(ii) Investigate the continuity of the following function f(2) ={ if z +0 if z =0 (iii) Is f(z) differentiable? (iv) Is f(z) analytic?

Q: Test the continuity of the following functions at the indicated points of each x < 1 (1) f(x) = { 3x…

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Q: Determine the intervals of continuity for the function: f(x) = over the interval of 0 < x < 2n…

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

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Q: Find the value of the constant k so that f(x) is continuous for all real values of x where f(x) is…

A: Solution

Q: 1. Examine the continuity and differential possibilities of function f (x)= |x² – 1| in x= 1.

A: Given To check for continuity at x=1 , we find the limit to the left and right of x=1

Q: Find the constant a such that the function is continuous on the entire real line. [ 5x², x 2 1 x < 1…

A: The given function is: f(x)=5x2   ,   x≥1ax-3,  x&lt;1 To find constant 'a' such that the function…

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

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Q: 1. Given that x +1 2x2-1 g(x) Show that the g(x) function is continuous at a = 4.

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Q: Discuss the continuity of the function. f(x, y, z) = 1/(√x2 + y2 + z2)

A: Given rational function is

Q: The function f(x) is everywhere continuous and differentiable. Prove that if the function f(x) has…

A: Given that f(x) is continuous and differentiable everywhere and f(x) has two real roots.

Q: Discuss the following theorem: If a function is differentiable at, then is continuous at. Provide…

A: Assume the point as x = c. Thus, if a function is differentiable at c , then is continues at c. Let…

Q: 1. What is the definition of what it means for f(x) to be continuous at x = a? Discuss the ways that…

A: In this question, we need to define the meaning of f(x) to be continuous at x=a and different ways…

Q: 1. For what value of k is the function h(x) continuous on the whole real line? HINT: Use definition…

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Q: Find the constant a such that the function is continuous on the entire real line. 4sin x if x < 0…

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Q: 1. f(x) = x2- 6x + 3 is continuous on the interval [- 3,2].

A: As per the company guidelines we can solve first question. I hope that the given solution will help…

Q: D. Discuss the continuity of the function g (x): 1 %3D on the interval [-1,2]. x2-4

A: Given, To discuss continuity of the function g(x)

Q: x+1 x 0 6-х the indicated test point. If it is discontinuous, specify the type of discontinuity.

A: We have to solve

Q: b. Test the continuity of the following function 1, V1-x,0sIs1, at x = 0,1 (x-2, x<0 x21

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Q: 2. Test continuity at x = 0 and x = 3 for the function (3 + 2x; - 3 3

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Q: x +1 1. Let f(x) = prove that f is continuous at 0. x – 1 -

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Q: x2 – 4x + 3 - f(æ) = 5. Let x - 3 How could we define f(3) x = 3? so that J is continuous at

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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: Question 1: Discuss the continuity of the function f (x) = xtan (÷) : x = 0 given that x + 0 and…

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Q: 19.D. Show that the function defined by g(x) g² sin (1/x), x * 0, %3D = 0, 0, is differentiable for…

A:

Q: 3. Determine whether f is differentiable or not differentiable at 1. (x – 1 if x > 1 (-x² _if x <1…

A: If a function is differentiable at any point x=a ,then its L.H.D. =R.H.D.

Q: 2. For the given function, f, determine if the function is discontinuous at a given point. State…

A: Function is f(x)=xx a=0 We have to check whether the function is continuous or discontinuous at…

Q: Discuss the continuity of the function f(r) at a = 0.

A:

Q: (b) The function f is defined by f(x) = V3 sin x + cos x Where x E R,-n <x< (iv) Define the largest…

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Q: How can you sometimes decide from examining ƒx and ƒy that a function ƒ(x, y) is differentiable?…

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Q: 4x-8 i- -2 x +3a

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Q: 6- if f(x) is a continuous function at x = 0, then a = 1- cos 4x x0 V16+ Vx -4 3.

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Q: (c) Simplify the function f(x) = |x - 1|+ |x + 1|. Test the continuity of this function at x 1. At…

A: f(x)=|x-1|+|x+1|, for all x∈ R, It can be written as f(x)=-(x+1)-(x-1),       x&lt;-1(x+1)-(x-1),…

Q: x2- 1 -1Sx<0 1- If f(x) 1 x=0 is it continuous at x=0 %3D 2х 0<x<1

A: Here the given function is  f(x)=x2-1if -1≤x&lt;01if x=02xif 0&lt;x≤1 We need to find weather the…

Q: Consider a function f continuous and differentiable on R with f(2. 5) = -3. 1 and f'(2. 5)=-0. 2.…

A: We have to find

Q: Discuss the continuity of the function g(x) = 1/(x^2-4) on the interval (0, 2] and [−1, 3].

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Q: Q2/B/ check the function continuity at x-0, for the fimction f(x)%= 1+ 4 2/3

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Q: For the function f(x)=sin(x)/x defined on [-1,1], there is some -1<c<1 such that f′(c)=0.f′(c)=0.…

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Q: Prove that the function f(x) = cos x is strictly decreasing on the interval [0,1]. %3!

A: 5) Given that, fx=cosx

Q: Let ff be a differentiable function such that f(-2)=9f(−2)=9 and f'(x)=x-2.f′(x)=x−2. What is the…

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Q: 1. Explain what it intuitively means for function f(x,y) to be continuous in some domain D.

A: To explain the meaning of the given statement.

Q: 5. Find the linearization L(x) of f at a = 0 f(x) = = 1 = 0 2+x

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Q: 6. Determine the value of the constant a for which the function g(x) is continuous at x = 1 x* + 5x…

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Q: 7. Find all values of x on the graph of f(x)=(x-1)´(2x+1) at which the tangent line is horizontal.…

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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: f(x)={x2−3x     x≤5          2x            x>5 (a) Is f continuous at x=5  ? yes, or no (b) Is f…

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Q: x? + 2x + 1, if x 1. is continuous at x = 1 or not by using the definition of continuity.

A: f(x) = x2+2x+1  , if x≤14-x           , x &gt;1for a function to be continious at x = alimx→a+f(x) =…

Q: Q. 1: Determine the func- tion is continuous or dis- continuous at x = -1 and x = 1.

A:

Q: Determine the continuity of the function at the given points. forx =-1 %3D f(x = - 2, for x -1 at x…

A: If function f is continuous at x=a then limx→af(x)= f(a)

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: