🗊Презентация Mathematical Induction

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Mathematical Induction, слайд №1Mathematical Induction, слайд №2Mathematical Induction, слайд №3Mathematical Induction, слайд №4Mathematical Induction, слайд №5Mathematical Induction, слайд №6Mathematical Induction, слайд №7Mathematical Induction, слайд №8Mathematical Induction, слайд №9Mathematical Induction, слайд №10Mathematical Induction, слайд №11Mathematical Induction, слайд №12Mathematical Induction, слайд №13Mathematical Induction, слайд №14Mathematical Induction, слайд №15Mathematical Induction, слайд №16Mathematical Induction, слайд №17Mathematical Induction, слайд №18Mathematical Induction, слайд №19Mathematical Induction, слайд №20
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Mathematical Induction, слайд №1
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Question 0. A continuous function f is defined on the interval [−1,1], and f 2(x) = x 2 for each x from the interval [−1,1]. Question 0. A continuous function f is defined on the interval [−1,1], and f 2(x) = x 2 for each x from the interval [−1,1].
Описание слайда:
Question 0. A continuous function f is defined on the interval [−1,1], and f 2(x) = x 2 for each x from the interval [−1,1]. Question 0. A continuous function f is defined on the interval [−1,1], and f 2(x) = x 2 for each x from the interval [−1,1].

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Question 0+. A function f is defined on the interval [−1,1], and f 2(x) = x 2 for each x from the interval [−1,1]. Question 0+. A function f is defined on the interval [−1,1], and f 2(x) = x 2 for each x from the interval [−1,1].
Описание слайда:
Question 0+. A function f is defined on the interval [−1,1], and f 2(x) = x 2 for each x from the interval [−1,1]. Question 0+. A function f is defined on the interval [−1,1], and f 2(x) = x 2 for each x from the interval [−1,1].

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Mathematical Induction Let Sn, n = 1,2,3,… be statements involving positive integer numbers n. Suppose that 1. S1 is true. 2. If Sk is true, then Sk +1 is also true.
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Mathematical Induction Let Sn, n = 1,2,3,… be statements involving positive integer numbers n. Suppose that 1. S1 is true. 2. If Sk is true, then Sk +1 is also true.

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Question 1. Using the Principle of Mathematical Induction show that Question 1. Using the Principle of Mathematical Induction show that
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Question 1. Using the Principle of Mathematical Induction show that Question 1. Using the Principle of Mathematical Induction show that

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Mathematical Induction, слайд №6
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Mathematical Induction, слайд №7
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Mathematical Induction, слайд №8
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Question 1b. Using the Principle of Mathematical Induction show that Question 1b. Using the Principle of Mathematical Induction show that
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Question 1b. Using the Principle of Mathematical Induction show that Question 1b. Using the Principle of Mathematical Induction show that

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Mathematical Induction, слайд №10
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Question 3a. Calculate the following sum Question 3a. Calculate the following sum
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Question 3a. Calculate the following sum Question 3a. Calculate the following sum

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Question 5. Using the formula for the derivative of inverse function derive explicit formulae for the derivatives of arcsin x, arccos x, arctan x, and arccot x. Question 5. Using the formula for the derivative of inverse function derive explicit formulae for the derivatives of arcsin x, arccos x, arctan x, and arccot x.
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Question 5. Using the formula for the derivative of inverse function derive explicit formulae for the derivatives of arcsin x, arccos x, arctan x, and arccot x. Question 5. Using the formula for the derivative of inverse function derive explicit formulae for the derivatives of arcsin x, arccos x, arctan x, and arccot x.

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Mathematical Induction, слайд №13
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Question 6. Use the Cauchy criterion to show Question 6. Use the Cauchy criterion to show
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Question 6. Use the Cauchy criterion to show Question 6. Use the Cauchy criterion to show

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Mathematical Induction, слайд №15
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Picture of the Week
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Picture of the Week

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Question 4. Let f (x) be a differentiable function such that the derivative is a continuous function and f (f (x)) = x for any x. Furthermore, let f (0) = 1, and f (1) = 0. Question 4. Let f (x) be a differentiable function such that the derivative is a continuous function and f (f (x)) = x for any x. Furthermore, let f (0) = 1, and f (1) = 0. a) Is it possible that there exists a number a such that
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Question 4. Let f (x) be a differentiable function such that the derivative is a continuous function and f (f (x)) = x for any x. Furthermore, let f (0) = 1, and f (1) = 0. Question 4. Let f (x) be a differentiable function such that the derivative is a continuous function and f (f (x)) = x for any x. Furthermore, let f (0) = 1, and f (1) = 0. a) Is it possible that there exists a number a such that

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b) Is it possible that there exists a number a such that b) Is it possible that there exists a number a such that
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b) Is it possible that there exists a number a such that b) Is it possible that there exists a number a such that

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c) Let x1 be a solution of the equation f (x) = x. Find c) Let x1 be a solution of the equation f (x) = x. Find
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c) Let x1 be a solution of the equation f (x) = x. Find c) Let x1 be a solution of the equation f (x) = x. Find

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Mathematical Induction, слайд №20
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