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43. x = t², y = t³ 3

43. x = t², y = t³ 3

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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CAS 40. (a) Graph the curve r(t) = (sin 3t, sin 2t, sin 3t). At how ture has a local or absolute maximum? many points on the curve does it appear that the curva- (b) Use a CAS to find and graph the curvature function. siss Does this graph confirm your conclusion from part (a)? nuo bus for 2 CAS 41. The graph of r(t) = (t - sin t, 1 -cos t, t) is shown in Figure 13.1.12(b). Where do you think the curvature is largest? Use a CAS to find and graph the curvature function. 101419001 101 For which values of t is the curvature largest? metric curve x = f(t), y = g(t) is 42. Use Theorem 10 to show that the curvature of a plane para- K = 44. x = a cos wt, 45. x = e' cos t, |xÿ - yx| 2 [x² + y ² ] ³/² where the dots indicate derivatives with respect to t. (saite (1) sosy lipolay ont 43-45 Use the formula in Exercise 42 to find the curvature. 43. x = t², y = t³ y = b sin wt X y = e' sin t thew mal pla CAS 54. Is there ar osculating [Note: Yo ing, and 46. Consider the curvature at x = 0 for each member of the family of functions f(x) = e. For which members is K(0) largest? 55. Find equ curve of = x² Z= 56. Show t r(t) = conclu 57. Show the a the s unit 58. The con fyi the 59. Sh no 60. S
Transcribed Image Text:CAS 40. (a) Graph the curve r(t) = (sin 3t, sin 2t, sin 3t). At how ture has a local or absolute maximum? many points on the curve does it appear that the curva- (b) Use a CAS to find and graph the curvature function. siss Does this graph confirm your conclusion from part (a)? nuo bus for 2 CAS 41. The graph of r(t) = (t - sin t, 1 -cos t, t) is shown in Figure 13.1.12(b). Where do you think the curvature is largest? Use a CAS to find and graph the curvature function. 101419001 101 For which values of t is the curvature largest? metric curve x = f(t), y = g(t) is 42. Use Theorem 10 to show that the curvature of a plane para- K = 44. x = a cos wt, 45. x = e' cos t, |xÿ - yx| 2 [x² + y ² ] ³/² where the dots indicate derivatives with respect to t. (saite (1) sosy lipolay ont 43-45 Use the formula in Exercise 42 to find the curvature. 43. x = t², y = t³ y = b sin wt X y = e' sin t thew mal pla CAS 54. Is there ar osculating [Note: Yo ing, and 46. Consider the curvature at x = 0 for each member of the family of functions f(x) = e. For which members is K(0) largest? 55. Find equ curve of = x² Z= 56. Show t r(t) = conclu 57. Show the a the s unit 58. The con fyi the 59. Sh no 60. S
10 hansemenige od nas grups 9 SO dit adi mori But ds/dt = |r'(1)| from Equation 7, so is Therefore 15.11 EXAMPLE 3 Show that the curvature of a circle of radius a is 1/a. DOLLS DRIT D THIN SOLUTION We can take the circle to have center the origin, and then a r(t) = a cos ti + a sin tj and and r'(t) = -a sin ti + a cos tj CT()| OF ROL k(t) = T(t) = tegen berupor ed r'(t) [r'(t)} por el ba T'(t) = -cos ti - sin tj Chos- (0)5 This gives |T'(t) = 1, so using Formula 9, we have = - sin ti + cost j Boome bellso ai (1)|T'(t) | k(t) = dioome balls |r'(t) | k(t) = 1 VALE SUNDE a | r'(t)| = a parametrization betegs two of Ivary S The result of Example 3 shows that small circles have large curvature and large circles have small curvature, in accordance with our intuition. We can see directly from the defi- nition of curvature that the curvature of a straight line is always 0 because the tangent vector is constant. sariq Although Formula 9 can be used in all cases to compute the curvature, the formula given by the following theorem is often more convenient to apply. 10 Theorem The curvature of the curve given by the vector function r is ABAUDI r'(t) × r"(t)vilcups is usady segnal in | \r'(t) ³¹
Transcribed Image Text:10 hansemenige od nas grups 9 SO dit adi mori But ds/dt = |r'(1)| from Equation 7, so is Therefore 15.11 EXAMPLE 3 Show that the curvature of a circle of radius a is 1/a. DOLLS DRIT D THIN SOLUTION We can take the circle to have center the origin, and then a r(t) = a cos ti + a sin tj and and r'(t) = -a sin ti + a cos tj CT()| OF ROL k(t) = T(t) = tegen berupor ed r'(t) [r'(t)} por el ba T'(t) = -cos ti - sin tj Chos- (0)5 This gives |T'(t) = 1, so using Formula 9, we have = - sin ti + cost j Boome bellso ai (1)|T'(t) | k(t) = dioome balls |r'(t) | k(t) = 1 VALE SUNDE a | r'(t)| = a parametrization betegs two of Ivary S The result of Example 3 shows that small circles have large curvature and large circles have small curvature, in accordance with our intuition. We can see directly from the defi- nition of curvature that the curvature of a straight line is always 0 because the tangent vector is constant. sariq Although Formula 9 can be used in all cases to compute the curvature, the formula given by the following theorem is often more convenient to apply. 10 Theorem The curvature of the curve given by the vector function r is ABAUDI r'(t) × r"(t)vilcups is usady segnal in | \r'(t) ³¹
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