For each of the questions below, indicate if the statement is true (T) or false (F). (a) If f(x) is continuous on a closed interval [a, b] and L f(x)dx > 0, then f(x) > 0 for all x E [a, b]. True False (b) A continuous function on a closed interval is integrable. True O False (c) A continuous function on an open interval is integrable. O True O False (d) Every continuous function has an antiderivative. O True False (e) If f(x) and g(x) are integrable on [a, b] and g(x) < f(x) for all x E [a, b], then [ g(x)dx < [° f(x)dx . O True False O O O O

Q: Find the x-values (if any) at which f (x)= 1s not continuous. x²+ 3x O fx) is not continuous at x =…

A: It is known that to check the continuity, we can equate the denominator part with 0.   Here, we…

Q: For the continuous function f(x), if f’(-4) does not exist, then x=-4 is a critical point.

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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: Find the values of α and β such that the function f(x) =αcosx + 2 if x < 0  and f(x) = βe3x + αx2…

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Q: Is the following statement correct? If the function is continuous on R then the constants a and b…

A: It is not necessarily correct that every continuous function on R take the value 2&amp; -2…

Q: If f(x) is continuous in the interval [-2, 1], ƒ(−2) = −4, and f(1) = -1, can we use the…

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

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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: If a function f(x) is not defined at x-c, then it is not continuous at a close interval [a, c). * O…

A: We know that function is continuous at c if the 1) f(c) is defined 2)limit of f(x) exist, Here…

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: Suppose X is continuous with f(x) = x/2 from 0 to 2. What are the values of f(1) and P(X=1)? 1/2 and…

A: Given,f(x)=x2 ; 0&lt;x&lt;2

Q: Suppose that f (x) > 0 for all x, f(x) has a continuous derivative and that е — 1 f(-3) = 0 and f(4)…

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Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

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Q: If f"(c)<0, where c is a critical value and f(x) is a continuous function over an open interval I…

A: The answer is that f ''(c) tells us the concavity of f(x) at x = c. If f(x) is concave up, i.e., if…

Q: It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function…

A: To solve an equation fx=0, if f is continuous on an interval a,b such that fa and fb are of opposite…

Q: (x²-x for x +0 If f(x) : and if fis continuous at x = 0, then k= 2x f(0) = k -1/2 O 1/2 O 1

A: f(x) = (x2 - x)/2x limx→0 x2-x2xlimx→0 x(x-1)2xlimx→0 x-12= -12

Q: Assume that ƒ is continuous on [0,1] and that f(0) = 2 and f(1) = 1. Show that the graph of y = f(x)…

A: given  f is continuous on [0,1] f0=2,f1=1y=x2+1

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…

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Q: Suppose f(x) = x. g(x) for some function g. Which of the following is true? Of need not be…

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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: 7. Suppose f is a continuous function on [a, b] and Sf(x)g(x) dx =0 for every continuous function g.…

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Q: Iffis continuous and f(x) < 0 for all xE[a,b], then what is true about f(x) dx? a Choose the correct…

A: Solution

Q: 1 - is continuous at a. f If f(x) 0 for all x in the interval I, and f is continuous at a in I then…

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

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

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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: x²-1 for x# 1 - If flx) = show that flxx) is continuous at x = 1 x-1 2 for x = 1,

A: topic - limits and continuity

Q: Suppose f(x) is continuous everywhere, and has the following values: 10 20 30 40 f(x) 9. 13 2 3 What…

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Q: Suppose f '' is continuous on (−∞, ∞). (a)  If f '(−3) = 0 and f ''(−3) = 7,  what can you say about…

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Q: Suppose that a continuous function g satisfies: | g(x)dx = 12 and g(2x)dx = 27. Find æg(3x²)dx.

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Q: 86. If f is continuous and f (x) dx = 4, finc

A: Given that: ∫09f(x)dx=4

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: Answer the following statements true or false. 8.1 The sum of continuous functions is continuous.…

A: Here we have to state whether the statement is true or false.

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

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Q: If f(x) is a continuous function such that it is increasing on [1, 3], then Si f(x)dæ is less than…

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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: . Is f(x)= Vx + 1 continuous at x == -1? Yes, because all functions are continuous No, because the…

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Q: For every continuous f:[0,+0) →R _function , if a>0 2fx'f(x*)dx =[ xf(x)dx Demonstrate that this…

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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: 6. gx) VX- 3 at x = 4 三 7 g(x) at X = 8 三 e1 gCX)= n4ーメ at x =0

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Q: Find a constant b such that h is continuous at x = 2, where x +1 for |x| 2 With this choice of b,…

A: It is given that the continuous function  hx=x+1for x&lt;2b-x2for x≥2 It is also given that function…

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: Suppose that a continuous function g satisfies: g(x)dx = 42 and | g(2x)dx = 12. 2 | #g(3x²)dx. Find

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Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

Q: .3 Suppose that f is a continuous function and that f(x) dx = 6. Find f(3x)dx.

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Q: 1. If y = f(x) is a continuous function for all 0 <x< 6, and f(0) = 1 and f(6) = 7, then which of…

A: We use the given graph of the function y=f(x), to answer the given question.

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.