Let f be the function defined by: sin x, x < O x2, 0 2 - For what values of x is f NOT continuous?

Q: Let f(x) = sin(sin(sin.x)) for for all x in R (f(x) x-0 g (x), and g(x) = (sinx), VxER, then lim is…

A:

Q: Let f(x) be given by the function f(x) = { 9(x)+a, 13 – b cos x, where a and b are constants x > 0'…

A: To show that f(x) is continuous at x=0

Q: Determine the values of a, b, and c so that the function (sin(a + 1)x + sin x x0 is continuous for…

A:

Q: Let the function ƒ : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. %3D 1 Prove that f is…

A: The solution is given as

Q: Q2. Let f:R- R be a function such that f(x) = x sin| f is defined at x, , where the value of d…

A: Topic- Limits

Q: Consider the functions f(x) = cos(x) and g(x) = x--1 over the interval

A: Definite integration can be used to find the volume of a solid. A small volume element is integrated…

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…

A:

Q: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

Q: Let f and g be continuous on the closed interval (0, 5). Suppose that f(x)dx = A, f(x)dx =B, and…

A:

Q: Determine if the function f below is continuous at x = 1: [1– VI– x, x 1

A: We have to find

Q: Let xª sin , x + 0; f(x) ={ 0, x = 0, where a > 0 is a parameter. 1. For which values of a is f(x)…

A:

Q: 1. Let the function f: R R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Differentiablity analysis problem....

Q: Find the values of a and b that make f(x) continuous for all real x. be* + a + 1, ax² + b(x +3), a…

A: The function f(x) is said to be continuous if the left-hand limit is equal to the right-hand limit…

Q: Let f be a function which is continuous on the interval [a, b]. The definite integral of f on [a, b]…

A: n is nothing but (b-a)/∆X mean,    n=(b-a)/∆X

Q: Let f be the function defined by: sin x, x <0 x2, 0 <x < 1 2 - x, 1<x < 2 x-3, x22 f(x) = %3D For…

A:

Q: Find a and b such that f(x) is continuous at x=1. [x – 2] + a x 1

A: A detailed solution is given below.

Q: Is the function continuous on the interval 0, 27? 1+(sin z)? O yes O no cannot be determined

A:

Q: Find a value of x, if any, at which f (x) = Vx – 125 is not continuous. If f(x) is continuous for…

A:

Q: 2. Suppose f20, f is continuous on [a, b], and f(x) dx = 0. Prove that f(x) =0 for all x e [a, b).…

A:

Q: Determine the domain of the function and prove that it is continuous on its domain using Theorems…

A: Consider the given function: fx=xsinx Here, the objective is to determine the domain of the function…

Q: Discuss whether the following functions are continuous at the specified point(s). sin(x) (a) f(x) =…

A: To determine whether the following functions are continuous at the specified points or not. (a)…

Q: Define h(x) = x³ sin(1/x) for x + 0 and h(0) = 0. Prove that h' is not differentiable at x = 0. .

A:

Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

A:

Q: Consider the following piecewise defined function: sin(ax) -1 < x < 0 4x f(x) = a2 –(b+1)x+b x²+x-2…

A: Given function is f(x)=sin(ax)4x,-1≤x&lt;0x2-(b+1)x+bx2+x-2,0≤x&lt;1x2-1+a ,1≤x≤2…

Q: Let f be a continuous function on R such that f(x+y)=f(x)+(y) ¥x, yER. Show that f(x)3cx for some…

A:

Q: Let f(x) = x?sin (-) for x + 0 and f(0) = 0. c) Show that f'is not continuous at x = 0.

A: Given function  f(x)=x2 sin1x      for x≠00                  for x=0 we have to show that f' is not…

Q: Let f: RR be a function satisfying f(x + y) = f(x) + f(y) for all x, ye R. If the function f(x) is…

A: Note :   Let r&gt;0 and c∈R ,then vr(c)={x∈R : x-c&lt;r }=(c-r ,c+r) ,which is called r…

Q: a. Let f(x) be a function satisfying |f(x)| < x² for –1 < x < 1. Show that f is differentiable at x…

A:

Q: Let f(x) = sin(rx). Suppose that h and k are two functions such that H (x) = cos(xx), k'(x) = f(x+…

A: Given that fx = sinπx . Also given that h and k are two functions such that h'x = cosπx , k'x = fx+1…

Q: Let f be the function defined by f (x) = x cos x – sin x. What is the absolute maximum value of f on…

A: The given function is: f(x)=x cos(x)-sin(x) ;                   x∈-π2, 2π

Q: Which of the following functions is continuous on the interval (-0, +)? …* f(x) = Vx² – 1 (A f(x) =…

A:

Q: Let f be the function defined by f (x) = r ln I for ¤ > 0. On what open interval is f decreasing? A…

A: Explanation of the answer is as follows

Q: Find all values of the constant a such that f is continuous for all real numbers. ах x 2 0 f(x): tan…

A:

Q: Find the constant a such that this function g(x) is continuous on the entire number line.

A: Given,

Q: ind the absolute maximum and minimum values of f on the given interval. a) f(x) = on [0, 2] g2+1 on…

A:

Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

A:

Q: Let f: (0,0) → (0, 0) be a function defined by f (x) =- for all x E (0, ∞). Prove that fis…

A:

Q: Prove that the function f(x)=sinπ|x| is continuous at x = 0 but not differntiable at the same point.

A:

Q: Consider the function f (x) = cos(x) for x ≤ c f (x) = ax +b for x > c (i) Give the condition on…

A: Solve the following

Q: ) For a, 3 ER and B > 0, let f(x) ={} x° sin(|x|-#), when x # 0 when x = 0 0, ange of a and B (and…

A: Continuity and differentiability

Q: (2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies…

A:

Q: B/ For what value of a is fa continuous at every x? x2-1, x<3 fo 2ах, x2 3

A:

Q: A function f is defined by f (x1) = / Cos(x – t) os(x-t) dt cos(t) f (x) : 01 Simplify f(x) and…

A:

Q: Let f(x) be continuous on [0, oo) and twice differentiable on (0, co). If f'(x) > 0 and f"(x) 1,…

A: Let f(x) be a continuous function on [0,∞) and differentiable at 0,∞. The given double derivative…

Q: 3. Consider the function g(x) defined by { x² sin(1/x) if x > 0, g(x) = if x <0. (a) Show that g(x)…

A:

Q: sin(bx) x0. Find a value of b such that f is continuous, or explain why no such value exists.

A: Given, f(x)=sin(bx)xx&lt;0exp(x+5)x≥0