(a Wilat Is me lengtiI UI the track? (6) Which student is running faster? Compute the distance covered by each student. 5. Consider the curve that is the graph of the function y = x2/3 on [-1, 1]. (a) Show that y is not differentiable at 0. Conclude that the parametrization c(t) = (t, t), te [-1, 1], is not differentiable. (b) Show that c(t) = (cos' t, cos2 t), te [-n, T], is a differentiable parametrization of the given %3D %3D curve. (c) Prove that the parametrization in (b) is not smooth. 6. Prove that the statement we made in Example 3.27 is true; that is, by "unfolding" the helix, we obtain a straight-line segment (see Figure 3.27). Exercises 7 to 14: Find the length of the path c(t). 7. c(t) = (sin 2t, cos 2t), t E [0, 5] 9. c(t) = e' cos ti + e' sin tj, 0 < t < n 8. c(t) = (2t3/2 , 2t), from (0, 0) to (2, 2) 10. c(t) = t³i +t²j, –2 < t < l %3D %3D %3D 11. c(t) = ((1+t), (1+t)³/2), 1 e [0, 1] 12. c(t) = (e2' e-2ª, /8t), t e [0, 1] 13. c(t) = (2t – t²)i + 13/2j +k, from t = 1 to t = 3 14. c(t) = cos? ti + sin? tj, from t = 0 to t = 2n %3D 15. Show that the length of a logarithmic spiral c(t) = (e“ cos t, eat sin t), where a < 0 and t > 0, is finite. %D %3D 16. Find the length of the catenary curve given by c(t) = (t, a cosh (t/a)), a > 0, te [-a, a]. 17. Is it true that the curve y = 2 sin x, x E [0, 27 ] is twice as long as y = sin x, x e [0, 27 ]?
(a Wilat Is me lengtiI UI the track? (6) Which student is running faster? Compute the distance covered by each student. 5. Consider the curve that is the graph of the function y = x2/3 on [-1, 1]. (a) Show that y is not differentiable at 0. Conclude that the parametrization c(t) = (t, t), te [-1, 1], is not differentiable. (b) Show that c(t) = (cos' t, cos2 t), te [-n, T], is a differentiable parametrization of the given %3D %3D curve. (c) Prove that the parametrization in (b) is not smooth. 6. Prove that the statement we made in Example 3.27 is true; that is, by "unfolding" the helix, we obtain a straight-line segment (see Figure 3.27). Exercises 7 to 14: Find the length of the path c(t). 7. c(t) = (sin 2t, cos 2t), t E [0, 5] 9. c(t) = e' cos ti + e' sin tj, 0 < t < n 8. c(t) = (2t3/2 , 2t), from (0, 0) to (2, 2) 10. c(t) = t³i +t²j, –2 < t < l %3D %3D %3D 11. c(t) = ((1+t), (1+t)³/2), 1 e [0, 1] 12. c(t) = (e2' e-2ª, /8t), t e [0, 1] 13. c(t) = (2t – t²)i + 13/2j +k, from t = 1 to t = 3 14. c(t) = cos? ti + sin? tj, from t = 0 to t = 2n %3D 15. Show that the length of a logarithmic spiral c(t) = (e“ cos t, eat sin t), where a < 0 and t > 0, is finite. %D %3D 16. Find the length of the catenary curve given by c(t) = (t, a cosh (t/a)), a > 0, te [-a, a]. 17. Is it true that the curve y = 2 sin x, x E [0, 27 ] is twice as long as y = sin x, x e [0, 27 ]?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Can you help with 7 and 15 please?
![(a
Wilat Is me lengtiI UI the track?
(6) Which student is running faster? Compute the distance covered by each student.
5. Consider the curve that is the graph of the function y = x2/3 on [-1, 1].
(a) Show that y is not differentiable at 0. Conclude that the parametrization c(t) = (t, t),
te [-1, 1], is not differentiable.
(b) Show that c(t) = (cos' t, cos2 t), te [-n, T], is a differentiable parametrization of the given
%3D
%3D
curve.
(c) Prove that the parametrization in (b) is not smooth.
6. Prove that the statement we made in Example 3.27 is true; that is, by "unfolding" the helix, we
obtain a straight-line segment (see Figure 3.27).
Exercises 7 to 14: Find the length of the path c(t).
7. c(t) = (sin 2t, cos 2t), t E [0, 5]
9. c(t) = e' cos ti + e' sin tj, 0 < t < n
8. c(t) = (2t3/2 , 2t), from (0, 0) to (2, 2)
10. c(t) = t³i +t²j, –2 < t < l
%3D
%3D
%3D
11. c(t) = ((1+t), (1+t)³/2), 1 e [0, 1]
12. c(t) = (e2' e-2ª, /8t), t e [0, 1]
13. c(t) = (2t – t²)i + 13/2j +k, from t = 1 to t = 3
14. c(t) = cos? ti + sin? tj, from t = 0 to t = 2n
%3D
15. Show that the length of a logarithmic spiral c(t) = (e“ cos t, eat sin t), where a < 0 and t > 0,
is finite.
%D
%3D
16. Find the length of the catenary curve given by c(t) = (t, a cosh (t/a)), a > 0, te [-a, a].
17. Is it true that the curve y = 2 sin x, x E [0, 27 ] is twice as long as y = sin x, x e [0, 27 ]?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff920003c-b80d-4d46-9199-437e1336b936%2F9cf02e31-3b14-4510-a122-943be0490189%2Fzdqfc1m.jpeg&w=3840&q=75)
Transcribed Image Text:(a
Wilat Is me lengtiI UI the track?
(6) Which student is running faster? Compute the distance covered by each student.
5. Consider the curve that is the graph of the function y = x2/3 on [-1, 1].
(a) Show that y is not differentiable at 0. Conclude that the parametrization c(t) = (t, t),
te [-1, 1], is not differentiable.
(b) Show that c(t) = (cos' t, cos2 t), te [-n, T], is a differentiable parametrization of the given
%3D
%3D
curve.
(c) Prove that the parametrization in (b) is not smooth.
6. Prove that the statement we made in Example 3.27 is true; that is, by "unfolding" the helix, we
obtain a straight-line segment (see Figure 3.27).
Exercises 7 to 14: Find the length of the path c(t).
7. c(t) = (sin 2t, cos 2t), t E [0, 5]
9. c(t) = e' cos ti + e' sin tj, 0 < t < n
8. c(t) = (2t3/2 , 2t), from (0, 0) to (2, 2)
10. c(t) = t³i +t²j, –2 < t < l
%3D
%3D
%3D
11. c(t) = ((1+t), (1+t)³/2), 1 e [0, 1]
12. c(t) = (e2' e-2ª, /8t), t e [0, 1]
13. c(t) = (2t – t²)i + 13/2j +k, from t = 1 to t = 3
14. c(t) = cos? ti + sin? tj, from t = 0 to t = 2n
%3D
15. Show that the length of a logarithmic spiral c(t) = (e“ cos t, eat sin t), where a < 0 and t > 0,
is finite.
%D
%3D
16. Find the length of the catenary curve given by c(t) = (t, a cosh (t/a)), a > 0, te [-a, a].
17. Is it true that the curve y = 2 sin x, x E [0, 27 ] is twice as long as y = sin x, x e [0, 27 ]?
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