Explanation Given: (a) The parametric equation, x = t , y = 2 t + 1 (b) The parametric equation, x = cos θ , y = 2 cos θ + 1 (c) The parametric equation, x = e − t , y = 2 e − t + 1 (d) The parametric equation, x = e t , y = 2 e t + 1 (a) Consider the parametric function, x = t …...…... (1) and y = 2 t + 1 …...…... (2) Substitute the value of x = t in equation (2), y = 2 ( x ) + 1 = 2 x + 1 The above equation is equation of straight line, y = 2 x + 1 . Assume different values of t and find the different values for x and y by the parametric equation x = t and y = 2 t + 1 , t − 2 − 1 0 1 2 x = t − 2 − 1 0 1 2 y = 2 t + 1 − 3 − 1 1 3 5 Plot all the points and join. The graph obtained is, The arrow on the curve indicate the orientation of the curve. From the above graph, the orientation of the curve is up for the interval ( − ∞ , ∞ ) . That means the curve has a definite orientation for all real number. From the above graph, the curve of the parametric function is straight line and that curve is smooth for all real number. (b) Consider the parametric function, x = cos θ …...…... (3) Since, the rage of the cosine function is, [ − 1 , 1 ] . So, the value of x varies between [ − 1 , 1 ] . y = 2 cos θ + 1 …...…... (4) Substitute the value of cos θ = x in equation (4), y = 2 ( x ) + 1 = 2 x + 1 The above equation is equation of straight line, y = 2 x + 1 . This function is defined for [ − 1 , 1 ] because the value of x varies between [ − 1 , 1 ] . Assume different values of t and find the different values for x and y by the parametric equation x = cos θ and y = 2 cos θ + 1 , t − π − π 2 0 π 2 π x = cos θ − 1 0 1 0 − 1 y = 2 cos θ + 1 − 1 1 3 1 − 1 The graph is plotted with the help of table is, The arrow on the curve indicate the orientation of the curve. From the above graph, the orientation of the curve is up for certain interval and down for certain interval. That means the orientation of the curve is oscillate. Therefore, from the above graph, the orientation of the curve changes at x = − 1 and x = 1 or x = ± 1 . That means, x = ± 1 cos θ = ± 1 θ = n π Where, n belongs to integer. Differentiate the equation (3) with respect to θ , d x d θ = d cos θ d θ = − sin θ At θ = n π , d x d θ = − sin n π = 0 Differentiate the equation (4) with respect to θ , d x d θ = d d θ ( 2 cos θ + 1 ) = − 2 sin θ At θ = n π , d x d θ = − 2 sin n π = 0 From the above graph, the curve of the parametric function is straight line and that curve is not smooth because of d x d θ = 0 and d y d θ = 0 at θ = n π

11th Edition

ISBN: 9781337275361

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 10.2, Problem 37E

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Any difference between the curves of the given parametric equations. Also, check is the graphs, orientations and the smoothness of the curves are same or not.

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Chapter 10.2, Problem 36E

Chapter 10.2, Problem 38E

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8:38 *** TEMU TEMU -3 -2 7 B 2 1 & 5G. 61% 1 2 -1 Based on the graph above, determine the amplitude, period, midline, and equation of the function. Use f(x) as the output. Amplitude: 2 Period: 2 Midline: 2 ☑ syntax error: this is not an equation. Function: f(x) = −2 cos(πx + 2.5π) +2× Question Help: Worked Example 1 ☑ Message instructor Submit Question ||| <

8:39 *** TEMU 5G 60% A ferris wheel is 28 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. What is the amplitude? 14 meters What is the equation of the Midline? y = 16 What is the period? 4 meters minutes The equation that models the height of the ferris wheel after t minutes is: f(t): = ƒ (3) = ·−14(0) + 16 syntax error: you gave an equation, not an expression. syntax error. Check your variables - you might be using an incorrect one. How high are you off of the ground after 3 minutes? Round your answe the nearest meter. ||| <

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Chapter 10 Solutions

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Any difference between the curves of the given parametric equations. Also, check is the graphs, orientations and the smoothness of the curves are same or not.

Solution Summary: The author explains that the parametric function's curves are smooth in parts (a), (b), and (d).

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Any difference between the curves of the given parametric equations. Also, check is the graphs, orientations and the smoothness of the curves are same or not.

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