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1,2,3,4,5,6,7,8,9,10,11
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Слайды и текст к этой презентации:
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Содержание слайда: The Taylor Formula
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Содержание слайда: Question 1. If
Question 1. If
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Содержание слайда: Question 1. What is the greatest value of b for which any function f that satisfies the properties (i), (ii), and (iii) must also satisfy f (1) < 5?
Question 1. What is the greatest value of b for which any function f that satisfies the properties (i), (ii), and (iii) must also satisfy f (1) < 5?
(i) f (x) is infinitely differentiable for all x;
(ii) f (0) = 1, and
(iii) for all
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Содержание слайда: Question 2. Use the Taylor formula to show that e is irrational.
Question 2. Use the Taylor formula to show that e is irrational.
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Содержание слайда: Question 3. Find
Question 3. Find
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Содержание слайда: Example. The geometric series
Example. The geometric series
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Содержание слайда: Question 5. Which of the following series converge?
Question 5. Which of the following series converge?
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Содержание слайда: Question 5. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height rh, where 0 < r < 1.
Question 5. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height rh, where 0 < r < 1.
Suppose that the ball is dropped from an initial height of H meters.
a) Assuming that the ball continues to bounce indefinitely, find the total distance that it travels.
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Содержание слайда: A bouncing ball – total distance travelled
A bouncing ball – total distance travelled
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Содержание слайда: Question 7. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height r h, where 0 < r < 1.
Question 7. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height r h, where 0 < r < 1.
Suppose that the ball is dropped from an initial height of H meters.
b) Calculate the total time that the ball spends bouncing.
Hint: A ball having zero velocity falls ½ gt2 meters in t seconds.
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Содержание слайда: A bouncing ball – total bouncing time
A bouncing ball – total bouncing time
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Содержание слайда: Question 7. c) Suppose that each time the ball strikes the surface with velocity v, it rebounds with velocity – kv, where 0 < k < 1.
Question 7. c) Suppose that each time the ball strikes the surface with velocity v, it rebounds with velocity – kv, where 0 < k < 1.
How long will it take for the ball to come to rest?
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