Презентация - Mathematical Induction (20 слайдов)

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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].

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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].

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

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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 6. Use the Cauchy criterion to show Question 6. Use the Cauchy criterion to show

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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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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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