1. Let f:R R be defined by {1 if x rational number, 12 if x irrational number f(x) = Show that f is discontinuous at all x in R.

Q: 7- The function f(z) = z is: a) Analytic on C b) Analytic only at zo = 0 c) Not analytic

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Q: 2. If p and q are positive integers, f is a function defined for positive numbers and attains only…

A: Introduction :

Q: Define functions H and K from R to R by the fol- lowing formulas: For every x ER, H(x) = (x– 2)² and…

A: Observe the given functions as follows.

Q: Let f(x) = cos (x). %3D If it is true that f" (x) = x(f'(x))ª for some constant a, find the value of…

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Q: 3. Compute the value of F.dr if F = -yi + xj and C is a semicircle from (-1,1) to (1,-1).

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Q: 8. Given the following function f, define f(1) so that f is con- tinuous at x = 1, if possible: Зх —…

A: Given: fx=3x-1if x<1x2+1if x>1

Q: 2. (a) Suppose that f is a differentiable function such that |f'(x)| < c for every x. Show that for…

A: To Prove: a. If f is a differentiable function such that f'(x)≤c for every x.Then for any real…

Q: Suppose that f is differentiable on some open interval containing c, and (c,f(c)) is a point of…

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Q: Show that if f:R → R is odd and the restriction of fto the interval [0, co) is strictly decreasing,…

A: The odd functions are functions that return their negative inverse when x is replaced with –x. This…

Q: What, if any, are the removable discontinuities of f(x) = ? How do you remove them?

A: Give that: The function f(x) = 1x .

Q: 8. Find all the points in the interval [1,5] for f(x) = -x² + 6x – 6 such that the function verifies…

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Q: 2. ASSUME f(x) IS CONTINUOUS AND DIFFERENTIABLE ON AN OPEN INTERVAL AS GRAPHED. WHICH OF THESE…

A: Since you have posted multiple question in this question so I have solved first question as per our…

Q: 16. Suppose that f is differentiable at c and that f'(c) + 0. Show that there exists a ð > 0 such…

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Q: If we apply Rolle's Theorem to the function f(x) = 2x² – 16x – 5 on the interval [2, 6], how many…

A: Given,          fx=2x2-16x-5 on the interval 2,6.We know that,         Rolle's theorem states that…

Q: 18. Suppose that f is a differentiable function such that f(g(x))= x, and f'(x)= |3D 1 1+(f(x))².…

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Q: Let f be a function defined on JO, 1[ by : | 0, if x is irrational f(x) ={1/q, if x=p/q; where p and…

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Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

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Q: Which is the correct conclusion about the function h(x) ={V4- x2 if x - 3 if h has a removable…

A: Given function is: hx=-x-2    ;  if x&lt;-24-x2     ;  if -2≤x&lt;2x-3  ;  if x&gt;2 We have to…

Q: Use Theorem 2 to show that 3, if x <-10 2, if – 10 sxso F:R → R,F(x) = -2, if 0< x< 10 -3, if x2 10…

A: so we have to show that given function is discontinuous at 2 and -2 (a)   Discontinuous  at -2…

Q: Prove that the following function is continuous at 0. x, if x is rational f(x) = {0, if 0, if x is…

A: Given function is f(x) = x,if x is rational0,if x is irrational

Q: 3.Show, by means of the example f(t) = sin t, that f ∗ f is not necessarily nonnegative.

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Q: Define the following real function: f(x) = 2 - x, x is rational and f(x) x2, x is irrational. What…

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Q: Define f:R → R by f (x) = 5x if x is rational and f (x) = x² + 6 if x is irrational. Prove that f is…

A: Given, f:R→R by f(x)=5x if f(x) is rational and f(x)=x2+6 if x is irrational.To prove that f is…

Q: 0 if x V2. Prove or disprove that f is continuous.

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Q: Let f, g:R R defined by if x 1 (x2 and g(x) = 3x + 2. %3D Find f-1 and g-1.

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Q: Suppose that f: R (0, ∞) and that f'(x) = f(x) ‡ 0. Prove that (f-¹)(x) = 1/x for x > 0.

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Q: Define functions f and g from R to R by the fol- lowing formulas: For every x E R, 2x° + 2x = 2x and…

A: It is given that, f(x) and g(x) are defined from R to R by the following formulas:

Q: Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. b) Prove that f is differentiable…

A: Given fx=x2x is rational0x is irrational To prove f is differentiable at exactly one point, namely…

Q: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

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Q: For the function f (x) = |x| defined on [-1,1], there is some -1<c<1such that f' (c) = 0. O True…

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Q: Suppose that the function f: [0, 1]  in R is continuous and thatf (x)&gt;= 2     if 0 &lt;= x &lt;…

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Q: Suppose that a function h satisfies −x^2 ≤ h(x) ≤ x^2 for all x. Prove that h is differentiable at x…

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Q: If f : [0, 1] → R is defined as if æ is rational f(x) = 1-x if a is irrational. Ecuss the Riemann…

A: Here the f is bounded on [0, 1]. Let Pn be the partition of [0, 1] defined by Pn = (x0 , x1, x2, ……

Q: If f is a function defined from R to R, is given by f(x) = 5x - 3 then f -(x) is given by %3D -- 20…

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Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

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Q: X if x e Q If f (x) is defined on [0,1] by the rule f (x) = < (1-x , if x 4 Q Prove that fof (x) = x…

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Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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Q: (3) Consider the function defined as : R -> R>0 if laT 1 2 p(x) 0 if a> np(nx), and let Kn(x) be the…

A: Consider the given function which is defined as,

Q: Prove, if f(x)= V1+x² –In +1+ 1 is increasing and concave on x X E (0;+∞) .

A: f is increasing if first derivative is positive in the interval  and f has concave if second…

Q: If f(x) =x²-x over x-1 and g(x) =x is it true that f=g?

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Q: If we apply Rolle's Theorem to the function f(x) = 2x- 16x 6 on the interval [2, 6|, how many values…

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Q: If we apply Rolle's Theorem to the function f(x) = 2x² - 12x - 4 on the interval [1, 5], how many…

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Q: 1. Let, f(x) = sin x- cos x; [-T, 7]. Find: (a) the intervals on which f is increas- ing, (b) the…

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Q: 2. Assume additionally that f E C[a, b]. Show that there is E (a, b) such that | F(x)g(x) dæ = f(€)…

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Q: If we apply Rolle's Theorem to the function f(x) = 2x² - 8x - 1 on the interval [0, 4], how many…

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Q: 15. If f+g is differentiable at a, are f and g necessarily differentiable at a ? If f g and f are…

A: Let f be any function that is not differentiable  at the point a and define as: gx=-fx for all real…

Q: Give an example of a function f : R → R that is continuous at x if x is an irrational number, but it…

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Q: Suppose that f is analytic on a domain D andVz € D,f'(z) = af(z) for some constant a. Show that f(2)…

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Q: 5.11. Prove that the function f(z) = Arg(z) is discontinuous at each %3D point on the nonpositive…

A: Solution

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.