustration aids in relating the mathematical aspects to geometric representation.
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