Prove that the curve y: (-,-)→ R³ is regular with factor H and determine whether the image y (t) = (2 cos? t – 3, sint – 8,3sin?t + 4) is included in 1- Is it a straight line or not?

Q: Graph the curve r = 1 - cos θ in the Cartesian xy-plane

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Q: 42. Find the family of curves that is orthogonal to the fam- ily defined by the equation y? = cx and…

A: 42. given curve is                                           derivative of the curve is…

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Q: 4. Prove that the following curves are intersecting with a right angle: 5y-2x+y-xy= 0 2y+5x+x-x'y² =…

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Q: Find the Cartesian equation of a curve defined parametrically by 5t and y =10t for all real values…

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Q: x = 2t2 – C : 3t + 1 y = et²-1

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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

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Q: A Cartesian equation for the parametric curve x=24sin?e, y=21cos20, n ses 31/2 is given by O…

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Q: 27. Show that the curve with parametric equations x = t cos t, y =t sin t, z = t lies on the cone z²…

A: To Show: i. Show that the curve with the parametric equationx=tcost, y=tsint, z=t  lies on the…

Q: Verify Green's Theorem where F1=y^2-7y , F2=2xy+2x and the curve C is a circle x^2+y^2=1

A: Given: F1=y2-7y , F2=2xy+2x and C is a circle i.e., x2+y2=1 As, we have:…

Q: Use the graphs of x = f(t) and y = g(t) to sketch the parametric curve x = f(t), y = g(t). Indicate…

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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: Prove that the curve y:(-,) → R³ is regular with factor H and determine whether the image y (t) = (2…

A: To determine : the      included in a plane in   or not. Here,…

Q: use the given graphs of x = ƒ(t) and y = g(t) to sketch the corresponding parametric curve in the…

A:

Q: Denote that G is a parametric curve x(z) = z2-z-2 y(z) = z²-=2+z Calculate whether if G is concaving…

A: Given parametric curve G: xz=z2-z-2yz=z3-12z2+z We have to calculate whether if G is concave up or…

Q: Find the slope of the tangent to the parametric curve c(t) = (cos t, sin t) at the point (-1/2,-V3).

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Q: Find the tangent line(s) to the parametric curve given by a = t-9t and y =t at (0, 9) %3D %3D

A: We have to find the tangent line

Q: Graph the lemniscate curve r2 = sin 2θ in the Cartesian xy-plane

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Q: Consider the curves:r = 4sin 0 and r = 2 sin(0) + 2cos (0).

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Q: Verify Green's Theorem in the plane for F = -3y³i+ (3x3 + cos y)ĵ where C is the circle x² + y² =…

A: In this we have to verify green theorem.

Q: A curve C is given parametrically by x = cos(t) - 2 and y = sin(t) + 3, where 0 ≤t≤ 2π Find and…

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Q: Find an equation of the tangent(s) to the curve with parametric equations tsint at the point…

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Q: Find the exact value of x²y³z ds, where C is the curve with parametric equations x = e-t cos(5t), y…

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Q: Consider the parametric curve C that is defined by C: x=t^3, y=t^2+t, −1≤t≤3. (b) Is the curve…

A: Consider the parametric curve C that is defined byC: x=t^3, y=t^2+t, −1≤t≤3. (b) Is the curve…

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: Consider a particle moving rectilinearly according to x = 13 - 312 -45t m, where t is in sec. Draw…

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Q: 9. Find the exact length of the parametric curve for the given interval. x = 4t 2 - 3, y = 3t, 0 St5…

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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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Q: Find a parametric description for the curve y=4-x^2 from (-2, 0) to (2, 0) such that t=0 corresponds…

A: Given: The curve y=4-x2 from (-2,0) to (2,0)

Q: Suppose a curve is traced by the parametric equations 3(sin(t) + cos(t)) = x у 3 18 — 6 соs"(€) — 12…

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Q: Find a parametric description for the curve x=y^2-9 from (0, 3) to (-5, -2) such that t=0…

A: We need to parametrize the curve x=y2-9 from 0,3 to -5,2 We need to put either x=t or y=t and see a…

Q: Consider the curve C parametrized by x = 19 – t and y = 17 for all real t. %3D Then C is a O…

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Q: Graph the Plane Curve defined by: I = 3 cos(t) - t + 5. y = sin(t). for t in [0, 37]. x Clear + Undo…

A: Plot the points for different values of t from [0,3π]. Draw the curve passing through the points.

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

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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 an equation of the tangent(s) to the curve with parametric equations x = 1 + √t, y = e^(t^2)…

A: Given parametric equation of curve x=1+t   y=et2we want to find equation of tangent at  ( 2, e)

Q: Verify Green's Theorem in the plane for F = -3y³i+ (3x3 + cos y)j where C is the circle x2 + y? =…

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Q: Make a graph of the curve defined by the parametric equations X = t -5 on the interval 0 sts 3 y=t?

A: Given x=t-5 y=t2   0≤t≤3

Q: A curve is described parametrically as x = e' sint, y = e'cost, – o <t< ao. Find the equation of the…

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Q: 2. Find parametric equations for the tangent line to the curve r(t) = (t8,t³,t7) at (1,1,1).

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Q: Prove that the function U(r, 0) = tan- ( ), 0<r<1,0 s0< 2n is harmonic inside the circle Iz| 1.…

A: Given Ur,θ=2πtan-12rsinθ1-r2, 0&lt;r&lt;1, 0≤θ&lt;2π---1. The Laplacian equation in polar…

Q: ) Find the equation of the tangent line (in terms of x and y) to the parametric curve at the given…

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Q: Find the length of the parametric curve defined by x=r°, y=t- for 0st<1 7.

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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.