(1 point) For a plane curve r(t) = (x(t), y(t)), k(t) = (x' (t)y' (t) –...
X) 13.4.21 Find an equation for the circle of curvature of the curve r(t)-21 + sin(t) j at the point (z,1). (The curve parameterizes the graph of y = sin | 2x | in the xy-plane.) An equation for the circle of curvature is (Type an equation. Type an exact answer, using π as needed.) X) 13.4.21 Find an equation for the circle of curvature of the curve r(t)-21 + sin(t) j at the point (z,1). (The curve parameterizes the...
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin t Cost 5 Cos (a) (2 marks) Show that y is unit speed (7 marks) Find, at each point on the curve, the principal tangent T, principal normal (b) N, binormal B, curvature K, and torsion 7. (c) (3 marks) Show directly that T, N, B satisfy the Frenet-Serret frame equations (d) (3 marks) Show that the image of y lies in a plane...
3. [3 marks] Show that for a plane curve described by r = c(t)i + y(t)j, the curvature k(t) is I'Y' - YX| (x2 + y2)3/27 where a prime denotes differentiation with respect to t. 4. [2 marks] Let f(x, y) = xy +3. Find (a) f(x + y, x - y); (b) f(xy, 3.22y).
component functions denoted by y(t) ((t), y(t), z(t). The plane curve t) = (x(t), y(t)) represents the projection of γ onto the xy-plane. Assume that γ, is nowhere parallel to (0,0,1), so that γ is regular. Let K and K denote the curvature functions of y and 7 respectively. Let v,v denote the velocity functions of γ and γ respectively. (1) Prove that R 2RV. In particular, at a time t e I for which v(t) lies in the ay-plane,...
EXERCISE 1.63. The unsigned curvature of a plane curve γ(t)-(x(t), v(t)) can be computed with Proposition 1.46 by considering it to have a vanishing third component function: γ(t) (x(t),y(t),0). Use this method to compute the curvature function of the parabola y(t) (t, t2). How can the signed curvature be determined from this approach? EXERCISE 1.63. The unsigned curvature of a plane curve γ(t)-(x(t), v(t)) can be computed with Proposition 1.46 by considering it to have a vanishing third component function:...
For the curve r(t), find an equation for the indicated plane at the given value of t. 55) r(t) (3 sint+6i+ (3 cos 20t) - 1j+ 12tk; osculating plane at t 2.5m. 12 12 60 +1) + 13 B) y-1) + 169 =0 13 169 12 -6) +. 60 9131)+30) 0 =0 (206-2 56) rt) (t2-6)i+ (2t-3)j+9k; osculating plane at t A) x+y+ (z+9)-0 C) x+ y+(z-9) 0 6. B) z =9 D) z =-9 For the curve r(t), find...
The curvature of vector-valued functions theoretical Someone, please help! 2. The curvature of a vector-valued function r(t) is given by n(t) r (t) (a) If a circle of radius a is given by r(t) (a cos t, a sin t), show that the curvature is n(t) = (b) Recall that the tangent line to a curve at a point can be thought of as the best approx- imation of the curve by a line at that point. Similarly, we can...
Question 2, non-calculator Question 1, calculator The curve C in the x-y-plane is given parametrically by (x(t), y(t), where dr = t sine) and dv = cos| t The Maclaruin series for a function f is given by r" for 1 sts 6 a) Use the ratio test to find the interval of convergence of the Maclaurin series for f a) Find the slope of the line tangent to the curve C at the point where t 3. b) Let...
The total curvature of the portion of a smooth curve that runs from s so to s can be found by integrating k from so to s,. If the $1 curve has some other parameter, say t, then the total curvature is K K ds-dtK|v dt, where to and ty correspond to so So and s1 a. Find the total curvature of the portion of the helix r(t) = (3 cos t)i + (3 sin tj-tk, 0 sts 4m b...
for the curve r(t) find an equation for the indicated plane at the given value of t 56) r(t) (t2-6)i+ (2t-3)j+9k; osculating plane at t=6 A) x+ y+(z+9)=0 C)x+y+ (z-9)-0 56) B) z-9 D) z -9 (3t sint+3 cos t)i + (3t cos t-3 sin t)j+ 4k; normal plane at t 1.5r.. A) y=-3 57) r(t) 57) B) y 3 C)x-y+z-3 D) x+y+z=-3 56) r(t) (t2-6)i+ (2t-3)j+9k; osculating plane at t=6 A) x+ y+(z+9)=0 C)x+y+ (z-9)-0 56) B) z-9 D)...