A particle moves in the xy-plane in such a way that its position at time t is r(t) = (t – sin t) i + (1 – cos t) j. a. Graph r(t). The resulting curve is a cycloid. b. Find the maximum and minimum values of |v| and Jal. (Hint: Find the extreme values of |v and Jal first and take square roots later.)

Q: 4. Find the moving trihedral of the curve f(t) at the indicated value: †(t) = cos? t î + sin² t ĵ…

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Q: 2. Let y be the curve in R° given by t (t, 2 cos t, 2 sin t) for 0 < t< 47. Compute y? ds

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Q: ne if the curve (In t. t In t. 1) is tangent to the surface rz- uz? + cos Tu =1 at (0.0. 1).

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Q: Determine the open t-intervals on which the curve is concave downward or concave upward.  x = 4 cos…

A: given,             x = 4 cos t, y = 2 sin t, 0 &lt; t &lt; 2∏

Q: 1. Consider the curve C: r(t) = (t - sin t)i + (1– cos t)j, 0StS 2. %3| | (a) Find an equation of…

A: Since solve the first question for you. If you wanting any specific question to be solved then…

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Q: Find f F. dr for F = 2yi – (sin y)j on the curve counterclockwise around the unit circle C starting…

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Q: (b) (i) Parametrise the graph of the C function y = f(x), for x = [a, b], as the track of a regular…

A: b) (i) Hint: A mathematical process known as parameterization involves representing the state of a…

Q: Consider the curve in the xy-plane repre- sented by = e' and y = The slope of the line tangent to…

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Q: Find the length of the curve. r(t) = cos(6t) i + sin(6t)j + 6In(cos(t)) k, 0 <t< n/4

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Q: The graph of f(t) = tan (2t + 4TT) intersects the t axis on the interval T0, TT] at points %3D

A: Consider the given: ft=tan2t+4πAt t-axis, ft=0,  tan2t+4π=0           2t+4π=0+πn…

Q: 13. Find the value of t in the given interval at which the given curve is not smooth. r(t)=(5cost-…

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Q: The slope of the tangent to the curve r = 3 sin 40 at 0 = 0 is %3D Şelect one: O a. -1 3D1 O b. 0 C.…

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Q: Q7) Set up the integral representing the length of the curve (sint, cos² t), cos"e). - – s1s. <t< 2

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Q: Determine the length of the curve 7(t) = (3t, sin t, cos t) on the interval 0 < t < 2n. %3D А. L =…

A: As we know that, The expression for the length of the vector-valued curve is: L=∫abx't2+y't2+z't2dt…

Q: Find the length L of the curve R(t) = e" cos(2t) i + e* sin(2t) j+2e"k over the interval [1, 4]. L =

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Q: Q.18 A curve 'C'is defined as x= a cos³0, y = a sin°0 in [O, t/2]. What will be the point P on curve…

A: Explanation of the answer is as follows

Q: The slope of the tangent line to the graph of z- y? = In(zy) at the point (1,1) is Select one: O a.…

A: Given the equation of the curve x2 -y2 =ln(xy) We have to find the slope at (1,1) So, the slope is…

Q: Find a path c(t) that traces the parabolic arc y = x2 from (0, 0) to (3, 9) for 0 ≤ t ≤ 1.

A: Given parabolic arc y=x2 from 0,0 to 3,9 for 0≤t≤1. The path ct that traces the parabolic arc has to…

Q: If f'(x) = /cosec*x-1 then what is the arclength of the curve y= f (x) on со ? 3 A) 1 В) 3 C) 2 D)…

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Q: Find the unit tangent of the following curve at the specified point. r(t) = 2ti + 1 · sin(2t)j+3 ·…

A: We have to find the unit tangent vector at t=pi/3

Q: 3 For which values of a € R the tangent plane to the graph of f(x, y) = 1 + sin(ax + y²) at the…

A: The given graph is                                f(x,y)=1+sin(αx+y2)

Q: 7. Consider the function f: R → R³ given by: t+1 f(t) = sin(cos(t)), tet) t2 +4° Find f'(t) for all…

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Q: а. z = t', y = ; 0 < t < 1 b. x = cos (2t), y = sin°(2t); 0 <t< 4

A: In the given question we have to find the exact length of the given curves.

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Q: Sketch the curve. And indicate the direction of increasing T at T =0

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Q: 2. Find the length of the curve r(t) = e'i + e' sin 2tj + e' cos 2tk, 0st< 2n.

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Q: Let C be the curve parameterized by Y: [0, b] R³, y(s) = (12 cos(s²), 12 sin(s), 9 s²). Find b>0…

A: The given problem is to find the value of b for which the given length of curve of the given…

Q: Find the length of the curve. r(t) = cos(4t) i + sin(4t) j + 4ln(cos(t)) k,  0 ≤ t ≤ ?/4

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Q: Consider a χ2-curve with df = 16. Obtain the χ2-value that has area a. 0.025 to its left. b. 0.975…

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Q: Consider the curve where a =0 (a) Compute(t) at t = [0, +∞). (b) Find the tangent line at the point…

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Q: 3 4. Find the arc length of the graph of the function f(x) =;(x² + 1)ī over the interval [0, 1]. =…

A: y =f(x) =23(x2+1)32f(x)=23(x2+1)32f'(x)=23*32(x2+1)12(2x)f'(x)=(2x)(x2+1)12 The Formula for arc…

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Q: Find the length of the curve. r(t) = cos(6t) i + sin(6t)j + 6In(cos(t)) k, 0stsa/4

A: In this question we have to find the length of the curve.

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Q: Find the lengthof the curve r= a sin² (0/2), 0 s0s T, a>0

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Q: Q2: Gruph the nector fundion Tt)= Find the equat in for the tanget f functron fexy) = SinCxy) at…

A: a) Given,  rt=cost, sint, t Here  xt=cost,yt=sint,zt=t

Q: Find T, Ñ and B for the curve 7(t) = (2 cos(4t), 2 sin(4t), 5t) at the point t = 0 Give your answers…

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Q: Find T, Ñ and B for the curve 7 (t) = (3 cos(3t), 3 sin(3t), 5t) at the point t = 0 Give your…

A: Space Curves and T, N, B vectors The unit tangent vector T, the unit normal vector N and the unit…

Q: Find T, N and B for the curve F(t) = (2 cos(5t), 2 sin(5t), 3t) at the point t -0 Give your answers…

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Q: Find the length of the curve 7 (t) = (4 cos(t), 4 sin(t), 3t) for – 1 <t< 4

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Q: Find the length of the curve. r(t) = cos(5t) i+ sin(5t)j + 5ln(cos(t)) k, 0<t< n/4

A: r(t)=cos(5t)i+sin(5t)j+5ln(cos(t))kWhere, 0≤t≤π/4

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A: Given,  ct = cost, sint  and t ≥ 0 We have to find the interval of t-values such that ct = cost,…

Q: Find the length L of the curve R(t) = 4 cos(5t) i – 4sin(5t) j + tk over the interval [3, 6]. L =…

A: Given:

Q: et +e- find the length of the curve h(x) = from r = 0 to z = In 2.

A: The standard formula for arclength will be used in solving this problem. The typeset solution is…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.