Find the length of the curve given by c(t) = (t*, t², tº), for t E [0, 1].

Q: Suppose C=D0, so that 23 24y 5x3 = 0. Find all points (x,y) at which the tangent line to the…

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Q: Find the point at which the curve Sx=t-4 ly=t-4t has a horizontal tangent. (x, y) = |

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Q: Find the tangent to r(t) = (t +1, t², e') when t= 3.

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Q: dy Find dx for the curve: x = t, y= 2/t, t>0

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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: Find dy/dx and d^2y/dx^2 X=t-e^t, y= t+e^-t For which values of t. Is the curve concave upward?

A: Given that, x=t-e^t, y= t+e^-t

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: 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: 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: Find the arc length of the curve x = t2 + 1, y = 4t3 + 3 on the given interval    0 ≤ t ≤ 2.

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Q: Plot the curve c(t) = (cos t, esin t ) for 0 ≤ t ≤ 2π

A: The plot of the curve for t within 0 to 2pi is shown in figure below.

Q: Calculate F.dr, where C is the curve given by ti + 123+ tk, with t = [0, 1] and F(x, y, z) = 4xei +…

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Q: Show that at any point (x,y) on the curve y=bex/a , the length of the subtangent is constant and the…

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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: Find the interval on which the curve dt 3 +t + 212 y = is concave upward.

A: Given: y=∫1x13+t+2t2dt

Q: Consider the curve defined by xy + y2 = 21 Determine dy/dx for this curve at the point (4, 3)

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Q: Find dy/dx and dy/dx. x = e°, y = te-t dy %3D dx %3D For which values of t is the curve concave…

A: Given: x=ety=te-t Find dydx and d2ydx2 as shown below.

Q: Find the maximum height above the xy-plane of a point on r(t) = (e1 , sint, t(4 - t))

A: Maximum height above the x-y plane will be the maximum value of z .So we will take z parameter of…

Q: Find an equation of the tangent plane to the graph of F(r, s) at t F(r. s). - (3r4 1 + 1), (-1, 1.…

A: Introduction: Let z=f(x,y) denote the equation of a surface S in R3, and P=(a,b,c) denotes a point…

Q: Find the equation of the tangent plane to z=e^y+x+x^4+6 at the point (4,0,267).

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Q: Find an equation of the tangent to the curve at the point corresponding to the given value of the…

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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: Q3/Find the tangent to the curve x+ xy+ y' = 1 at (1,-1).

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Q: For the curve y = f(x) = x², 0 ≤ x ≤ 2, its length is 2 L = 1+ dy dx dx =

A: Given curve y=f(x)=x2,0≤x≤2 The objective is to find the length of the curve.

Q: Let H(x) = S V1+t² dt. Find the equation of the tangent line to the curve y = H(x) at x = 0. ex

A: It is given that, H(x)=∫ex11+t2dt we have to find the equation of tangent line on curve y=H(x) at…

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: . On what interval is the curve y = . dt +t+ 2 concave downward?

A: For y to be concave downward, it's second derivative y''  &lt; 0 We will use the Leibniz rule to…

Q: a particle moves along a curve x = t^3+1, y=t^2, z=2t+5 , where t represents the time . its velocity…

A: Velocity is obtained by differentiating the position function

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: Q1: Use Green's theorem to evalute f. (xy + y2)dx + x²dy by the curve y = x and y = x2.

A: Given problem:- Use Green's theorem to evaluate ∫c (xy + y2 )dx + x²dy  by the curve y = x and y =…

Q: Find the angle between the tangents to the curve R(t) = ti + 2tj – t°k at the points t = +1

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Q: Evaluate the line integral. { 4) ( y2 dx + x dy; C is the curve x =t2 - 1, y = 4t, 0 st s 1

A: The given line integral is ∫Cy2 dx + x dy; C is the curve x=t2-1, y=4t, 0≤t≤1

Q: Find the equation of the tangent plane to z = ey + x + x² + 7 at the point (5, 0, 38). = Z |

A: To find the equation of tangent plane to z=ey+x+x2+7  at the point 5,0,38In general the equation of…

Q: For the curve x=3cos(t), y=4sin(t), find the value of d^2y/dx^2 at the point where t=π/5.

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Q: At the point (a, b) the curve T =t-4t, y=t2 +2t, t eR has a vertical tangent line. What is the sum a…

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Q: The slope of the tangent line to the curve r = t2 + 1, y = t+t at t = 1 is (A) 3/2 (B) 2 (C) 4 (D)…

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Q: A curve is written parametrically with x and y defined as functions of t: x=t2 +1, y = 4r° +3. Find…

A: Take the derivative of x and y with respect to t then use the formula for the arc length.

Q: (b) Use a graphing utility to find dx/dt, dy/dt, and dy/dx at the parameter t 6. dt TO -/-

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Q: Find the length of the arcc of the curve x=t2, y=t3 from t=0 to t=4.

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Q: For the implicit curve y + xy + x2 = 3, find a point on the curve where the tangent line is…

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Q: Find the tangent plant to f(r, y) = x² + 3ry + 5y² at the point (1,2).

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Q: Find all points (x,y) on the curve C:x(t) = t² + 6t + 12, y(t) = 5t – 8 where the tangent line to…

A: We need to find the point (x,y) on the curve C,whose tangent line to the curve is vertical

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.

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Q: Find the length L of the curve R(t) = 4ť²i + t° j+ 4k over the interval [3, 6]. L

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Q: On what interval is the curve t2 dt t2 +t+2 y = concave downward?

A: The given curve is: y=∫0xt2t2+t+2dt Using fundamental theorem of Calculus says: ddx∫axf(t)dt=f(x)…

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