F. dr. (Check the orientation of the curve before applying the theorem.) Use Green's Theorem to evaluate Jc F(x, y) = (y2 cos(x), x² + 2y sin(x)) Cis the triangle from (0, 0) to (1, 3) to (1, 0) to (0, 0)

Q: 1. Find an equation for the tangent plane to the surface x cos z + yře*z =4 at Po(3, –1, 0)

A:

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

A:

Q: Describe the shape of the curve x = 3 cos t, y = 3 sin t .

A: Given:

Q: I. Let C be the parametric curve with parametric equations x=t(t² + 1)-¹/2 te [-3,3] ly: = 2t - 2…

A: There are two points at where the tangent line is horizontal

Q: Consider the parametric curve x(t) a) Sketch and identify this curve. b) Set up an integral to…

A:

Q: e Green's Theorem to evaluate [(-y²)dx + (x²)dy if the curve C is a triangle with rtices at (0,0),…

A: Top:- application of integration

Q: I the slope of the tangent line to the parametric curve X = 3 cos (t) y = 2 sin(t) ㅠ R 4 at the…

A: Given query is to find the slope of the tangent.

Q: A. Sketch the two curves: 1. r1 = 3 cos 0. 2. r2 = 1+ cos 0.

A: Use polar system to sketch this curve.

Q: 5. Given that r(t) = 3i + e* sin tj + e cos tk Find equations of the normal plane and osculation…

A: We have to  find equation of normal and oscillating plane of above curve.

Q: 3. Let f(x, y) = xy+ y². Find the equation of the tangent line in parametric form in the x-direction…

A: Solve the following

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: III. Find the parametric equations of normal line to the surface given by f(x, y) = tan 2 V + y? at…

A:

Q: x(t) y(t) = 3 sin t 8 cos t 0<t< 2n a) What does the curve described by these parametric equations…

A: Recall: Equation of line: y-y1=mx-x1 ; where m=dydx or  y2-y1x2-x1 Length of the curve…

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

A: Given:

Q: Find the length of the curve x = cos t, y= sin t, 0 <IST. 4/2 4 IT 2/2 IT 2 2

A: Given that: x=cos2t, y=sin2t, 0≤t≤π

Q: Evaluate Fdr where F (x, y) = ( arctan y, 1+y and the curve C is given by C : x = 2 cos t, y = 2 sin…

A:

Q: 1. Evaluate xyz dz, where C is the curve 7(t) =t cos(t)î + sin(t)ĵ – cos(t)k, 0 < t <1.

A:

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: Find the unit tangent vector to the curve defined by r(t) = (4 cos(t), 4 sin(t), 2 sin²(t)) π at t =…

A:

Q: Parameterize the curve r(t) = (2+ cos 3t, 3 – sin 3t, 4t) using arc length, s, as a parameter. Use t…

A: Given the curve rt=2+cos3t, 3-sin3t, 4t---1. Differentiating, we get vt=r't=-3sin3t,…

Q: Q1 Consider the plane curve defined by the parametric equations x = p + t sin(t),y= p² + t cos(t).…

A:

Q: Evaluate (x² + y² ) ds , where C is the curve represented b r (1) = (1 – sin t) i + (1 – cos t) j ,…

A:

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

A:

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

A: Parametric equations for a line:

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

A: The tangent line to a parametric curve given by c(t)=&lt; x(t) , y(t) , z(t) &gt; at some point…

Q: 1/B) Find the length of the cardioid 'curve: r = 1 + cose

A:

Q: Find an equation y = f(x) for the parametric curve c(0) = (cos(0), 10 cos(0) + 3 sin? (0) and…

A:

Q: 1. Find the second derivative of the parametric equations x = a cos t and y = b sin t 2. Use second…

A:

Q: x = -ln(t + 1); y = sint € [0,1]

A:

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

A:

Q: Let C be the curve defined parametrically by x (t) = cos (t), y (t) = sin (t) and z (t) = 3t, where…

A: The given integral:∫CF.drWhere Fx,y,z=x2+y2,2yx,z227And the curve C is given by:xt=cos t ; yt=sin t;…

Q: Let C1 be the curve with parametric equations x = cos t, y = sin t – t, where t E [0, 5). (a) Find…

A: This question can be solved using the concept of equation of tangent and length of curve when the…

Q: Calculate the length of the curve γ (t) = (cos(t), sin(t), t), t ∈ [0, 2π].

A:

Q: Let C1 be the curve with parametric equations x = Cos t, y = sin t – t, where t e [0, ). (a) Find…

A:

Q: length of the parametric curve defined by x = t³ +2, y = 2tº/2, for 1 <t < 3. e the arc

A:

Q: Find the length of the curve y = In (cos x), 0sxs 4 The length of the curve is (Type an exact…

A:

Q: 2. the area of the curve r-a cos 2 0

A: The curve contains two petals,

Q: cos(u) dx. int_ccC [2x*cos(y) \ dx \ - | x^2 sin(y) \ dy] x? sin(u) dul if C is the curve x^4+y^4=1,…

A:

Q: 1 A curve C is defined by the parametric equations x = and y = for 1z 0. VI +1 t + 1 dy (a) Find dx…

A: First find slopes then us equation of tangent line

Q: 16. Let C be the curve given by r(t) (t cos(t), t sin(t), t). Show that C lies on the cone x2 + y? =…

A:

Q: Sela'y-J) dy. Cis the arc of the curve y- Vx from (1. 1) to (4, 2)

A:

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

A: Answer

Q: Find the arc length of the curve y = e® from x=0 to x=1. VIte-1 VIte+1 - (v2 + ¿ln a. V2+1 Ob.…

A: Consider the equation: y=ex……1 from the limit x=0 to x=1.

Q: 1. Sketch the parametric curve x = (cos t) (1+ cos t), y= (sin t) (1 + cos t) , 0 < t < 27 .

A: we have been given the parametric curve x=(cos t)(1+cos t) and y=(sin t)(1+cos t) where 0≤t≤2π

Q: Find equation of (a) the tangent plane to the given surface at the specified point. y = e^x cos (z)…

A:

Q: Consider the curve, y with parameter (, and function f in 3 dimensional Euclidean space with…

A:

Q: (x(t), y(t)) = (2 cos(t), 1+ sin(t)). 2. Answer the following questions about the curve given…

A:

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

A: We find parametric equations for the tangent line to the curve with the given parametric equations…

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.