- 2r2 + In 2 , 0<ı<ı In f(r) = sin-(a + 2r) 3 1<1 be the function. Which of the following statements are true? I. f(x) is continuous on [0, 1] for each a E R. II. f(x) is continuous for a = -3. II. f(x) has a jump discontinuity at x = 1 for a = -3. IV. f(x) has a removable discontinuity at х%3D 1 for a E (-3, —2).

Q: In I+1 2r2 + In 2 , 0<r<1 f(1) = sin (a + 2r) 3 ,1<r< be the function. Which of the following…

A: We will check each options

Q: (d) Suppose f : (a; b) → R is differentiable and its derivative f' : (a; b) → R is continuous.…

A: SOLUTION D PART

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A:

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A:

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…

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A: given , f(x)=x+22-3      for x&lt;1asinx-1+6      for x≥1 we have to determine the value of "a" such…

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A: We have to apply the comparison test

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A: Differentiablity analysis problem....

Q: 4- If a function is differentiable at zo, then it is : a) Differentiable on C b) Continuous on C c)…

A: To answer the given question about the complex function.

Q: 1- Discuss the continuity of the following functions at given poin { V9r4+x² 5x2+3x+1 (a) f(x) = if…

A: It is given that

Q: If f(x) is a continuous function on [3,7] and F(x) is an anti-derivative of f(x) such that F(3)=7…

A: Given :f(x) is continuous function 3,7Fx is anti-derivative of fxF3=7 and F7=3To determine…

Q: 1- Discuss the continuity of the following functions at given points { V9r4+x² 5x2+3x+1 ,if a 0 (a)…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Suppose h' is continuous x>0. IF x = 0 and h'(x) = 3√tan (2x)-1 it exists. i at Find h (0), 'F

A: For the solution of the problem follow the next step.

Q: Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute…

A: Option (C) is correct. Because the given function is continuous in the interval [0 4].

Q: VI. Suppose h' is continuous at x = 0 and h'(x) =tan(2x) - 1 if x > 0. Find h(0), if it exists.

A: Let's find.

Q: Зx2 — 1 Q7- Let f(x){ cx + d Vx + 8 if x 1 Determine e and d so that f is continuous everywhere (as…

A:

Q: In I +1 2.x2 + In 2 , 0 < x <1 Let f(x) = 4 - sin-(a + 2x) ,1<x < Which of the following statements…

A: first we check it is continuous ya not at x=-1by using definition of limit then we find limit value…

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:

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A:

Q: If W) is a continuous function on [4,7] and F(x) is an anti-derivative of f(x) such that F(4)=9 and…

A:

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A:

Q: Suppose that f is a continuous function for which f(1)÷0 and f(t) dt=0. What can we conclude about…

A:

Q: If f is continuous on [-1, 1] and f(x)dx = 0, then there exists some ce [-1,1] for which f(c) = 0.

A: Given: f is a continuous function on the interval -1, 1 and ∫-11f(x) dx=0. Prove: There exists some…

Q: Suppose a function f is continuous on [0, 1], except at a = 0.25, and that f(0) = 1 and f(1) = 3. Is…

A:

Q: 15.1. Using the results of the preceding exercise, show that the function g, defined on R to R by…

A:

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A:

Q: At what value(s) of x is the function below discontinuous? x+4, if x1 O x=1, only O x=-1, only At…

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

A:

Q: - if f(x) is a continuous function at x O, then a = 1-cos4x よく0 x? { a If f(x) = r= 0 r>0 V16+ VF -

A: Then to find the value of a if f(x) is continuous at x=0   Since for any continuous function at…

Q: 3. Consider the function g(x) defined by | 22 sin(1/x) if x > 0, g(x) = –x² if x < 0. (a) Show that…

A:

Q: Suppose that f(1)=8, f(4)=−4, f′(1)=−3, f′(4)=2, and f′′ is continuous. Find the value of ∫from 1 to…

A:

Q: 10.C. If f is defined and continuous on R to R and if f(xo) > 0, then is f positive on some…

A:

Q: Suppose f is known to be differentiable and contains a local extremum at c, then which of the…

A:

Q: b) Show that the function sin for æ e (0, 1] { f (x) = for æ = 0 is absolutely continuous on [0, 1].

A: Function is called continuous function if the left and right hand limit values at each and every…

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A: 1/f is rational function this is discontinous where either f is discontinous or f is zero  But f is…

Q: 50 if x E Q 1 if x € R\ Q Let f(x)= Prove that y=f(x) is not continuous at any points a E IR.

A: Take p ∈ Q and let (xn) be a sequence of irrationals converging to p. Then f (p) = 1 but f (xn)→ 0…

Q: 23. Suppose f(x) ={9 (бх + 5, хSо (2x+ 5, x> 0 Which of the following statements is NOT true? i.fis…

A: 23.fx=6x2+5,x≤02x+5,x&gt;0 Note:     Multiple questions are given, we are answer with only one…

Q: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

Q: 1+ cx, if x 2 Given the function: f(x) = C - x, What should be the value of c so thatf is…

A:

Q: Find the values of a and b such that the following function a² cos(x − 2) — 26(³-x), if x 2, is…

A: Topic = Limits

Q: 2. Consider the function f with domain (-2, 0): 3|22 – 1| if – 2<r<1 2.r2 + 2x – 4 f(x) = 3 if r = 1…

A:

Q: On which of the following interval is f(x) S O (0, +∞0) ○ (2, +∞) O [2, +∞0) 0 (-∞, +∞0) 1 x-2…

A: We will the fact that root value should be positive.

Q: (5) Prove that if f is continuously differentiable on [a, b], then |f(a) – f(b)| a – b max f'(x)|…

A:

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

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A:

Q: 18.Q. Is it true that function on R? to R is continuous at a point if and only if it is both upper…

A: exercise 18.Q

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.