Consider the parametric curve r(t) = (t", t²). Determine which of the following graphs is the correct image of this curve: A B Also, compute the velocity and acceleration vectors at t = 1, and use this information to determine the direction of motion for the curve. The correct graph is C v(1) = a(1) = The diregrion of morion is Loft Right

Q: Find the equation of the tangent line to the curve defined by the parametric equations x(t) = 2t3 -…

A: Topic- application of derivatives and tangent equation

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: Find parametric equations for the tangent line of the curve r(t) = (3+ 3/7, to – 5t2 + 4, 3t³ + 2t²…

A: Find the parametric equations fot the tangent line

Q: Evaluate the arc length of the curve generated by the parametric equations below for t (-1,1) x(t)=…

A: The expression for the arc length of the parametric curve is; L=∫abx't2+y't2dt   Given: xt=3+3t2 ;…

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…

Q: Given the the parametric curve Į æ(t) = ecost y(t) = t³ – 243t

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Q: Consider the parametric curve given by (a) Find dy/dx and d²y/dx² in terms of t. dy/dx = |d²y/dx² =…

A:

Q: Find the points on the parametric curve c(t) = (3t2 − 2t, t3 − 6t) where the tangent line has slope…

A: Given curve: So,

Q: Consider the parametric curve C: x 1+cost, y= -3+2sint, 0stS (1) Sketch the curve and indicate the…

A:

Q: Consider the parametric curve x = Vt , y= 3t –1, for t20. Find the slope of the tangent line to this…

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

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

A:

Q: Consider the curve C given in parametric equations as t(t – 1)2 In (4t² + 1) %3D C : for t e R. (a)…

A: Given problem is :

Q: Consider the parametric equations x(t) = 1 + 3t2, y(t) = 4 + 21³, -1 <t <1. • What is the length of…

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Q: Find the slope of the curve whose parametric equation is given by y = t³ + t²+ 1/t and x = t⁴ + 3t²…

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Q: Find the equation for the line tangent to the parametric curve: t3 – 25t 25t2 – 14 at the points…

A:

Q: A curve is defined by the parametric equations z (t) = e-* and y (t) = e*. What is in terms of t ? A…

A:

Q: find the slope of the parametric curve x=-6t-5, y=3t^3, at the point t=3

A: To find the slope we have to find dy/dx For that we find dx/dt and dy/dt first

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

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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: Consider the parametric curve defined by 1 a(t) %3D y(t) = t² || What is the slope of the line…

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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: Find the length of the curve defined by the parametric equations t, y = 5 In((t/6)? – 1) I = from t…

A: Given, x = 56t , y = 5lnt62-1 = 5lnt2-36-5ln(36)and 7≤t≤10

Q: Find the exact slope of the tangent line to the parametric curve at the point where y = 2 sin(t) t =…

A: x=8 cos (t) y=2 sin (t)

Q: consider the parametric curve x=t2 , y=9t-t3 . find dy/dx at t=2.

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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: The following curve is in parametric coordinates: x = 2t + 1; y =t3 - 3t; find the points (in…

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Q: Find the points on the parametric curve c(t) = (3t 2 − 2t,t 3 − 6t)where the tangent line has slope…

A: This question is taken from the tangent and normal in which we have to find points on the given…

Q: 2. Find the second derivative for the following set of parametric equations. = t° – 4t3 y = t?

A: We have to find the second derivative of the given equation.

Q: Find the exact length of the given parametric curve. X = = 8t³, y = 12t², 0 <t< 2

A: Consider the parametric equations

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: Find an equation of the tangent(s) to the curve with parametric equations x = t cost, y = tsin t…

A: Given- x=tcost , y=tsint To find - To find the equation of tangent to the curve with given…

Q: Consider the curve described by the parametric equations ¤ = t² +4 and y = 3t + 4t. Determine the…

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Q: (ь) Find the value of a for which the tangent to the parametric curve = 4t° +t? + at, y = t – 6 at t…

A: Given the two parametric curves x=4t3+t2+αt   and y=t4-6. We need to  find the value of α for which…

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

A:

Q: Find parametric equations for the tangent line at t=3 for the motion of a particle given by…

A: To work out the equation of the tangent in terms of t

Q: Find the length of the curve C defined parametrically by C: r(t) = e' cos t, y(t) = e sin t, 0<t< T

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Q: Find the length of the parametric curve x = 2t + 1, y = 3 + 2tž, 0 < t <%.

A: Find the Length of the curve

Q: Find the equation for the line tangent to the parametric curve: at the points where t = 5 and t =…

A: Solution is given below:

Q: Find the asymptotes of the curve defined by the parametric equationsx=φ(t)=1t−2,y=ψ(t)=tt−1.

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Q: Find the equation of the tangent line to the following parametric equation at the given point. x =…

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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: Consider the parametric curve C: x=t³ + 1, y = t² – t,0 <t < 3. Find the equation of the tangent(s)…

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

A:

Q: x = +4r2 +4t +1 y =1+3 at the point P(1, 1).

A:

Q: Find an equation of the tangent(s) to the curve with parametric equations E = t cos t, y =t sin t nt…

A: follow next step

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: Consider the parametric curve given by equations:  x(t)= t2 + 30t - 21 y(t)= t2 + 30t - 37  How…

A: At t = 0, determine x(t) and y(t). x0=02+300-21=0+0-21=-21   y0=02+300-37=0+0-37=-37   Thus, the…

Q: Sketch the curve given by the parametric equations x= t³ - 1 and y= t² + t.

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Q: Consider the parametric curve given by the equations:  x(t) = t2 + 37t - 23 y(t) = t2 + 37t + 48…

A: We need arc length to find the units of distance are covered by the point P(t) = (x(t), y(t))…

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

A:

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.