Q1) check the continuity of fX) at X--1, 0,1.2,3 -1

Q: Suppose that f(x, y) = at which %3D - {(x, y) | 3 < x < 4, 3 < y < 5}.

A: we have ,  f(x,y)=xy at which (x,y)|3≤x≤4,3≤y≤5 we have to calculate,  ∫∫Df(x,y)dxdy

Q: Suppose that f: [0,1]–R is a function satisfying the conclusions of the intermediate value theorem.…

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Q: Show that the functi on When x g 0 f(x)= +1 0 . When x = 0 is discontinuous at x = 0.

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Q: Define the function f(x, v) = 4x³-x³y-4y5 3x5-3y5 at the origin so that it becomes continuous there.

A: A function is said to be continuous at (a,b) if the the value of function at (a,b) is equal to…

Q: Check the differentiability of the function f(x, y) = √xy | at the point (0,0). Also check the…

A: As per the question we are give a bivariate function : f(x,y) = √|xy| And we have to check for its…

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

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Q: Show that f(x, y) = point (5,-3). 3x+2y x+y+1 meets the three conditions for continuity at Add-ins…

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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: Show that f(x) = is uniformly continuous by definition on the interval (-1,1) . 4 - *

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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: Use the comparison theorem with f(x) = = 1 √x to show that [₂²= 1 √x-1 is divergent.

A: See this solution in below

Q: Let f(x) = x²+x+ 2 = Z₂ (x) prove that f(x) irreduciable Poly

A: The given function is  f(x)=x2+x+2.

Q: Define f(x)=x^3 for all x. Verify the ε-δ criterion for continuity at each point x_0.

A: X3 is continuous at every point in R.

Q: Let f(r) = 1-5r2. Use the limit definition of the derivative to show that f'(z) = -10r. %3D

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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: prove using the IVT. √2 exists. Hint: Consider f (x) = x2.

A: we have to show √2 exists. Now let

Q: 3. f(x) = v2x – 5 at x = 3 -

A: (3) Given function is fx=2x−5 We have check whether the function fx is continuous or not at x=3 For…

Q: Verify that the Intermediate Value Theorem applies to the indicated interval and find the calue of c…

A: Given function is

Q: 21. Discuss the continuity of: for for 0sx 2

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Q: Q2) Discuss the continuity at x=2. When: x> 2 x = 2 3x + 1 f(x) : x² + 3 %3D 2x + 1 x< 2

A: The given functionf(x)=3x+1x&gt;2x2+3x=22x+1x&lt;2to check continuity at x=2

Q: What is the continuous extension of f(x) = x3-1 at point x-1. %3D x2-1'

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Q: ) Verify the continuity of the function f(x, y) 26– 2x -- y at the point (2,-3).

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Q: (b). Find the bound for absolute error at x = 2.9 on the interval [x, x,], where x, = 2.25, x, =…

A: On the interval x0,xn where x0=2.25, x1=2.5, x2=2.75, x3=3 with n=3 along with the function…

Q: Discuss the continuity of f(x) = {Vx, x > 1 1, x<1 at x = 1.

A: A function f is said to be continuous at some point a if the value of the function at that point is…

Q: Verify that Rolle's Theorem can be applied to the function f (z) =2³ – 10z2 +27z – 18 on the…

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Q: Suppose that ƒ(x) = x2 and g(x) = | x | . Then thecompositions( ƒ ∘ g)(x) = | x | 2 = x2 and (g ∘…

A: Given: The composite of the two functions are given as,

Q: Find all values of a and b that make -а - х, x<-2 f(x) = {bx? + 1, Vx+b, -2<x<1 x21 continuous…

A: The function is given by: f(x)= -a-x , x&lt;-2           bx2+1 , -2≤x&lt;1         x+b   , x≥1 Also…

Q: Prove that f (x) = x2 takes on the value 0.5 in the interval [0, 1].

A: Here given

Q: 1. Let f(x) = 4x − 4x 2 . Verify that f(x) satisfies the conditions of Rolle’s Theorem on the…

A: Note:- According to the company policy we can attempt ?️ only one question so please upload the next…

Q: (b) Verify mean value theorem for f(x)=x² - 5x+6 over the interval [1,3].

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Q: Suppose that f(1) = 5, ƒ(4) = -8, f'(1) = 4, f'(4) = -2, and f" is continuous. Find the value of |…

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Q: 4. Verify Rolle's Theorem for f(x) = x² - 2x +1 on [-1,3]

A: Given the function f(x)=x2−2x+1 On the interval [−1,3]

Q: Q2) Discuss the continuity at x=2. When: x> 2 X = 2 x< 2 Зх + 1 f(x) = } x? + 3 %3D 2х + 1

A: Our objective is to discuss the continuity at x=2.

Q: 5. Show that the functions x,x? -,x³ -x x are mutually orthogonal in the interval [-1,1].

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Q: x + 6|, if x - 1 x² – 3x

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Q: f. Is the domain of f(x,y) = In (3x2 + 3y²) bounded or unbounded? Bounded Unbounded

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Q: Find m so that f(x) is continuous at x = 2. x +4 if x 2 -

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Q: Find the values of a and b such that the following function f(x) = 2a cos(x + 1) - b.5x+2, if x -1,

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Q: Determine if f(x)=x² is uniformly continous on the interval I = (-∞, +)

A: See the attachment.

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A: Intermediate Value Theorem: It states that if f is a continuous function whose domain contains the…

Q: Show that the equation x - 17x +c = 0 has at most one solution in the interval [-2, 2]. Let f(x) = x…

A: The given equation is : fx=x3-17x+c The given equation has at most one solution in the interval…

Q: Prove: f(x)=5x y is continuous at x=2

A: A function fx is continuous at a point x=a, if limx→afx=fa Given function is

Q: Identify the removable discontinuity. f(x) =  x3 + 2x2  over  x + 2

A: Given query is to find identify the removable discontinuity.

Q: Suppose that the function f: R → Ris continuous and that its image f (R) is bounded. Prove that…

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Q: Verify Rolle's theorem for f(x) = x² - 9x +6 over the interval [0, 3].

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

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Q: Verify that the Intermediate Value Theorem applies to the Indicated interval and find the value of c…

A: The given function is;  f(x)=x2+4x+2, [0,5], f(c)=23

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)