Consider a particle moving rectilinearly according to x = t3 - 3t2 -45t m, where t is in sec. Draw the x-t, v-t and a-t curves for the interval 0 sts 8 sec. Determine the distance travelled during the first 8 seconds.

Q: Find the orthogonal trajectories for the family of curves, sin? y C In x.

A:

Q: Compute F. dr for the oriented curve specified. Jc F(x, y) = (6, y), quarter circle x2 + y? = 1 with…

A:

Q: Find equations for the horizontal and vertical A)y= at (1, 똥) B) z = 0 at (0, ) D) y = 2 at (2, *)…

A:

Q: Find an equation of the tangent to the curve given by = evi y = t – In(?) at the point corresponding…

A:

Q: Compute F• dr for the oriented curve specified. F(x, y) = (9, y), quarter circle x² + y2 = 1 with x…

A: Given:

Q: Let m be a positive real number and C be the curve given by: r(t) = m(-cost + 3)i – m(sint + 3)j+…

A: If r(t) = (f(t) , g(t) , h(t))then L = ∫ab r→'(t)  dtwhere  r→'(t)  = f'(t)2 + g'(t)2 + h'(t)2

Q: Calculate the arc length of the curve f(x)=(2/3)(x²+1)ª2) in the interval [–1,1]

A:

Q: Find the length of the curve over the given interval. r = 4 sin θ, [0, π]

A:

Q: Which of the following functions parametrizes the portion of the curve x=y³-4 that starts at (-4,0)…

A: In this question , we can check one by one all option .

Q: Determine whether the statement "The curve given by x = t3 and y = t2 has a horizontal tangent at…

A: For horizontal tangent dy/dx=0

Q: Find an equation for the tangent line to the curve given parametrically by x(t) 2t, y(t) 4t2 +t+ 3 =…

A: we want to find the tangent lines to theparametric equations given byx=f(t)  y=g(t)The equation of…

Q: Compute f ds for the curve specified. f(x, y, z) = z2, r(t) = (2t, 3t, 4t) for 0 <ts 2

A:

Q: . Sketch the curve parametrized by r(t) = (ltl + t, ltl - t)

A: The given curve is rt=t+t, t-t. Sketch the graph of the above curve as shown below.

Q: the detined b is Constant. Curiature a(t) =| 1 Cost, 1 sint, o of the curve d:laa)tR? where Find

A: We are given : α(t)=1rcos t,1rsin t,0 ⇒α'(t)=-1rsin t,1rcos t,0 Now, the Unit tangent Vector is :…

Q: ) Find the orthogonal trajectories of the family of curves r = 3C sin 0, where C being parameter.

A:

Q: Show that the tangent at P(x,y,) the curve Vx+ Vy = Va is yy,² +xx, ² = a²

A:

Q: Find the slope of the tangent line to the parametric curves x=3t+2, y= 2t^2+ 3t-3 at t=11 and then…

A: The given question as belonging to the derivative in which we have to find the derivative or the…

Q: Find a curve through the origin in the xy- plane whose length is given by 1/2 L = [; (1+ *)"* dx 4

A: The arc length, L of a curve y=f(x) in the interval a,b is calculated using the formula below.…

Q: Find 4 and 4y axz dx y=te* dy- -(t-1)et y t is Fov which values the curve con cave upwavd. U BRANDS

A:

Q: Find the equation of the tangent line to the parametric curve at (1,3). 2=1+ Int y =t² +2

A:

Q: Find the tangent line(s) to the parametric curve given by a = t5 – 4t3 y = t? at (0, 4).

A: Find the tangent line(s) to the parametric curve given by x = t^5-4t^3 , y = t^2 at (0,4).

Q: 5. Find the length of the curve given by y= Vt* – 1 dt and 1<x<2.

A:

Q: Find the tangent plane to the equation z = 3e²²-2y at the point (4, 8, 3)

A:

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: Find the moment of inertia, with respect to the line x=1, of the area enclosed by the loop of the…

A:

Q: Compute the arc length of the indicated portion of the curve: 7(1) = (7 +2r²)î + (21² – 4)ĵ + (7 –…

A:

Q: Sketch the motion of a particle with position as: x = 5 sint y = ²/-nstsn - St≤n as tvaries in the…

A: We need to plot the motion of the particle, whose position is defined as: x=5sint, y=t2 in the…

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

A:

Q: Find the length of the curve y = S Vcos t dt, 0 <x<n.

A: Given; The curve y=∫0xcost dt,      0≤x≤π .

Q: Find an equation of the tangent(s) to the curve with paramnetric equations ¤ = 1+ VE, y = e" at the…

A:

Q: * Find the length of the curve.7 (ähäs 3) f(x) = In(sec x) %3D 4 In(3 + 1) O In(3 + 2) O In(2 + 2) O…

A: Hence option (d) is correct.

Q: Find the Length L of the curve R(t)= -2cos(4t)i + 2sin(4t) j + t k over the interval [2,6]. L =

A:

Q: What is the length of the curve I= cost +tsint, y=sint - tcost, (0 <t< 2n) ?

A:

Q: Find the equation of the tangent line to the curve y = et at the point (0, 1). Give your answer in…

A: Given,            The curve is y=exx2+1 and the point is 0,1.To find : Find the equation of the…

Q: Find the arc length of the curve x = Length = -2y³/2 from y = 0 to y = 10.

A: See the details solution in below

Q: Starting from the point (2, –1, 2), reparametrize the curve r(t) = (2 – 2t, -1+ 2t, 2 – 3t) in terms…

A:

Q: Find the arclength of the curve z = 7 cos(3t), y = 7 sin(3t) with 0 < t< 12

A:

Q: x = sin´t, y = cosʻt, 0<t<3™

A: Given x=sin2t, y=cos2t, 0≤t≤3π

Q: Find the length of the curve x = 8 cos t + 8t sint,y= 8 sin t – 8t cos t ; 0 < t %3D -

A: The length of the parametric curve is given by  L=∫abx't2+y't2dt.

Q: Find the derivative 4cos4t sin4t O2cot4t O Other: -1/2 cot4t of the parametrically defined curve (t)…

A:

Q: Find the length of parametrized curve given by  x(t) = 6t2 + 12t, y(t) = -4t3- 12t2 - 9t where t…

A: Consider the parametric curve given by: x(t)=6t2+12t and y(t)=-4t3-12t2-9t for t from 0 to 1. The…

Q: . Find SF dř where F(x,y,z) = (3y?e*, 6ye* -, 1+) and C is the curve F(t) = (sin t, ,2-) for 0 stsn.

A:

Q: he length of the curve y = sin(3X) from x = 0 to I= is given by

A:

Q: Find the exact length of the curve. y = In(sec x), 0s x s a/3

A: This question is based on application of Derivative.

Q: Calculate the arclength of the following curve on the given interval. y%=2x3/2+1O on the interval…

A:

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: Use Green's Theorem to evaluate F. dr, C where F = (Va + 6y, 4x + /ỹ C consists of the arc of the…

A:

Q: Find F. dř for F = 5yi – (sin y)j on the curve counterclockwise around the unit circle C starting at…

A:

Q: Find the arc length of the curve y = t2 - 1 dt for 4 sxs6

A: