(a) Under what conditions is it possible to consider a parametrized curve as the graph of a function? (Hint: If x = f(t) and y = g(t) for a ≤ t ≤ b, what might one let f(t) be?)

Q: 1. Consider the parametric curve given by x = e-t, y = 1 – e-2t. Eliminate the parameter, sketch the…

A: Sketch the graph of the given parametric curve and determine the limits on values of x and y.

Q: x = t + t and y = t* + 2t²

A: If the parametric equation of a curve are x=f(t) and y=g(t), then the slope of the tangent line at…

Q: 4) Find the critical points of the parametric equations x = t* - 1 y = t3 - t

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Q: Suppose parametric equations for the line segment between (1,9) and (0, Sx(t) = a + bt ly(t) = c+ dt…

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Q: - For what value(s) of t does the curve given by the parametric equations x= -² -1 and y=t* +2r² –…

A: As per our guidelines we are allowed to solve one question at a time . Kindly repost the rest…

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

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Q: x = 2t2 – C : 3t + 1 y = et²-1

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

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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

Q: Q2 - A) Find the parametric equations for the line that is tangent to the given curve : r(t) = (a…

A: To find The parametric equations for the line that is tangent to the given curve

Q: 5. Find the length of the parametric curve x = t + y = v12t +5, 1<t<2. %3D

A: We have to find the curve length.

Q: Find a parametric equation for the path of a particle that moves halfway around the circle…

A: We have to find the parametric equation for the path of a particle.

Q: (2) Consider the curve given parametrically by 1=1+2/2 cost, y = tant +2/2 sin t, for dy and show…

A: This is the application of derivatives.

Q: Consider the parametric curve segment (t, t2), t ∈ [0, 1]. What is the firstorder derivative of the…

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Q: Eliminate the parameter f in the parametric equations z(t)=√t+1, y(t) = Sketch the graph of the…

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Q: Suppose parametric equations for the line segment between (7, 3) and (3, 0) have the form: S z(t) =…

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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,

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

A: Given:

Q: A curve is given by the parametric equations x(t)=t2-4t and y(t)=t2+2t-3. The line tangent to the…

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Q: 3. Consider the parametric equations x = 1+ cos(t), y = -2+ sin(t), 0sts2M. 3.1. Find an equation…

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Q: A curve C has parametric equations 1 x = t t+±1. 1-t² dy in terms of t. dx (i) Find (ii) Show that…

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

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Q: 15. Graph the parametric curve -3 <t < 3-3sts3. Identify all intercepts of the x and y axes. (t – 1,…

A: (15) The parametric equation is given as t2-1,3t+2 over interval -3≤t≤3 Sketch the graph and…

Q: Consider the graph of r(t) = 2t^2i + t^2j + t^3 k Determine parametric equations of the tangent line…

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Q: 3. Consider the parametric equations x = 1+ cos(t), y = -2+ sin(t), 0S <2M. 3.1. Find an equation…

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Q: 1. For the parametric equations: x = 4t3, y= 2 +60t – 8t?, find: a) All values of t at which the…

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Q: Suppose parametric equations for the line segment between (3, – 4) and (0, 3) have the form: S ¤(t)…

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Q: 4) Find the critical points of the parametric equations x = t* – 1 y = t3 -t

A: Topic:- application of derivatives

Q: 4. Determine the length of the parametric curve given by the following parametric equations. (a) x =…

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Q: 2 Make a graph of the parametric equations x(t) = t - 3 on the interval 0sts 4 y(t) = ) = t ³ - 5 t

A: Given parametric equations  X(t) = t2-3  Y(t) =t3-5t

Q: [6]. Consider the following set of parametric equations: x = r° - 3t y = 312 – 9 On which intervals…

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Q: 6. Find a set of parametric equations for the line tangent to the space curve at the given point. 2…

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Q: A curve C is defined by the parametric equations x(t) = 3 + t“ and y(t) = t° + 5t.Which of the…

A: Given,         A curve C is defined by the parametric equations xt=3+t2 and yt=t3+5t.

Q: Suppose parametric equations for the line segment between (2, 8) and (– 2, 0) have the for s ¤(t) =…

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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 a parametric curve C that satisfies all of the following conditions a)-d). a) y−…

A: As there are only two intercepts on x-axis there has to be a tangent, but it cannot be a vertical…

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: Consider the smooth parametric curve C₁ defined by equations t³-t²- 2t Xx y = = (161²- 48t +35) NIW…

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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: -10 -8 -6 -4 4 6 8 10 -10- Which of the following sets of parametric equations correctly represents…

A: Given: The curve as To determine: Parametric equation of given curve.

Q: 2. Consider the curve C given in parametric equations as t(t – 1)2 In (4t2 + 1) C: for t e R. (a)…

A: Tangent and normal line

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: 9.) a. Find the parametric equations for the tangent line to the Curve at' the Specific point. X =t°…

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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: 1. Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the…

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Q: Suppose that a parametric curve is given by x = f(t), y = g(t) for 0 ≤ t ≤ 1. If f 0 (t) > 0,…

A: Given: x=ft……1y=gt……2   where, 0≤t≤1.   To prove: If ft&gt;0, then the graph of the curve represents…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.