III (1 + 1) 3. 0:[-8, 1] –→ [-2, 1], ¢(1) =1/3 2. 4:[-1, 1] –→ [0, 1], 0(t) = t² 4. p:[1,2] [0, 3], $(t) = t2 -1 %3D 5. 0: [0, 1] → (1, e], ¢(1) = e' 7. ¢:[-2, 1] –→ [0, 2], $(1) = || 6. 0:[-1, 1] [-1/4, n/4], 0(t) = arctan t %3D 8. Let c(t) = (t – 2, 3 – 1 – 12), 1 e [0, 1]. Is the reparametrization p: [0, 3] → [0, 1], given by 0(t) = 1-1/3, orientation-preserving or orientation-reversing? 9. Using the Mean Value Theorem, show that a differentiable function o: [a, B] → [a, b] is one-to- one if o'(t) > 0 for all t e (a, B) (or o'(t) < 0 for all t e (a, B)). 10. Explain why a continuous and bijective function o: [a, B] → [a, b] must map endpoints to endpoints. Show that this statement is no longer true if o is not continuous. Exercises 11 to 16: Check whether the curve c(t) is simple or not, closed or not, simple closed or not. 11. c(t)= (sin t, cos t, (t- 27)²), t e [-27, 67] 12. c(t) = (sin t, cos t, (t- 27)2), t e (-27, 47] 13. c(t) = (t sin t, t cos t), t e [0, 27] 14. c(t) = (sin 2t, t cos t), t e [0, 7/2] %3D 15. c(t) = (t –1-1, 1 +t-'), 1 E [1, 2] 16. c(t) = (t² – t, 3 – V12 – t), t e [0, 1] %3D 17. Find a parametrization of the part of the curve y = Vx² + 1 from (-1, /2) to (1, /2). Is your parametrization continuous? Differentiable? Piecewise C? C'? 18. Find a parametrization of the curve x2/3+y2/3 = 1. Is your parametrization continuous? Dif- ferentiable? Piecewise C'? C'? %3D 19. Consider the following parametrizations of the straight-line segment from (-1, 1) to (1, 1). State which parametrizations are continuous, piecewise C' and C'. (a) c1(1) = (1, 1), –1 <1< 1 (-72, 1) if -1<1<0 (12, 1) (b) c2(t) = if 0
III (1 + 1) 3. 0:[-8, 1] –→ [-2, 1], ¢(1) =1/3 2. 4:[-1, 1] –→ [0, 1], 0(t) = t² 4. p:[1,2] [0, 3], $(t) = t2 -1 %3D 5. 0: [0, 1] → (1, e], ¢(1) = e' 7. ¢:[-2, 1] –→ [0, 2], $(1) = || 6. 0:[-1, 1] [-1/4, n/4], 0(t) = arctan t %3D 8. Let c(t) = (t – 2, 3 – 1 – 12), 1 e [0, 1]. Is the reparametrization p: [0, 3] → [0, 1], given by 0(t) = 1-1/3, orientation-preserving or orientation-reversing? 9. Using the Mean Value Theorem, show that a differentiable function o: [a, B] → [a, b] is one-to- one if o'(t) > 0 for all t e (a, B) (or o'(t) < 0 for all t e (a, B)). 10. Explain why a continuous and bijective function o: [a, B] → [a, b] must map endpoints to endpoints. Show that this statement is no longer true if o is not continuous. Exercises 11 to 16: Check whether the curve c(t) is simple or not, closed or not, simple closed or not. 11. c(t)= (sin t, cos t, (t- 27)²), t e [-27, 67] 12. c(t) = (sin t, cos t, (t- 27)2), t e (-27, 47] 13. c(t) = (t sin t, t cos t), t e [0, 27] 14. c(t) = (sin 2t, t cos t), t e [0, 7/2] %3D 15. c(t) = (t –1-1, 1 +t-'), 1 E [1, 2] 16. c(t) = (t² – t, 3 – V12 – t), t e [0, 1] %3D 17. Find a parametrization of the part of the curve y = Vx² + 1 from (-1, /2) to (1, /2). Is your parametrization continuous? Differentiable? Piecewise C? C'? 18. Find a parametrization of the curve x2/3+y2/3 = 1. Is your parametrization continuous? Dif- ferentiable? Piecewise C'? C'? %3D 19. Consider the following parametrizations of the straight-line segment from (-1, 1) to (1, 1). State which parametrizations are continuous, piecewise C' and C'. (a) c1(1) = (1, 1), –1 <1< 1 (-72, 1) if -1<1<0 (12, 1) (b) c2(t) = if 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you help with 12 and 14 please?
![III (1 + 1)
3. 0:[-8, 1] –→ [-2, 1], ¢(1) =1/3
2. 4:[-1, 1] –→ [0, 1], 0(t) = t²
4. p:[1,2] [0, 3], $(t) = t2 -1
%3D
5. 0: [0, 1] → (1, e], ¢(1) = e'
7. ¢:[-2, 1] –→ [0, 2], $(1) = ||
6. 0:[-1, 1] [-1/4, n/4], 0(t) = arctan t
%3D
8. Let c(t) = (t – 2, 3 – 1 – 12), 1 e [0, 1]. Is the reparametrization p: [0, 3] → [0, 1], given by
0(t) = 1-1/3, orientation-preserving or orientation-reversing?
9. Using the Mean Value Theorem, show that a differentiable function o: [a, B] → [a, b] is one-to-
one if o'(t) > 0 for all t e (a, B) (or o'(t) < 0 for all t e (a, B)).
10. Explain why a continuous and bijective function o: [a, B] → [a, b] must map endpoints to
endpoints. Show that this statement is no longer true if o is not continuous.
Exercises 11 to 16: Check whether the curve c(t) is simple or not, closed or not, simple closed or
not.
11. c(t)= (sin t, cos t, (t- 27)²), t e [-27, 67]
12. c(t) = (sin t, cos t, (t- 27)2), t e (-27, 47]
13. c(t) = (t sin t, t cos t), t e [0, 27]
14. c(t) = (sin 2t, t cos t), t e [0, 7/2]
%3D
15. c(t) = (t –1-1, 1 +t-'), 1 E [1, 2]
16. c(t) = (t² – t, 3 – V12 – t), t e [0, 1]
%3D
17. Find a parametrization of the part of the curve y = Vx² + 1 from (-1, /2) to (1, /2). Is your
parametrization continuous? Differentiable? Piecewise C? C'?
18. Find a parametrization of the curve x2/3+y2/3 = 1. Is your parametrization continuous? Dif-
ferentiable? Piecewise C'? C'?
%3D
19. Consider the following parametrizations of the straight-line segment from (-1, 1) to (1, 1). State
which parametrizations are continuous, piecewise C' and C'.
(a) c1(1) = (1, 1), –1 <1< 1
(-72, 1) if -1<1<0
(12, 1)
(b) c2(t) =
if 0 <t < 1
(d) c4(t) = (t³, 1), -1<1<1
(c) c3(t) = (t/3, 1), -1 <1<1
%3D
(1, 1)
if - 1<t<0
(e) c5(t) =
(1-1, 1) if 0 <t < 1
20. Consider the curve c in R² given by c(t) = (1, 1²), t e [-1, 2]. State which of the following
maps o are reparametrizations of c. Describe the curve c(o(t)) for those o that are reparametrizations:
%3D
5(t) 32t
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff920003c-b80d-4d46-9199-437e1336b936%2F65783297-47d2-45e3-a13c-4c031bbf300e%2Fpa5a33p.jpeg&w=3840&q=75)
Transcribed Image Text:III (1 + 1)
3. 0:[-8, 1] –→ [-2, 1], ¢(1) =1/3
2. 4:[-1, 1] –→ [0, 1], 0(t) = t²
4. p:[1,2] [0, 3], $(t) = t2 -1
%3D
5. 0: [0, 1] → (1, e], ¢(1) = e'
7. ¢:[-2, 1] –→ [0, 2], $(1) = ||
6. 0:[-1, 1] [-1/4, n/4], 0(t) = arctan t
%3D
8. Let c(t) = (t – 2, 3 – 1 – 12), 1 e [0, 1]. Is the reparametrization p: [0, 3] → [0, 1], given by
0(t) = 1-1/3, orientation-preserving or orientation-reversing?
9. Using the Mean Value Theorem, show that a differentiable function o: [a, B] → [a, b] is one-to-
one if o'(t) > 0 for all t e (a, B) (or o'(t) < 0 for all t e (a, B)).
10. Explain why a continuous and bijective function o: [a, B] → [a, b] must map endpoints to
endpoints. Show that this statement is no longer true if o is not continuous.
Exercises 11 to 16: Check whether the curve c(t) is simple or not, closed or not, simple closed or
not.
11. c(t)= (sin t, cos t, (t- 27)²), t e [-27, 67]
12. c(t) = (sin t, cos t, (t- 27)2), t e (-27, 47]
13. c(t) = (t sin t, t cos t), t e [0, 27]
14. c(t) = (sin 2t, t cos t), t e [0, 7/2]
%3D
15. c(t) = (t –1-1, 1 +t-'), 1 E [1, 2]
16. c(t) = (t² – t, 3 – V12 – t), t e [0, 1]
%3D
17. Find a parametrization of the part of the curve y = Vx² + 1 from (-1, /2) to (1, /2). Is your
parametrization continuous? Differentiable? Piecewise C? C'?
18. Find a parametrization of the curve x2/3+y2/3 = 1. Is your parametrization continuous? Dif-
ferentiable? Piecewise C'? C'?
%3D
19. Consider the following parametrizations of the straight-line segment from (-1, 1) to (1, 1). State
which parametrizations are continuous, piecewise C' and C'.
(a) c1(1) = (1, 1), –1 <1< 1
(-72, 1) if -1<1<0
(12, 1)
(b) c2(t) =
if 0 <t < 1
(d) c4(t) = (t³, 1), -1<1<1
(c) c3(t) = (t/3, 1), -1 <1<1
%3D
(1, 1)
if - 1<t<0
(e) c5(t) =
(1-1, 1) if 0 <t < 1
20. Consider the curve c in R² given by c(t) = (1, 1²), t e [-1, 2]. State which of the following
maps o are reparametrizations of c. Describe the curve c(o(t)) for those o that are reparametrizations:
%3D
5(t) 32t
%3D
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