1. Determine whether these statements are always true or sometimes false. (a) The function f(x) |a-3 is continuous at r differentiable at a = 3. 3, but f(r) is no ( True False )

Q: The following statement is false, explain why. The function g(x) = x^2 on the interval [-1,1] has…

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Q: Under what circumstances can you extend a function ƒ(x) to be continuous at a point x = c? Give an…

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Q: Consider g(x) = x!/5. Is g(x) differentiable at x=0? Is g(x) continuous at x=0?

A: Given function

Q: 10 If f is a continuous function and F'(x)=f(x) for all x € R, then x)dx=?

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Q: b) Suppose that f(x) 1 if x is negative. Can ƒ be continuous at x = 0?

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Q: let                    x^2+1   if x is less than 1 f(x)=                   x+1        if x is…

A: Consider the given function fx=x2+1  if x<1x+1   if x≥1. The point is x = 1. Obtain the left hand…

Q: 16. Suppose that f is differentiable at c and that f'(c) + 0. Show that there exists a ð > 0 such…

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Q: 19. Determine the following function if continuous at 2 f(x) = (,) ;at x = -1 + 1 A. discontinuous…

A: Explanation of the answer is as follows

Q: 18. Suppose that f is a differentiable function such that f(g(x))= x, and f'(x)= |3D 1 1+(f(x))².…

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Q: If y = f(x) is an odd differentiable function defined on (-o, 0) such that %3D f'(3) = -2 then…

A: A function f(x) is said to be even if f(-x) = f(x) And odd if f(-x) = -f(x)

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: Suppose f' (x) = g (x) (x + 1) (x – 2)´, where g (s) is a continuous function which is always…

A: When f'(x)>0 then function increasing when f'(x)<0 then function decreasing

Q: If f(x) is continuous at x = c then f'(c) exists. True False

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Q: If f(x) is continuous at x=a , then f(x) is also differentiable at x=a.

A: " If F(x) is continuous at x=a then f(x) is differentiable at x=a." The condition for the function…

Q: 3. Define the function sin(2x – 2) x 3 3' - Determine if f is continuous at x = 1 and x = 3. If…

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Q: Is f(x) = -3 A. Yes B. Does Not Exist continuous at x = 3? C. No D. Cannot be determined

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Q: 8. Suppose that f is a continuous function such that f'(x) > 0 for all æ, f(0) = 0, and f(1) = 1. If…

A: f'(x)>0 for all x, f(0)=0 and f(1)=1.To Find: If ∫01f(x)dx=13, find the value of ∫01f-1(x)dx.

Q: Determine the values of x at which the function in Figure 18 is: (a) dis- continuous and (b)…

A: (a) Here from the graph, we can see that, The graph has a break at the point x =1, that is, Thus…

Q: 18. Suppose that f is a differentiable function such that f(g(x)) = x, and f'(x) = 1 1+(f(x))². Show…

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Q: A. Consider the function defined by [x² +1, if x<5 f (x)={ |6x– 4, if x25 a. Is f continuous at x =…

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Q: If f(g(x)) is continuous at 0, then g(x) is continuous at 0  true or false

A: Given that If f(g(x)) is continuous at 0, then g(x) is continuous at 0

Q: For all but two choices of a, the function (x) = x if xS a and x' ifx > a will be discontinuous at…

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Q: 1. Suppose f is a function which satisfies f(x + y) = f(x) + f (y) for all x,y e R. What is the…

A: Let f is a function which satisfies f(x+y) = f(x) + f(y)  for all x,y∈R   f(x+y) = f(x) + f(y) ,   ∀…

Q: 1. Determine whether the following functions are continuous at the given point. a) f(x) = 1 ; а — 3…

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Q: 2. Consider the function if x = in lowest terms q > 0, p, q € Q; f(x) = O if x e Q°,x = 0 Question:…

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Q: 3. Sketch the graph of a function that is continuous, but not differentiable at I= 1.

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Q: 1. Suppose that f : [0, 0) → R is continuous and f(x) # 0 for all x > 0. If we have (f(x))? = 2…

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Q: 35. Given that f(z) is a continuous even function, then A 0 B D

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Q: B) If f is differentiable at x, then prove that f is continuous at x Is the converse true? Why

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Q: If a function f is not differentiable at x = a E R, then f is not a differentiable function. True O…

A: The given statement is if a function f is not differentiable at x=a∈ℝ, then f is not a…

Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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Q: Consider f(x) = #3/5. Is f(x) differentiable at x=0? Is f(x) continuous at x=0?

A: The given function is,

Q: 17.Determine the following function if continuous at p(x) = 3x? – x +5 ;at x = 1 A. discontinuous at…

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Q: Suppose that f has a positive derivative for all values of x, and that f(1) = 0. Which of the…

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Q: True or false: if f(x) is continuous at x =a, then f(x) is differentiable at x =a.

A: We have to determine the given statement as true or false.

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

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Q: (d) A function not continuous at 0 such that (f(x))4 is continuous at 0.

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Q: Suppose that the function f: [0, 1] in R is continuous and thatf(x) &gt; 2      if 0&lt;=X &lt; 1.Is…

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Q: Let f be defined by the following. S2x-3 f(x) = 6 - x if x < 3 if x 2 3 (a) Is f continuous at x =…

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Q: If f (x) is a function that is differentiable at x = 2 , then f (x) is continuous at x = 2. True O…

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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…

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Q: . Is the function continuous on the interval (4, 10)? (x + 1 if x > 8 f(x) = E) + 7 if x < 8 A.…

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Q: If f is a continuous function and F'(x)=f(x) for all x € R, then - x)dx=?

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Q: Is the following statement correct? If the function f is continuous at x=1, then the constant a…

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Q: If a function f is continuous at 0, show that g(x) = xf(x) is differentiable at 0 and find the value…

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Q: Гa+ 4x ifx 1 Let f(x) (a) The value of a that will make f(x) continuous for all real numbers is a =…

A: Check the differentiability of a function when it is continuous at x= 1 .

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: x², if xs1 4. Let f(x) = . Find f. (1) andf'(1). Does f'(1) exists? 2x, if x>1 Show that the…

A: 4. fx=x2,if x≤12x,if x&gt;15. y=f(x)=x2-3x

Q: Suppose that f(x) is a continuous function on the interval [0, T] . If f(x)=(sin x)ª for 0<x<T, then…

A: we have to find the value of given function. the solution is following.