Let 7(t) < 3t4 – 5, t² + 2t2, °, 4 In(t) > Find a parametric equation of the line tangent to 7(t) at t = 1. In particular, parametrize the line so that the output at t = 1 is the same as the output of the original curve at that time, 7(1). x(t) 14 + 12t y(t) z(t) : ||

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

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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Q: Consider the smooth parametric curve 6 defined by the equations t (t²-3) { 3 (t²- 3)' = = where t€…

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Q: (a) Find the tangent line(s) to the parametric curve given by x(t) = t° – 4t", and y(t) = t. at the…

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

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Q: Let C be a curve with the parametric equations à = − ln(t + 1); y = sint, t = [0, 1]. a. Set-up the…

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Q: 3. Let R(t) = (t²,1 - t², t) and r(t) = cost. Determine the parametric equations of the line tangent…

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

A:

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: Find the equation for the line tangent to the parametric curve: x = t^3 - 9t y = 9t^2 - t^4   at…

A: Find slope and then use equation of tangent line

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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

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A: Given x=lnt+1 and y=cosh−12t We have to find the dydx when t=1 We know that dydx=dydtdxdt.....1…

Q: 16. Sketch the parametric curve x = Vt, y= sin t, 0<t< 2n. (b) (a) 1.0 (c) 101 1.0 0.5 0.5 0.5 05 10…

A: The graph of the parametric curve x=√t, y=sint , 0≤t≤2π is given as below,

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A: Given:

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A: Consider the curve Find the equation of the line tangent to the curve at t=2

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Q: x(t) = t², y(t) = t³ – 3t, on – ∞ <t < ∞ %3D а. Show that the parametric curve has two tangent lines…

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Q: A curve is defined by the parametric equations x (t) = at² and d² y y (t) = bt where a and b are…

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Q: Find the parametric equation for the line tangent to the r(t)=sinti+(t2-cost)j+etk curve t=0 at the…

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Q: Consider the smooth parametric curve defined by the equations = t (t²-3) { = 3 (t²-3) ¹ where t € R.…

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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: 1. Consider a fence whose base is the path with parametric equations |x(t) = cos³ t, |y(t) = sin³t,…

A: Given, A fence whose base is the path with parametric equations xt=cos3tyt=sin3t for 0≤t≤π2. Let the…

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Q: Obtain the gradient to a curve with the parametric equations given by I = In - and y = sinh 2t whent…

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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: A curve C is defined by the parametric equations x(t) y(t) = 8t³ – 2t² + 4. Write the equation of…

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Q: Consider the following parametric curve: x(t) = 3t^2 −6 and y(t) = 3t^4 −3t^2 +12t +10 Find the…

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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: A curve C has parametric equations X = tチ+1. dy (i) Find in terms oft. dx (ii) Show that there are…

A: We first find dy/dx

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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 the equation of the tangent line to the curve defined by the parametric equations x(t) = t3 +…

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Q: Which is the graph of the derivative dyldx of the parametric equations x(t)= 3cos t and y(t)=3sin t…

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Q: - Let C be the curve in R parameterized by r(t) = (t², t³, t^). (a) Compute r'(1). (b) Find a…

A: Given; r(t)=&lt;t^2,t^3,t^4&gt;

Q: 1. Consider the graph of r(t) = 2t i+ tj +t° k Determine parametric equations of the tangent line to…

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Q: Eliminate the parameter in x = -t +2t and y =t-4 and then identify the parametric curve and sketch…

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Q: The parametric equations of a curve are given by 1 = In (Vi +1) and y= cosh(2t). dy when t = 1. dr…

A: According  to question given thatx=ln(t+1)y=cosh-1(2t)

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A: According to question,Given Curve C,Ft = ln2-t, 1t-2, t+3

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: 38. Let r(t) = (1³ -1- 2)i + 4r* j. %3D (b) Find parametric equations for the tangent line to the…

A: Topic = Vector

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: x = V1 - i , y = t³ – 3t

A: We have to find derivative

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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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.