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- Find the length of the following curve using methods for parametric curves. x = 4 sin ty = 4 cos tt ∈ [0, π/3]Find the mean value of y with respect to x for which the curve has the parametric equation belowLet C be a curve with the parametric equation in the given. a. Set-up the integral that will give the length of C. b. Determine (algebraically) if C is concave or down at the point where t=0.
- Consider the following parametric curve:Find the area enclosed by one loop of this polar curve: r=3sqrt(cos2theta) from 0 to 2pi using the formula A=1/2 integral from 0 to 2pi (r)^2 for parametric curve.Consider the following parametric curve: x(t) = 3t^2 −6 and y(t) = 3t^4 −3t^2 +12t +10 Find the equation of the tangent line to to the point along the curve at(−3,−2).
- Sketch the plane curve r(t) = a cos3 ti + a sin3 tj and find its length over the given interval [0, 2π] .Consider the parametric curve segment (t, t2), t ∈ [0, 1]. What is the firstorder derivative of the curve at t = 0? Show that exactly the same curve segment can be re-parameterized so that the first-order derivative at t = 0 is different.22-Sick leave of employees in a factory before and after Covid-19 was investigated in a year. Which of the following is the table value in the hypothesis of whether there is a difference between the sick leaves of the employees? (The data do not satisfy the parametric assumption). a) 4 B) 3 NS) 6 D) 7 TO) 5
- Use Green's theorem to find the k number that satisfies (k + 3) ∫C((k/2)*(x^k)*(y^2))dx + ((yx^(k + 1)) + y^2) dy = 7680, where curve C is given below.Suppose a curve is traced by the following parametric equations as t runs from 0 to pi. x=5(sin(t)+cos(t)) y=54−20cos2(t)−40sin(t) At what point (x,y) on this curve is the tangent line horizontal?Sketch a graph of the parametric curve defined by: x = t^2 and y = t^3 - 4t for − 2 ≤ t ≤ 2. Include a table on how to calculate the points of the graph.