Problem 1QC: What is the curvature of the circle r() = 3sin,3cos? Problem 2QC: Use the alternative curvature formula to compute the curvature of the curve r(t) = t2,10,10 Problem 3QC Problem 4QC Problem 5QC Problem 6QC Problem 7QC Problem 1E: What is the curvature of a straight line? Problem 2E: Explain the meaning of the curvature of a curve. Is it a scalar function or a vector function? Problem 3E: Give a practical formula for computing the curvature. Problem 4E: Interpret the principal unit normal vector of a curve. Is it a scalar function or a vector function? Problem 5E: Give a practical formula for computing the principal unit normal vector. Problem 6E: Explain how to decompose the acceleration vector of a moving object into its tangential and normal... Problem 7E: Explain how the vectors T, N, and B are related geometrically. Problem 8E: How do you compute B? Problem 9E: Give a geometrical interpretation of the torsion. Problem 10E: How do you compute the torsion? Problem 11E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 12E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 13E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 14E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 15E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 16E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 17E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 18E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 19E: Curvature Find the unit tangent vector T and the curvature for the following parameterized curves.... Problem 20E Problem 21E: Alternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the... Problem 22E: Alternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the... Problem 23E: Alternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the... Problem 24E: Alternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the... Problem 25E: Alternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the... Problem 26E: Alternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the... Problem 27E Problem 28E Problem 29E Problem 30E Problem 31E Problem 32E Problem 33E Problem 34E Problem 35E: Components of the acceleration Consider the following trajectories of moving objects. Find the... Problem 36E: Components of the acceleration Consider the following trajectories of moving objects. Find the... Problem 37E: Components of the acceleration Consider the following trajectories of moving objects. Find the... Problem 38E: Components of the acceleration Consider the following trajectories of moving objects. Find the... Problem 39E Problem 40E Problem 41E: Computing the binormal vector and torsion In Exercises 2730, the unit tangent vector T and the... Problem 42E: Computing the binormal vector and torsion In Exercises 2730, the unit tangent vector T and the... Problem 43E Problem 44E Problem 45E Problem 46E: Computing the binormal vector and torsion Use the definitions to compute the unit binormal vector... Problem 47E: Computing the binormal vector and torsion Use the definitions to compute the unit binormal vector... Problem 48E Problem 49E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 50E: Special formula: Curvature for y = f(x) Assume that f is twice differentiable. Prove that the curve... Problem 51E: Curvature for y = f(x) Use the result of Exercise 50 to find the curvature function of the following... Problem 52E Problem 53E Problem 54E: Curvature for y = f(x) Use the result of Exercise 50 to find the curvature function of the following... Problem 55E Problem 56E: Curvature for plane curves Use the result of Exercise 55 to find the curvature function of the... Problem 57E: Curvature for plane curves Use the result of Exercise 55 to find the curvature function of the... Problem 58E: Curvature for plane curves Use the result of Exercise 55 to find the curvature function of the... Problem 59E: Curvature for plane curves Use the result of Exercise 55 to find the curvature function of the... Problem 60E: Same paths, different velocity The position functions of objects A and B describe different motion... Problem 61E: Same paths, different velocity The position functions of objects A and B describe different motion... Problem 62E: Same paths, different velocity The position functions of objects A and B describe different motion... Problem 63E: Same paths, different velocity The position functions of objects A and B describe different motion... Problem 64E: Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature.... Problem 65E: Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature.... Problem 66E: Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature.... Problem 67E: Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature.... Problem 68E: Curvature of ln x Find the curvature of f(x) = ln x, for x 0, and find the point at which it is a... Problem 69E: Curvature of ex Find the curvature of f(x) = ex and find the point at which it is a maximum. What is... Problem 70E Problem 71E: Finding radii of curvature Find the radius of curvature (see Exercise 70) of the following curves at... Problem 72E: Finding radii of curvature Find the radius of curvature (see Exercise 70) of the following curves at... Problem 73E: Finding radii of curvature Find the radius of curvature (see Exercise 70) of the following curves at... Problem 74E: Designing a highway curve The function
r(t) =
whose graph is called a clothoid or Euler spiral... Problem 75E: Curvature of the sine curve The function f(x) = sin nx, where n is a positive real number, has a... Problem 76E: Parabolic trajectory In Example 7 it was shown that for the parabolic trajectory r(t) = t, t2, a =... Problem 77E: Parabolic trajectory Consider the parabolic trajectory x=(V0cos)t,y=(V0sin)t12gt2, where V0 is the... Problem 78E Problem 79E: Zero curvature Prove that the curve r(t)=a+btp,c+dtp,e+ftp, where a, b, c, d, e, and f are real... Problem 80E Problem 81E: Maximum curvature Consider the superparabolas fn(x) = x2n, where n is a positive integer. a. Find... Problem 82E: Alternative derivation of the curvature Derive the computational formula for curvature using the... Problem 83E: Computational formula for B Use the result of part (a) of Exercise 82 and the formula for to show... Problem 84E Problem 85E: Descartes four-circle solution Consider the four mutually tangent circles shown in the figure that... format_list_bulleted