Find the exact value of x²y³z ds, where C is the curve with parametric equations x = e-t cos(5t), y = e-t sin(5t), z = e-t, 0 sts 2 .

Q: Find the exact value of 0.049768 x³y²z ds, where C is the curve with parametric equations x = e¯t…

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Q: Find the slope of the tangent line to the parametric equations: x = 3t2 - 3, y = 2t + 1 at t = 1. %D…

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Q: Evaluate the arc length of the curve generated by the parametric equations below for t (-1,1) x(t)=…

A: The expression for the arc length of the parametric curve is; L=∫abx't2+y't2dt   Given: xt=3+3t2 ;…

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A: The given curve is rt=t2+1,4t12,et2-t. Notice that the given curve satisfies t=1(If t=1 is plugged…

Q: Identify the length of the arc having parametric equations x = t3 and y = 2t2 from t = -2 to t = 0

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Q: Find parametric equations for the line tangent to the helix

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Q: Find a rectangular equation for the curve with the given parametric equations. x = 2 sin(t),y = 2…

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Q: 4. Find the length of the curve having a parametric equation of x = a cos 3 0, y = a sin 20 from 8 =…

A: We are asked to find the length of the curve whose parametric equations are as follows:…

Q: The equation of the tangent line to the parametric equations, x =t +sint, y 4t-cost att 0 is y = -1…

A: As we know that; The slope of the tangent line to the given curve is equal to the first derivative…

Q: Find the length of the curve with parametric equations x = e' · cos t, y = e' sin t, with 0 <t <…

A: Given x = et.cos(t) y = et.sin(t)

Q: Find the equation of the tangent line (s) to the following set of parametric equations at the given…

A: Given parametric equations of a curve.

Q: 3. Consider the parametric curve C given by * = te', y =t + cos t. (a) Find the equation of the…

A: Given: The parametric equation of the curve C is  x=tet, y=t+cost (a) To find: The equation of…

Q: A curve is defined as the parametric equations: 1 x = acose, y=-asin 0, 0<0< 2n dy where a is a…

A: on solving this we get

Q: Find the parametric equations of the tangent line LT to the curve C:r (t) = 2 cos(t)i + sin(t)j +…

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Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A: Given:

Q: A curve is defined by the parametric equations: x(1) = e' cos t, y(t) = e' sint d²y dy and dr of the…

A: It is given that, x(t)=et cost,y(t)=et sint   It is known that, dydx can be found by chain rule and…

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

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Q: Find the length of arc of the curve given by the parametric equations x = t°, y=t² from the point…

A: Arc length is given byL=∫abdxdt2+dydt2dt

Q: Determine the exact length of the curve for the parametric equation x = tsint and y = tcost for 0 ≤…

A: To find the exact length (arc length) of the curve for the parametric equation x=t sin(t), y=t…

Q: Sketch the curve defined by the parametric equations x = sin πt/2, y = t, 0≤ t ≤ 6.

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Q: A curve is defined by the parametric equations sin z dz, CoS Z dz, t 2 1. y= X= Find the length of…

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Q: Find the length of the parametric curve C = {(x(t),y(t)) : t E I} x(t) = cos (t) , y(t) = sin³ (t),…

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Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A: Parametric equations for a line:

Q: 7. The curve at right is given by the parametric equations t2 z = v2t. x = In t, y = Find its length…

A: Given that, x=lnt,y=t22 and z=2t. That is, r→t=lnt,t22 ,2t. It is known that, the length of the…

Q: 4. Find the slope of the curve r = cos 20 at the origin 5. A particle moves along a path whose…

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Q: x²y³z ds, where C is the curve with parametric equations x = e -t cos(4t), y = e-t sin(4t), z = et,…

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Q: consider the parametric curve x=t2 , y=9t-t3 . find dy/dx at t=2.

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Q: Find the parametric equations of the tangent line LT to the curve C:r(t) = 2 cos(t)i + sin(t)j+ 2tk…

A: A parametric equation of a curve is given by: C: r(t) = 2cos(t)i + sin(t)j + 2tk Now, we have to…

Q: 20. Evaluate Vx2 + y? ds where C is the curve with parametric equations x = 4 cos(2), y = 4 sin(t),…

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Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

A: The given parametric equation is: x=5 ln t  ,   y=4t   ,      z=t3 Given point is:  (0, 4, 1) From…

Q: 4. Find the parametric equation of the tangent line to r(t) = 4costi – 3tj %3D at Po(2, –n)

A: Given: To find the parametric equation of the given tangent line as follows, r(t) = 4 costi- 3tj…

Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Find an equation for the projection of…

A: Equation of projection of curve on xy plane will not contain the term of z.

Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

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Q: Find the exact length of the given parametric curve. X = = 8t³, y = 12t², 0 <t< 2

A: Consider the parametric equations

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: Find the length of the parametric curve x = e^t + e^-t, y = 5-2t on the interval [0,3]

A: Find the length of the parametric curve x = e^t + e^-t, y = 5-2t on the interval [0,3].

Q: Find parametric equations of the line tangent to the graph r(t) = (t cos 7t)i - (t+ t)j+ sin(t)k at…

A: Given that rt=t+cos 7ti-t2+tj+sintk   Tangents to the curve is given by Tt=ddtrt Tt=ddtt+cos…

Q: (ь) Find the value of a for which the tangent to the parametric curve = 4t° +t? + at, y = t – 6 at t…

A: Given the two parametric curves x=4t3+t2+αt   and y=t4-6. We need to  find the value of α for which…

Q: Find the length of the curve whose parametric equations are x = 3t^2 and 3t^3, from t = 0 andt = y =…

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Q: Find the length of the arc of the curve having the parametric equations x = t³3 and y = from t = 0…

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Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Find an equation for the projection of…

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Q: Find an equation of the tangent(s) to the curve with parametric equations x = 1+ vt, y = e at the…

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Q: Find the length of the parametric curve I = cos(t) y = 1 + sin(t) from t = 0 to t =

A: Given- x=cost and y=1+sint To Find- The length of the parametric curve from t=0 to t=π. Identity…

Q: A curve is defined by the parametric equations z (t) = at and y (t) = bt, where a and b are…

A: Given,  We have  xt = at   and yt = bt  ;  where a and b are constants.  We have to find the length…

Q: Find T(t) and a set of parametric equations for the tangent line to the helix given by 7(t) = 2…

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Q: r(t) = (e-t cos t, e-t sin t, e-t)

A: Given  rt=e-tcost,e-tsint,e-t First find derivative of given point dxdt=-e-tcost-e-tsint…

Q: 3. Find the exact length of the parametric curve x = sin-1t, y = In /1 – t2, 0<t<

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Q: For the curve with parametric equations x = 4 sin (t) – 1 y= 2 cos (1) + 3 dy find the point dx at…

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Q: Find the length of the curve C defined parametrically by C:x(t) = e' cos t, y(t) = e' sint, 0<tS#…

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Q: Find parametric equations for the tangent line to the curve with the given parametric equations at…

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Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.