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3. (12 points) Consider the curve C defined by r(t) = (4 sint, -4 cost,0) with...
3. (12 points) Consider the curve C defined by r(t) = (4 sint, -4 cost,0) with t€ (0,2) (a) Compute the length of the curve C. (b) Parametrize fit) with respect to are length measured from t = 0. (c) Determine the curvature of C.
12. Consider the curve given by ř(t) (3 cos(t),4t, 3 sin(t) (a) Which of the images below is the plot of the curve? IV 20 50 (a) Compute the arc length of the curve from t = 0 to t = 3. (b) Find the unit tangent vector T(t). (c) Compute the curvature of the curve at any value of t. 12. Consider the curve given by ř(t) (3 cos(t),4t, 3 sin(t) (a) Which of the images below is the...
Question 6 14 pts Consider the curve C defined by the parametric equations: x f(t) y= g(t) = sint -t costt (d) Which picture shows the curve C? Recall the curve C is defined by : x= f(t) cos t g(t) = sint - t y 20 20 10 10F 0 -10 -10 -20 -20 -20 10 -20 10 C 20 -10 0 10 (i) (ii) X 20 20 10 10 0 0 10 -10 -20 -20h -20 10 -20...
Please answer all parts with full, clear solutions so i can understand :) :) Q2 (6 points) If C is a smooth plane curve with parametrization r r(t),t E [a, b], then the curvature K(t) of C at the point r(t) is defined to be the magnitude of the rate of change -ll dT of the unit tangent vector with respect to the arc length. That is, = ds () [2p] Show that K(t) = ||F (C) xr" (t)|| r...
12. Let a curve be defined parametrically by x(1) = 3cost, y(t) = 3 sint, z(1)- 21. a) Find the equation of the tangent line to the curve att b) Find the curvature of the curve att
(a) Sketch the curve r(t) = (e cost, e sint) in R2 and compute its are length for 0 < t < 87. For the sketch, use of software is acceptable, but the graph should be drawn by hand and the right features should be present.] (b) The vector v makes an angle of with the positive -axis. Write the vector v in component form. Furthermore, write the equation of the line lt') passing through the origin with direction vector...
Consider the parametric curve F(t) = (2+1 - 2)i+2 4 13 (a) (10 points) Evaluate SF(t)dt. (b) (10 points) Show that the arc length parameter measured from the point (2,0) is given by s = 4 tan-(t). (c) (10 points) By substituting t = tan (4) verify that F(t) parameterizes a circle of radius 2. What is the curvature?
3. Let C be the curve r(t) = < sint, cost, t>,0 sts 1/2. Evaluate the line integral S ry ryds. 1/V2. 1/2, V2, 0,
Given ř(t) =< 2 cost, t, 2 sint > as a trace of a moving object. (a) Find the curvature of K(t). (b) Find the arc length when 0<t <31. (c) Find the unit normal and binormal vectors of F(t).
(22 - y2 + 2)ds, here C is the curve r(t) = (3 cost, 3 sint, 4t) with 0 <t<2.