11. Compute the length of the curve r(t) = (2t, In(t), t²) over the interval 1

Q: - Compute 5y dr a dy, where C is the arc of the parabola y = from (-2, 4) to (0, 0).

A: The line integral of the function with respect to x, y is solved using the formula…

Q: Find an equation for the line tangent to the curve at the point defined by the given value of t. x-…

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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: 1. Find the length of the curve given by e' (sin t+cos t) , yY = e'(cos t – sin t), x = 1<t< 4 .

A: The length of a parametric curve is given by L=∫abx't2+y't2dt Now, we have…

Q: Compute the length of the curve over the given interval. r(t) =〈2t, ln t,t^2〉, 1 ≤ t ≤ 4

A: it is known that the length of the curve can be calcualted by the integral formula…

Q: 7. Find the length along the parametric curve defined by v(t) = (t, In(cos(t))) from t = 0 andt=.…

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Q: b) Evaluate S(2y + 4z)dx + (2æ + 3z)dy + (4x + 3y)dz, where C is the curve given by r(t) = for 0 <t…

A: given ∫C2y+4zdx+2x+3zdy+4x+3ydz where C is the curve given by rt=t3,2sinπt2,3cosπt2 for 0≤t≤1…

Q: What is the point of intersection of the tangent lines to the curve r(t) = sin(πt)i + 8sin(πt)j +…

A: Let us consider the given curve r(t) the tangent line is obtained as Find the derivative of the…

Q: Find the length of the curve. r(t) = cos(2t) i + sin(2t) j + 2 ln cos t k,     0 ≤ t ≤ π/4

A: Consider the length of the curve is, r(t)=costi+sintj+(lncost)k Then,…

Q: Find the length of each of the following curves: %3D %3D x (ey + e) from y = 0 to y = 1

A: The length of a curve means the arc of the length of a curve. The arc of the length of a curve in…

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

A: From the given problem :   rt=eti+etsin2tj+etcos2tk   So,   x=et, y=etsin2t , z=etcos2t .   As we…

Q: 4) Consider the curve given by: cos(xy) = x(y – 7) – 1 %3D Find y' dx a)

A: Given function is cosxy=xy-π-1 Chain rule of differentiation states that: ddxfgx=f'gx·g'x…

Q: The equation of the tangent line to the curve y = 1 at the point (3,-) is Vx +1 A. x+16y +11=0 B. x+…

A: Please refer the attached solution. None of the option is correct here . I have solved it…

Q: Evaluate dy/dx at t = 1 for the parametrically defined curve with X = t3 and y = 5 - t. O -1.5 O 1.5…

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Q: 10. For the asteroid given by = 4cos (t), y = 4sin' (t) on 0,2n|, find the slope of the tangent at…

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Q: Find T', N and B for the curve r(t) = (2 cos(5t), 2 sin(5t), 4t) at the point t = 0

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Q: Let C be the curve defined by T(t) = (2 + 2 sin(t), 3 – 2 cos(t)) for 0 < t < 4x=²) dx + (2xyev" +…

A: Given the integral is ∫Cy2exy2+4x-2dx+2xyexy2+2dy. Rewrite the integral as ∫CF·→dr→:…

Q: 1. Calculate the derivative dy/dx for the plane curve defined by the equations…

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Q: Suppose that a particle following the given path c(t) flies off on a tangent at t = to. Compute the…

A: Consider the path of the particle           c(t) = (2t2, t3-4t, 0) The velocity of vector is…

Q: compute the length of the curve over the given interval.  r(t) = (cost,sin t,t3/2), 0 ≤ t ≤ 2π

A: Derivative of a function is represented as the change in variable with respect to change in another…

Q: 3 Consider cune detined by y cos (T2) =UY +y+2 2 ) Verify point (2,-2) belongs to this curve, fond…

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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…

A: The derivative of the function is the measure of how fast the function is varying with respect to…

Q: 3. The length of the curve determined by the equations x =t and y=t from t=0 to t= 4 is a. b. 2[VF…

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Q: 1. Make the curve y = ax°+ bx² +cx +d through (0,3) and have at (1, 2) a point of inflection with a…

A: To make the curve.

Q: 9.40. Show that the 0-parameter curves on the surface x = (r cos 0)e¡ + (r sin 0)eg + f(0)eg are…

A: We have to show that the θ-parameter curves on the surface x are parallel.

Q: Q3) Compute for y(x) = cos'(4vT)+ 5x and x(t) = 4 at t=1

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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: The parametric curve x(t)=t−2t6 and y(t)=t7+2t2 is given. In which case is the tangent line at the…

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Q: Consider the parametric curve given by (a) Find dy/dx and d²y/dx² in terms of t. dy/dx d²y/dx² = = x…

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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: 1. Find the length of the curve r(t) = (sin 2t, cos! cos 2t) from t =0 to t 2n. %3D II

A: 1)- Given: r→t=sin 2t, cos 2t from t=0 to t=2π. The objective is to determine the length of the…

Q: (11) Find the tangent line(s) to the parametric curve given by * = t5 - 4t3 y = t2 at (0, 4)

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Q: 1. Find the length of the curve r(t) = (sin 2t, cos 2t) from t 0 to t= 2n. CoS %3D

A: Since you have posted multiple question in this question which are not interlinked so i have solved…

Q: 1. If F(x, y) = (3+2xy)i + (x² – 3y²)j. Evaluate F. dr, where C is the curve given by r(t) = e'…

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Q: 13. Find the point on the curve 7(t) = (2 sin t, 2 cos t, t), 0<ts 7, where the tangent line is…

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Q: computes curve length for 0

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

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Q: Compute 2z dz (2² – i)3 when y is any arc z = z(t) in C with |z(t)| > 2 for every t, with…

A: Let the required arc be ABC, which starts at z=3 and ends at z=2i: The shaded area shows a sector…

Q: Determines the length of the curve in the indicated interval x = 3 sin t+ 3t, y = 3cos t, 0stsm O 12…

A: Given curve is x=3sint+3t and y=3cost, 0≤t≤π. We have to determine the length of the curve on the…

Q: Consider the parametric curve segment (t, t2), t ∈ [0, 1]. What is the firstorder derivative of the…

A: We have the given parametric curve t,t2, t∈0,1x=ty=t2 for t∈0,1the first order derivative the curve…

Q: 2. Let y be the curve in R3 given by t (t, 2 cos t, 2 sin t) for 0 <t < 47. Compute ds

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Q: Consider a curve parameterized by: c(t)=(sin^3(t),cos(t)) Compute dy/dx at the point ((sqrt(2)/4),…

A: Differentiate the given function as follows.

Q: Find the length of the curve. r(t) = cos(2t) i + sin(2t) j + 2 In(cos(t)) k, osts n/4

A: Given:

Q: Find the length of the curve defined by the following function for 0 <t < 2n. 7(t) = (– cos 2t)i +…

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Q: 1. Calculate the derivative dy/dx for the plane curve defined by the equations: x(t)=t²-4t,…

A: Given that: x=t2-4ty=2t3-6t

Q: The position of a particle in the xy-plane at time t is given by r(t) = (cos 2t) i+ (2 sin 2t) j…

A: Given: rt=cos2ti+2sin2tj For finding equation in x and y whose graph is the path of the particle, we…

Q: Evaluate 2.xyz dx + x²zdy +x²ydz over the path r(t) = (t², sin(nt/4), e'“-2') for 0 <t<2.

A: The given integral is: I=∫C 2xyz dx+x2z dy+x2y dzOver the path:  r(t)=t2,  sinπt4,  et2-2t    for…

Q: 7) Let x = t – sin(t) , y = 1– cos(t). Determine the equation of the line tangent at t = n.

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Q: 3. Find S. F · dr where F(x,y, z) = (3y²e*, 6ye* F(t) = (sin t, ,2-) for 0 <tsn. 1+3) and C is the…

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