Given that x (t) = 3 sin t, y(t) = 2 cos² t, convert the parametric equations into the Cartesian form.

Q: Consider the parametric equations x = 4 cos(0) and y = 2 sin(0). (a) Create a table of x- and…

A: We know that   cos(-θ)=cosθ  and   sin(-θ)=-sinθ

Q: Find the Cartesian equation of the curve: [2r/r2 = 1+ cos a]

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Q: Find the equation of the plane perpendicular to the curve r(t) = (sin(2t), t, cos(2t)) at the point…

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

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Q: A set of parametric equations is given as x = t+2 cos t and y =t+sin t. Find dy in terms of t. i. dx…

A: Given parametric equations arex=t+2cost and y=t+sintfindi) dydx interms of t.ii) d2ydx2 when t=π3

Q: Find an equation in rectangular coordinates for the curve r^2 = cos 2θ.

A: We know that, the relationship between polar and rectangular co-ordinates are given as r2=x2+y2…

Q: Eliminate the parameter t to rewrite the parametric equation as a Cartesian equation: x (t) = –t =…

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Q: Consider the curve given by the parametric equations = 3 cos 0 and y = 3 sin 6, 0<0<T. Which of the…

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

A: The equation of a line can be obtained by using the expression y-y1=m(x-x1). Here, m is the slope of…

Q: Sketch the parametric curve of the given parametric equations and indicate the orientation. 1. x=3t…

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Q: (b) Given that cos (t) Vos(21) sin' (t) Jos(2)" |cos(2t) dy Find for the parametric equation given…

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Q: 1.) Given the parametric equations: x=t'- 1 and y=f+t, find the 1ª and 2nd derivatives.

A: We will find the derivatives.

Q: Determine the number of cusps for the hypocycloid described by the parametric equations x(t) =…

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Q: Find the parametric equation of the line through a = 3 A co which is parallel to b = 2

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Q: Find a Cartesian equation of the curves. 1) r= 2 sin 0 2) r = 5 csc e

A: 1) r=2sinθ2) r=5cosecθwe have to convert these curves into cartesian equations

Q: The parametric equations that correspond to T= (2t 1) i- 3vtj+ sin 3t k is %3D

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Q: 19. Find y' in the parametric equations x =, t-1 t+1 A) (f – 1) 7-1 B) (1+ 1)? 2 +1 C) (1+1)? t +1…

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Q: Consider the parametric equations x= cos 1, y = sin t, 0sts 2n. a. What curve do the points of these…

A: slope = dy/dx = dy/dt /dx/dt

Q: Match the parametric equations with the correct graph. e0.9t t 0 cos(9t), y sin(9t), z e = ZA Z A 2…

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Q: Consider the following parametric equation. x = 7(cos 0 + 0 sin 0) y = 7(sin 0 – 0 cos 0) 11 What is…

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Q: Consider the parametric equations defined by x(t) = 3t² – 3 and y(t)= 2t +1, for t> 0. d'y at t= 1.…

A: Option A is correct option.

Q: What Cartesian equation is equivalent to the given parametric equations? x (t) = 4 sint ly (t) = 4…

A: Squarring then add

Q: Consider the following parametric equation. = 5(cos 0 + 0 sin 0) y = 5(sin 0 – 0 cos 0)

A: Given,          The parametric equation are x=5cosθ+θsinθ and y=5sinθ-θcosθ.

Q: Write the parametric equations x = 2 sin? 0, y = 9 cos? 0 in the given Cartesian form. y = with 0 <…

A: Simplification of the functions helps to find the parametric equation. Trigonometric identity…

Q: Find an equation (in polar coordinates) of the line tangent to the cycloid x = t - sin t, y = 1 -…

A: Given equations are,

Q: What curve is represented by the following parametric equations ? x = sin(t) ; y = cos(2t) ; 0 ≤ t ≤…

A: x = sin(t) ; y = cos(2t) We need to represent the curve determined by the given parametric…

Q: Consider the parametric equations x sin 2t and y -4+c ost. which of the following dy7 represents 2…

A: We are given that x=sin 2t y=4+cos tWe need to find dydx

Q: Find in terms of t for each of the following parametric equations. and dx dx a x=r - 3, y =r'+1 1+3…

A: 23 b. Consider the parametric equation x=t+3t and y=t2-4t. Differentiate x=t+3t with respect to t…

Q: For the parametric equation shown below, determine the value of (dy/dx) and (d²y/dx²). X = 3t – 4t +…

A: a x = 3t2 -4t +2 y = t3 -2t2 + 2t -5  dy/dx and d2y/dx2 =? dx/dt  = 6t -4 d2x/dt2 = 6 dy/dt = 3t2…

Q: Consider the following parametric equation. - 5(cos 0 + 0 sin 0) %3D y = 5(sin 0 – 0 cos 0)

A: Given,         The parametric equations are x=5cosθ+θsinθ and y=5sinθ-θcosθ

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

A: Consider the equation of helix given by r(t). Find the value for t , derivative at t and modulus of…

Q: Consider the parametric equations below. x = t?, y = t3 X =

A: Given data:     The first parametric equation is: x=t2. The second parametric equation is: y=t3.…

Q: Convert the following parametric curve into Cartesian form: x=2 sint, y = 3 sin t O 3x² + 2y² = 1 O²…

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Q: ) Suppose that the parametric equations x = 1, y = r', 12 0, model the position of a moving object…

A: Given

Q: Write the parametric equations x = 3 sin 0, y = 9 cos 0,

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Q: x = 4 cos t and y = 4 sin t 0sts 4

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Q: 3. Convert the following parametric equations to rectangular equation (a) x = cosh t, y= sinh t (b)…

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Q: The parametric equation of the x_axis is t=1 O x=1 O x=0 O x=t

A: To find the parametric equation of x axis.

Q: Graph the parametric equations below (x(t) = 3 cost + cos 3t ly(t) = 3 sint - sin 3t

A: To graph the parametric equations.

Q: If x(t) = 7 sint+ 9 and y(t) = 8 cos t + 3, eliminate the parameter to write the parametric…

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Q: Consider the following parametric equation. x = 3(cos 0+ 0 sin Ø) y = 3(sin 0 -0 cos 0) What is the…

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Q: Sketch the parametric curve for the following set of parametric equations x = 5 cos t + 2 , y = 2…

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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: Given the parametric equations x = 20 + sin 20, y =1- cos 20, dy whereð is the parameter, dx

A: here option D is true

Q: 26 Given the parametric equations x=4cos? r and y = 5sint. Find dy when t= dx

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Q: Consider the following parametric equation. x=7(cos(t) + tsin(t)) y=7(sin(t) - tcos(t)) what is…

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Q: Consider the parametric equations below. x = t2, y = t5

A: We have to find cartesian equation

Q: (b) Find the graph of the parametric equations: I = sin 0, y = 3 – 2 cos 20

A: Plugging values of θ, to find the corresponding values of x and y: When…

Q: 2. Given the parametric equation: x= 4 cos (t), y=9 sin (t), 0 <t< 2n, a) sketch the curve and…

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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.