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QUESTION 11 For what value(s) of t, the curve traced out by r(t) is not smooth?...
(7.5 points) Let C be the oriented closed space curve traced out by the parametrization r(t)...
(7.5 points) Let C be the oriented closed space curve traced out by the parametrization r(t) = (cost, sint, sin 2t), 0<t<27 and let v be the vector field in space defined by v(x, y, z) = (et - yº, ey + r), e) (a) Show that C lies on the cylinder x2 + y2 = 1 and the surface z = 2cy. (b) This implies that C can be seen as the boundary of the surface S which is...
Question 1. If C is a smooth curve given by a vector function r(t), a stsb,...
Question 1. If C is a smooth curve given by a vector function r(t), a stsb, show that Ser-dr = žlır(b)l2 – Ir(a)i?) Hint 1: Use the product rule and the fact that the dot product is commutative (this means a b = b.a for any two vectors a and b) to write: r(t).r' (t) Hint 2: Use FTC2 from first year calculus. 1 d 2 dt
Graph the curve C that is represented by r(t)-[t 2t also r'(0) and r() cos t], 0 2π. Graph (20 pt...
Question as above.
Graph the curve C that is represented by r(t)-[t 2t also r'(0) and r() cos t], 0 2π. Graph (20 pts) 2. t (10 pts) (c) Find the length of the curve traced by r(t)-[t sint tcost t], 0StS T. (10 pts) 4. Graph the curve: r- Pl. Graph also the velocity and accerlaration vectors at t=0 and I. Give the speeds at the two times. Give the expressions for the normal and tangential components of the...
3. Consider the vector-valued function: r(t) = Vt +1 i + pi a. State the domain...
3. Consider the vector-valued function: r(t) = Vt +1 i + pi a. State the domain of this function (using interval notation). b. Find the open intervals on which the curve traced out by this vector-valued function is smooth. Show all work, including r 't), the domain of r', and the other required steps. c. Provide a careful sketch of the path traced out by this function below. Include at least 3 points on the graph of this function. Assume...
Question 3 3 pts Find a vector function, r(t), that represents the curve of intersection of...
Question 3 3 pts Find a vector function, r(t), that represents the curve of intersection of the two surfaces. The cone z = V x2 + y2 and the plane z =6+y 12 1 12 a. r= 1 (61,42 – 36,42 +36) b. 1) = 1 (12t,t2 + 36,42 +36) +(12t,– 12,42 +12) ${12t,t2 – 36,2 +36) P(8) = 1 / 2 C. d. (6) 2 1
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...
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...
12.1.24 Question Help The tangent line to a smooth curve r(t) = f()i + 96)j +...
12.1.24 Question Help The tangent line to a smooth curve r(t) = f()i + 96)j + h(t]k at t= to is the line that passes through the point (f(t):(to)."(to) parallel to (to)the curve's velocity vector at to User (to) and (t) to find parametric equations for the line that is tangent to the given curve at the given parameter value t= to (1)-(31²)i + (4 + 3)j + (52) 10-3 What is the standard parametrization for the tangent line? yo...
(1 point) A parametric curve r(t) crosses itself if there exist t s such that r(t)-r(s). The angle of intersection is t...
(1 point) A parametric curve r(t) crosses itself if there exist t s such that r(t)-r(s). The angle of intersection is the (acute) angle between the tangent vectors r() and r'(s). The parametric curver (2 -2t 3,3 cos(at), t3 - 121) crosses itself at one and only one point. The point is (r, y, z)-5 3 16 Let 0 be the acute angle between the two tangent lines to the curve at the crossing point. Then cos(0.997
(1 point) A...
QUESTION 16 Find the arc length of the helix traced by r(t) = <p cost, p...
QUESTION 16 Find the arc length of the helix traced by r(t) = <p cost, p sint,pt> for Osts2.when the value of p-3. QUESTION 17 The equation passes through the points P (6,1,-1). Q(0, -2.0) and R(1.-1.7) is given by ax+by+czek Follow the steps below and find value of k. Lick Save and Submit to save and submit. Click Save All Answers to save all answers. A ) 17 F5 F4 F3 F2 % 7
6.14(a) Prove that the brachistochrone curve (6.26) is indeed a cycloid, that is, the curve traced...
6.14(a) Prove that the brachistochrone curve (6.26) is indeed a cycloid, that is, the curve traced by a point on the circumference of a wheel of radius a rolling along the underside of the x axis. (b) Although the cycloid repeats itself indefinitely in a succession of loops, only one loop is relevant to the brachistochrone problem. Sketch a single loop for three different values of a (all with the same starting point 1) and convince yourself that for any...
(7.5 points) Let C be the oriented closed space curve traced out by the parametrization r(t) = (cost, sint, sin 2t), 0<t<27 and let v be the vector field in space defined by v(x, y, z) = (et - yº, ey + r), e) (a) Show that C lies on the cylinder x2 + y2 = 1 and the surface z = 2cy. (b) This implies that C can be seen as the boundary of the surface S which is...
Question 1. If C is a smooth curve given by a vector function r(t), a stsb, show that Ser-dr = žlır(b)l2 – Ir(a)i?) Hint 1: Use the product rule and the fact that the dot product is commutative (this means a b = b.a for any two vectors a and b) to write: r(t).r' (t) Hint 2: Use FTC2 from first year calculus. 1 d 2 dt
Question as above.
Graph the curve C that is represented by r(t)-[t 2t also r'(0) and r() cos t], 0 2π. Graph (20 pts) 2. t (10 pts) (c) Find the length of the curve traced by r(t)-[t sint tcost t], 0StS T. (10 pts) 4. Graph the curve: r- Pl. Graph also the velocity and accerlaration vectors at t=0 and I. Give the speeds at the two times. Give the expressions for the normal and tangential components of the...
3. Consider the vector-valued function: r(t) = Vt +1 i + pi a. State the domain of this function (using interval notation). b. Find the open intervals on which the curve traced out by this vector-valued function is smooth. Show all work, including r 't), the domain of r', and the other required steps. c. Provide a careful sketch of the path traced out by this function below. Include at least 3 points on the graph of this function. Assume...
Question 3 3 pts Find a vector function, r(t), that represents the curve of intersection of the two surfaces. The cone z = V x2 + y2 and the plane z =6+y 12 1 12 a. r= 1 (61,42 – 36,42 +36) b. 1) = 1 (12t,t2 + 36,42 +36) +(12t,– 12,42 +12) ${12t,t2 – 36,2 +36) P(8) = 1 / 2 C. d. (6) 2 1
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...
12.1.24 Question Help The tangent line to a smooth curve r(t) = f()i + 96)j + h(t]k at t= to is the line that passes through the point (f(t):(to)."(to) parallel to (to)the curve's velocity vector at to User (to) and (t) to find parametric equations for the line that is tangent to the given curve at the given parameter value t= to (1)-(31²)i + (4 + 3)j + (52) 10-3 What is the standard parametrization for the tangent line? yo...
(1 point) A parametric curve r(t) crosses itself if there exist t s such that r(t)-r(s). The angle of intersection is the (acute) angle between the tangent vectors r() and r'(s). The parametric curver (2 -2t 3,3 cos(at), t3 - 121) crosses itself at one and only one point. The point is (r, y, z)-5 3 16 Let 0 be the acute angle between the two tangent lines to the curve at the crossing point. Then cos(0.997
(1 point) A...
QUESTION 16 Find the arc length of the helix traced by r(t) = <p cost, p sint,pt> for Osts2.when the value of p-3. QUESTION 17 The equation passes through the points P (6,1,-1). Q(0, -2.0) and R(1.-1.7) is given by ax+by+czek Follow the steps below and find value of k. Lick Save and Submit to save and submit. Click Save All Answers to save all answers. A ) 17 F5 F4 F3 F2 % 7
6.14(a) Prove that the brachistochrone curve (6.26) is indeed a cycloid, that is, the curve traced by a point on the circumference of a wheel of radius a rolling along the underside of the x axis. (b) Although the cycloid repeats itself indefinitely in a succession of loops, only one loop is relevant to the brachistochrone problem. Sketch a single loop for three different values of a (all with the same starting point 1) and convince yourself that for any...
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