Function g:R → R, defined by g(x) =x*+5 is a one-to-one function. True False

Q: f(t) = 3 – 9 ? -6, [-3, 3]

A: To find: The continuity of the function f(t)=3-9-t2 on the interval [-3, 3]

Q: Suppose f is a function defined and continuous on the open interval (a, b). Suppose g is a function…

A: Given that: f is a function defined and continuous on the open interval (a, b).g is a function…

Q: Prove that f(x) = (x – 1)2 + (x + 1)? is an even function. A function k(x) is an even function if…

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Q: For a function f: A --> B and subsets C and D of A and E and F of B, prove the following: f-1(E -…

A:

Q: Let f, g, h be differentiable functions. Show that (f g h )' (x) is equal to f'(x)g(x)h(x) +…

A: To show: fgh'(x)=f'(x)h(x)g(x)+f(x)h'(x)g(x)+f(x)h(x)g'(x)   It is known, that if we have two…

Q: Exercise 1. Let f(x) be the function 2х — 1 if x 2. - (a) Find the value of c that makes f(x)…

A: Given f(x)=2x-1if x<-1cxif -1≤x≤2x-2if x>2

Q: Sketch the graph of a function f such that f (x) ≥ 0 on [0, 1] and f (x) ≤ 0 on [1, 2], whose…

A: Consider the function Then and The graph of the function is shown below,

Q: How to determine whether the function is continuous or not at a given point x = a ? x ≤ 1 (3x5 - 1…

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Q: Let g be a function that is defined for all x, x # 2, such that g(3)% = 4 and the derivative of g is…

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Q: Prove the Theorem: A real valued function on a closed, bounded interval is continuous iff its graph…

A: If we assume that f(x) is unbounded at c , then bisecting the interval [a,b] would give us two…

Q: The definition of the first derivative of a function f(x) is: Select one: f(x+Ax)-f(x) a. f'(x) =…

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Q: a delta - epsilon proof that shows that the function f(x) = |x| is continuous on its do

A: Definition:

Q: snip

A: A function f is decreasing on an open interval I, if for any choice of x1 and x2 in I, with…

Q: 9. Discuss the continuity of the function - a³, 0 a

A: We have to check continuity

Q: Discuss the continuity of the following function: 2* (1- x), x0 x < 0 x = 0

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Q: Let f(x)=x(4√x −3) Determine the absolute maximum and minimum values of f on the interval [0, 4]

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Q: 7 Find the most general function F such that F'(): - 5V?.

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Q: Prove that if (c, f(c)) is a point of inflection of the graph of f and f" exists in an open interval…

A: It is given that (c,f(c)) is a point of inflection of the function f.

Q: Find a function such that f'(-1) = 1/2, f'(0) = 0, and f''(x) &gt; 0 for all x, or prove that such a…

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Q: Let g be a function with domain the rational numbers, defined byg(x) = 2/x - √,2for rational x. Is g…

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Q: Let f be a real-valued function continuous at the point a = (a1, a2) in R². Keep a2 fixe and define…

A: Let f be a real-valued function continuous at the point a=(a1,a2). Define g as g(x)=f(x,a2), where…

Q: Compute f(-4) for the function f defined below. x <0 f(x) = 1x2 - 9, x20 %3D

A: Given Function : f(x)=6xx&lt;0x2-9x≥0

Q: (a): Verify Mean value theorem for the function f(x) = 3x³ + x + 4 over the interval 0, 1].

A: “Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Let g be a function that is defined for all x, x 2, such that g(3) = 4 and the derivative of g is x²…

A: The interval where second derivative of function is positive, is the interval where the function is…

Q: (a): Verify Mean value theorem for the function f(x) = 6x3 + x + 2 over the interval [0, 1].

A:

Q: A function for which f(x) = f( - x) for all x in the domain of f is called a(n) ____function.

A: The types of functions help to fill the given blank. The definition that consists of the terms f(x)…

Q: Does there exist a function f such that f(0)  =-1, f(2)  =4, and f'(x) ≤ 2  for all  x?

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Q: Discuss the continuity of the function on the closed interval.                               h(x) =…

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Q: Discuss the continuity of the composite function h(x) = f(g(x)).

A: To discuss: The continuity of the function hx=fgx. Given: The functions are  fx=tan x and gx=x2.…

Q: (a): Verify Mean value theorem for the function f(x) = 2x³ + + 3 over the interval [0, 1].

A: Hair we use the result of mean value theorem for the given interval and given function.

Q: Prove that f(x) = (x- 1)2 + (x + 1)? is an even function. A function k(x) is an even function if…

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Q: Let f (x) = e-*x². Use calculus (derivatives) to find the absolute maximum and absolute minimum of f…

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Q: 3- x, 3. f(x) = [-1,4] 3+x, x > 0 1 4. g(x) [-1, 2] .2 - 4

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Q: A piecewise function f is defind by [1-x, x1.

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Q: Give an example of a function R -> R that is discontinuous with the usual topology but is continuous…

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Q: Find a function such that f'(-1)=1/2, f'(0)=0, and f''(x)&gt;0 for all x, or prove that such a…

A: Given that f"(x)&gt;0 for all x and f'(-1)=1/2, f'(0)=0 Since f"(x) exists for all x, so we can say…

Q: Test the continuity of the function f(x): 3 — х X +1 2<x <3 3< X< 4 F(x) = 2х — 1 X= 4

A: polynomial functions is everywhere continuous except its branch point , hence F(x) is everywhere…

Q: Let f be a real-valued function on R. Suppose that f(−1) = 3 and f'(x) ≤ 2 for all values of x. How…

A: f(-1)=3f'(x)≤2Integrating w.r.t x;∫ f'(x) dx ≤ ∫2 dxf(x)≤2x+Cf(-1)=33=2(-1)+CC=3+2C=5

Q: Show that Rolle’s theorem is false for the function f(x) = |x| on the interval [−2, 2]. How can a…

A: Given function f(x)=|x|interval [-2,2] We apply Rolle's theorem only when the function is continuous…

Q: Discuss the continuity of the function on the closed interval.                        Function g(x)…

A: Givengx=1x2−4  Interval [-1,2]

Q: Check the validity of Cauchy’s mean value theorem for the functions f(x)=x4 and g(x)=x2 on the…

A: We need to check the validity of Cauchy's Mean Value

Q: Let g be a function that is defined for all x, x # 2, such that g(3)% = 4 and the derivative of g is…

A: To find: Critical points of g where g'(x)=x2-16x-2,x≠2.

Q: Suppose that f:I->R is a continuous function and that |f| is a constant function. Prove that f is a…

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Q: If f is differentiable and f'(x)=x³-2x²-x+2, then fis increasing or

A: Given, x³−2x²−x+2=f' We equate the given function to zero for critical points. x³−2x²−x+2=0   Step…

Q: IV. Verify Rolle's theorem for the function: f(x) = x2 – 5x + 6 on [2,3] а.

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Q: How did the function f(x) = -x^3 + e^x - 8 satisfy the intermediate value theorem? Show that f has a…

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Q: The function of f is increasing on I if ______ when x1 < x2.

A: We have to find when function is increasing

Q: Verify the First Mean-Value Theorem for the function g(x) = x + 1 on the interval [1, 2]

A: g(x) = x + 1 is continuous in the interval [1, 2] It's differentiable in the interval (1, 2)

Q: Draw a function F(x), defined for all real numbers x, that satisfies the following properties.…

A: The first derivative is everywhere positive and the second derivative is everywhere negative

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!