Problem 1E: Find the length of the curve. 1. r(t) =t, 3 cos t, 3 sin t, 5 t 5 Problem 2E: Find the length of the curve. 2. r(t)=2t,t2,13t3, 0 t 1 Problem 3E: Find the length of the curve. 3. r(t)=2ti+etj+etk, 0 t 1 Problem 4E: Find the length of the curve. 4. r(t) =cos t i + sin t j +ln cos t k, 0 t /4 Problem 5E: Find the length of the curve. 5. r(t) = i + t2 j + t3 k, 0 t 1 Problem 6E: Find the length of the curve. 6. r(t) = t2 i + 9t j + 4t3/2 k, 1 t 4 Problem 7E: Find the length of the curve correct to four decimal places. (Use a calculator to approximate the... Problem 8E: Find the length of the curve correct to four decimal places. (Use a calculator to approximate the... Problem 9E: Find the length of the curve correct to four decimal places. (Use a calculator to approximate the... Problem 10E: Graph the curve with parametric equations x = sin t, y = sin 2t, z = sin 3t. Find the total length... Problem 11E: Let C be the curve of intersection of the parabolic cylinder x2 = 2y and the surface 3z = xy. Find... Problem 12E: Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x2 +... Problem 13E: (a) Find the arc length function for the curve measured from the point P in the direction of... Problem 14E Problem 15E: Suppose you start at the point (0, 0. 3) and move 5 units along the curve x = 3 sin t, y = 4t, z = 3... Problem 16E: Reparametrize the curve r(t)=(2t2+11)i+2tt2+1j with respect to arc length measured from the point... Problem 17E: (a) Find the unit tangent and unit normal vectors T(t) and N(t). (b) Use Formula 9 to find the... Problem 18E: (a) Find the unit tangent and unit normal vectors T(t) and N(t). (b) Use Formula 9 to find the... Problem 19E: (a) Find the unit tangent and unit normal vectors T(t) and N(t). (b) Use Formula 9 to find the... Problem 20E: (a) Find the unit tangent and unit normal vectors T(t) and N(t). (b) Use Formula 9 to find the... Problem 21E: Use Theorem 10 to find the curvature. 21. r(t) = t3 j + t2 k Problem 22E: Use Theorem 10 to find the curvature. 22. r(t) = t i = t2 j + et k Problem 23E: Use Theorem 10 to find the curvature. 23. r(t)=6t2i+2tj+2t3k Problem 24E Problem 25E: Find the curvature of r(t) = t, t2, t3 at the point (1, 1, 1). Problem 26E: Graph the curve with parametric equations x = cos t, y = sin t, z = sin 5t and find the curvature at... Problem 27E: Use Formula 11 to find the curvature. 27. y = x4 28. y = tan x 29. y = xex Problem 28E: To find: The curvature of y=tanx using Formula 11. Solution: The curvature of y=tanx is... Problem 29E: Use Formula 11 to find the curvature. 27. y = x4 28. y = tan x 29. y = xex Problem 30E: At what point does the curve have maximum curvature? What happens to the curvature as x ? 30. y =... Problem 31E: At what point does the curve have maximum curvature? What happens to the curvature as x ? 30. y =... Problem 32E: Find an equation of a parabola that has curvature 4 at the origin. Problem 33E: (a) Is the curvature of the curve C shown in the figure greater at P or at Q? Explain. (b) Estimate... Problem 38E Problem 39E Problem 42E: Use Theorem 10 to show that the curvature of a plane parametric curve x = f(t), y = g(t) is... Problem 43E: Use the formula in Exercise 42 to find the curvature. 43. x = t2, y = t3 Problem 44E Problem 45E Problem 46E: Consider the curvature at x = 0 for each member of the family of functions f(x) = ecx. For which... Problem 47E Problem 48E Problem 49E: Find equations of the normal plane and osculating plane of the curve at the given point. 49. x = sin... Problem 50E: Find equations of the normal plane and osculating plane of the curve at the given point. 50. x = ln... Problem 53E: At what point on the curve x = t3, y = 3t, z = t4 is the normal plane parallel to the plane 6x + 6y ... Problem 55E: Find equations of the normal and osculating planes of the curve of intersection of the parabolic... Problem 56E Problem 58E Problem 59E: Show that the curvature is related to the tangent and normal vectors by the equation dTds=N Problem 60E Problem 62E Problem 63E: Use ihe Frenet-Serret formulas to prove each of the following. (Primes denote derivatives with... Problem 64E: Show that the circular helix r(t) = a cos t, a sin t, bt), where a and b are positive constants, has... Problem 65E Problem 66E: Find the curvature and torsion of the curve x = sinh t. y = cosh t, z = t at the point (0, 1, 0). Problem 67E: The DNA molecule has the shape of a double helix (see Figure 3 on page 850). The radius of each... format_list_bulleted