59-62 - Graphs of Parametric Equations Sketch the curvegiven by the parametric equations.59. x = t cos t, y= t sin t, t2060. x = sin t, y = sin 2r3r3t61. x =1+1 +162. x = cot t, y = 2 sint, 0<1

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Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter8: Polar Coordinates And Parametric Equations

Section8.4: Plane Curves And Parametric Equations

Problem 61E

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Epicycloid If the circle C of Exercise 63 rolls on the outside of the larger circle, the curve traced out by P is called an epicycloid. Find parametric equations for the epicycloid. Hypocycloid A circle C of radius b rolls on the inside of a larger circle of radius a centered at the origin. Let P be a fixed point on the smaller circle, with the initial position at the point (a,0) as shown in the figure. The curve traced out by P is called a hypocycloid. a Show that parametric equations of hypocycloid are x=(ab)cos+bcos(abb) y=(ab)sinbsin(abb) b If a=4b, the hypocycloid is called an asteroid. Show that in this case parametric equations can be reduced to x=acos3y=asin3 Sketch the curve. Eliminate the parameter to obtain an equation for the asteroid in rectangular coordinates.
1. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x= e−2t cos(2t), y = e−2t sin(2t), z = e−2t; (1, 0, 1) 2. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = e−2t cos(2t), y = e−2t sin(2t), z = e−2t; (1, 0, 1) 3. Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t. (Enter your answer in terms of s.) r(t) = 4t i + (5 − 2t) j + (1 + 3t) k
Exercise1 :1) Sketch the curve defined by the parametric equations:x = 2t2 + t ; y = t − 2; 2) Find the caratesian equation for the curvex = 2t2 + t ; y = t − 2; Exercise2 :What curve is represented by the following parametric equations ?x = sin(t). ; y = cos(2t) ; 0 ≤ t ≤ 2π you can use cos(2t) = 1 − sin2(t) Exercise3 : A curve C is defined by the parametric equtions:a) Find the equation of tangent line at the point (0; 0 of this curve. b) At what points the curve has a horizontal tangent ? c) At what points the curve has a vertical tangent ? d) Determine where the curve is concave upward or downward ? Exercise4 :a) Compute dy/dx when:x = θ + cosθ ; y = θ − sinθ b) Compute d^2y/dx^2 , when:x = θ + cosθ ; y = θ − sinθ c) Find the tangent at the point θ = π/3 of:x = θ + cosθ ; y = θ − sinθ d) At what points the curve has a horizontal tangent ? when is it vertical ? Exercise5 :1) Find the area of the cycloid x = rcosθ ; y =…
1) Find a set of parametric equation to represent the graph of y= 1-x^2 using each parametera) t =xb) t =1-x2) Find a set of parametric equation to represent the graph of y=x^2 + 2parametera) t =xb) t=2-x3) Find a set of parametric equation to represent the graph of y=x^2 + 3 parametera) t=x^2 -2b) t =4x
1. Use the parametric equations x = sin(t) and y=sin^3(t) to answer parts a and b.a. Eliminate the parameter to obtain an equation (rectangular) in x and y. Graph the curve and describe the motion of a point on the curve as t increases.b. Determine the values t = t\:_{0} where the curve is not smooth.
7. a. Find the parametric equations for the surface generated byrevolving the curve y = sin x about the x-axis. b. Using the parametric equations from part a. set up but do NOTevaluate an integral that will give the surface area of that portion ofthe surface for which 0 ≤ x ≤ π. c. . Find the equation of the tangent plane to the parametric surfacein part a. at the point (x, y, z) = (pi/6, 1/2, 0)
(a) Find the exact area of the surface obtained by rotating the curve y = e^x about thex-axis over the interval 0 ≤ x ≤ 1.(b) Determine the length of the parametric curve given by the following set ofparametric equations.x = 3 cos t − cos 3t, y = 3 sin t − sin 3t, 0 ≤ t ≤ πYou may assume that the curve traces out exactly once for the given range of t.
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A simplified model of the Earth-Mars system assumes that the orbits of Earth and Mars are circular with radii of 2 and 3 respectively, and that Earth completes a complete orbit in one year while Mars takes two years. The position of Mars as seen from Earth is given by the parametric equations x= (3−4 cosπt) cosπt+2,y= (3−4 cosπt) sinπt. a. Graph the parametric equations for 0≤t≤2. b. Letting r= 3−4 cosπt, explain why the path of Mars as seen from Earth is a limacon

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59-62 - Graphs of Parametric Equations Sketch the curve given by the parametric equations. 59. x = t cos t, y= t sin t, t20 60. x = sin t, y = sin 2r 3r 3t 61. x = 1+ 1 +1 62. x = cot t, y = 2 sint, 0<1