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
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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