QUESTION 1 The parameterized curve {(cos(t), sin(t), t) EES:t [0,47[] } is a closed subset in...
1. Consider the curve i(t) = (t sin(t) + cos(t))i + (sin(t) – t)j + tk. (a) Find the length of the curve for 0 <t<5. (b) Is the curve parameterized by arc length? Justify your answer. (C) If possible, find the arc length function, s.
Let C be the curve parameterized by with 0 ≤ t ≤ 2π. a) Show that the curve C is contained in a plane and that it is a closed curve. You must explicitly give the equation of the plane that contains the curve. *( Reminder: the general equation of a plane is ax + by + cz = d.) b)Let P be the plane found in a). Calculate the area of the part of P delimited by curve C...
Question. Consider () - ( cos(t), sin(t)) for 0 +< 2. Parameterine this curve by are length. Chat
Question 17 Calculate the arc length of the curve r(t) = (cos: t)+ (sin t)k on the interval 0 <ts. Question 18 Find the curvature of the curve F(t) = (3t)i + (2+2)ż whent = -1. No new data to save. Last checked a
Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t< 27. (a) (5 pts] Show that this curve C lies on the surface S defined by z = 2xy. (b) (20 pts] By using Stokes' Theorem, evaluate the line integral F. dr C where F(x, y, z) = (y2 + cos x)i + (sin y +22)j + xk
1. Find the boundary and the interior for the following sets. Find the set of all accumulation points and the closure for the following sets. Classify each set as open, closed, or neither closed nor open. Use Heine-Borel theorem to determine whether it is a compact subset of R. A is closed/ open / neither closed nor open A is compact /not compact intB B is closed / open / neither closed nor open B is compact / not compact...
2. Let a curve be parameterized by x = integral for the length of the curve. t3 – 9t, y=t+3 for 1 <t< 2. Set up (but do not evaluate) the
T Find the length of the curve e' cos(t) e' sin(t) for 0 < t < 2 y (Hint: You can simplify the integrand by expanding the argument inside the square root and applying the Pythagorean identity, sinº (0) + cos²O) = 1.)
2. A dragon is flying around in a pattern given by the parametric curve r(t) (cos(t) cos((sin(t) sin(t) cos(t)j. cos(t) - cos sin(t)-sin(t) cos(t))j (a) Find a formula for the velocity of the dragon at time t (b) Find all the times at which the dragon's speed is zero. Explain your reasoning. c) Does the path of the dragon contain any cusps? Explain your reasoning 2. A dragon is flying around in a pattern given by the parametric curve r(t)...
5. Let a curve be parameterized by x = t3 + 9t, y=t+3 for 1 <t < 2. Set up and evaluate the integral for the area between the curve and the x-axis. Note that x(t) is different from the other problems.