Suppose that f’’ is continuous on [a,b] and that f has three zeroes in [a,b]. Show that f’’ has hat least one zero in (a,b).

Q: Use the definition of the definite integral as a limit of a sum to evaluate •1 (x2 – 4x – 2) dæ -2

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Q: Use the definition of the definite integral as the limit of a sum to evaluate (4x

A: Definition of Definite integral  The definite integral of a real-valued function f(x) with respect…

Q: Suppose that f is integrable on [-4, 7]. If the average of f on the interval [-4, 7] is 4.25, fin 4…

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Q: branch of z1/5 that is not continuous and positive on the imaginary axis?

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Q: Show that f1x2 Vin x is right-continuous at x = 1.

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Q: Use finite approximations to estimate the under the graph of the function f(x) = 4 – x² between x =…

A: Here given that a function and find lower sum with four rectangles of equal width. Here i attached…

Q: Every continuous function on a, b\, that has absolute extrema on endpoints, must have one .(a, b)…

A: b. False Consider a continuous function f (x) on [a, b], that has absolute extrema on the endpoints.…

Q: Express dx as a limit of left Riemann sums but do not attempt to eval- x2 uate the limit.

A: Given, ∫-11-6 5x2dx

Q: How are the continuous extensions often found?

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Q: Suppose that ƒ'' is continuous on [a, b] and that ƒ has three zeros in the interval. Show that ƒ''…

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Q: Show that f (x) = is not uni form continuous on (0,3) -3

A: To prove that 1x-3 is not uniformly continuous on 0,3

Q: Find the equation of the tangent line of f(x) = sqrt(x+1) at x = 0 using the definition of limits.

A: To find: The equation of tangent of line. Given: The function is f(x)=x+1. Concept used: Slope…

Q: Determine whether f is continuous on its domain. If it is not continuous on its domain, say v…

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Q: Show that the functions are continuous on their natural domains with examples?

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Q: Use the definition of the hyperbolic functions to prove that 2cosh (x)-1=cosh(2x).

A: As per the instruction, we will be answering only the first question. For (a),      2 cosh2(x)-1=2…

Q: Suppose f  is twice differentiable on R  and has threeroots. Show that  f " has at least one real…

A: According to Rolle's theorem, there is a zero of f'x in between two zeroes of f(x).

Q: If f is integrable on the interval

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Q: Express the limit as a definite integral.

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Q: If f is continuous on (-x, ∞), what can you say about its graph?

A: If f is continuous on (-oo oo) what can you say about its graph?

Q: Show that f is continuous on (∞ , -∞ )  .

A: we have to show that f(X) is continuous in (-∞,∞). f(x)=x2             if   x<1           x…

Q: Show that the function is continuous on  (∞ , -∞).

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Q: Identify the limit that represents the limit definition of the definite integral ∫39 6x3 dx using a…

A: From the given indefinite integral, observe that f (x) = 6x^3, the lower limit and upper limit is 3…

Q: Suppose /has a continuous derivative on the interval [0, 6]. Assume also that f' is increasing on…

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

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Q: Prove the limit statements in Exercise

A: To prove: limx→0x2sin1x=0

Q: Express the limit as a definite integral on the given interval. lim n - 00 -Дх, [1,5] (x*)2 + 6 dx

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Q: Express the limit as a definite integral on the given interval. x; In(1 + x3) Ax, [2, 3] lim 3 dx

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Q: Every continuous function on a, b, that has absolute extrema on endpoints, must have one critical…

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Q: Express the limit as a definite integral on the given interval. n x;* lim -ΔΧ, [1, 7] n→∞ dx i=1…

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Q: Suppose that the functions f and g are defined throughout an open interval containing the point x,,…

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Q: Show that f (x) =++1 1 - is not uni form continuous on (-1,0)

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Q: Which of the following is always true for integrable functions f and g?

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Q: If f is continous on [0,1) ,then f has absolute minimum on(0,1) True or false?

A: Extreme value theorem says that, if a function is continous on a closed interval [b, a] , then it…

Q: Verify the Mean Value Theorem for f(x) = In(6x + 1) on | 0, %3D

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Q: Use the definition of the definite integral as a limit of a sum to evaluate (x² – 4x - 2) dx -2

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Q: (a) Either prove that ƒ is continuous at x = 3 or classify the type of discontinuity, with…

A: " Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Write the integral 1 1dx -1 x2 as a limit.

A: To write the given integral as a limit.

Q: 8) Assume that 2 + Va if x >1 if x < 1 f(x) = Determine whether or not f is continuous at x = state…

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Q: Express the limit as a definite integral on the given interval. lim 8x; + (x;)² Ax, [3, 8] dx

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: What is the continuous extension of

A: This question is taken from the function and derivative now we have to solve the given function so.…

Q: Every function which is riemann integrable on [a, b] is continuous.true or false

A: To check if every function which is riemann integrable on [a, b] is continuous or not.

Q: Use the definition of the definite integral as the limit of the Riemann sum to find

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Q: Give an example of a function that is integrable on the interval [−1, 1] but not continuous on [−1,…

A: Let fx=x       Greatest integer function. Now, ∫-11x dx=∫-10x dx+∫01x dx=∫-10-1 dx+∫010…

Q: Express the limit as a definite integral on the given interval. lim [9(x,")³ – 4x†]Ax, [2, 6] n- 00…

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Q: 4. Assume that æ € [0, 1] and write the error in the approximation to the error function using the…

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Q: Give the definition of the definite integral as a limit using the right endpoint approximation.

A: Let's find.

Q: Every function which is Riemann 3/3 integrable on [a,b] is continuous. True False

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Q: Suppose  f is continuous on [0, ∞) and limx→∞ f (x) = 1 . Is itpossible that  ∫ 0∞f (x) dx is…

A: Consider a function f(x) which is continuous on [0, ∞) with limx→∞f(x)=1. The objective is to…

Q: 3. Show that 5 x + 7 is a continuous fimction

A: Let, f(x) = 5x+7 be defined over R, the set of all real numbers. Choose an arbitrary point c in R.…

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.