Show that if f(x) is integrable on every interval of real numbers, and if a and b are real numbers with a < b, then a. f(x) dr and f(x) dx both converge if and only if S f(x) dr and , f(x) dx both converge. b. fx) de + f(x) dx = f f(x) dx + f(x) dx when the integrals involved converge.

Q: Let's assume that f(x)= r²+c* I c-16 g(x, a) = a* x² + ar Find a such that the following error is…

A: Given that where            That is  Also, given that

Q: Given the points A: (3, — 2, 2) and B: (5, 2, 6), find the vector u = AB

A: The points are given by A3, -2, 2B5, 2, 6 To evaluate: The vector u→ = AB→.

Q: Let F(t) = (4t³ - 1)j + (5t+2)k. Find F'(t). 7"(t) O 288t 288t³j 05+288t³ O 36t³ +5 O None of these

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Q: Suppose that f'(x) > M for all x E [a, b]. Prove that f(b) > f(a) + M(b

A: Since  f'(x)≥M therefore the function is differentiable and hence continuous let f is continuous on…

Q: Question 3 If f is integrable on the interval [3, 9], then Lf (x)dx = [,'f (x)dx a) True b) False

A: Given query is to find the statement is true or false.

Q: Use the Second Fundamental Theorem of Calculus to find F'(x). • L₁ VE 4 + 4 dt Vt Step 1 F(x) = Note…

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Q: 31. Assume that f is continuous and 2x S f ₁ f(t) d² = 4 + x²" dt (a) Determine f(0). (b) Find the…

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Q: f(x)=xmodx for all x  in R. Discuss the derivability of f(x) at x=0

A: Given:fx=xxwe have to find derivability of fx at x=0

Q: Let f(x)=1/x. Show that f is bounded from above on any interval (α,1), where α>0, but the function…

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Q: ppose that f is a continuous function and the followi ƒ(2) = 2, ƒ(4) = −1, ƒ(6) = 6, ƒ(8) = = culate…

A: We have to choose the correct option from the given options according to the given information

Q: If that f is continuous on a sxsb. Let F(x) be any of f(x). Then, f(x)dx = F(b) - F(a). This theorem…

A: Recall: Mean value theorem: If f(x) is continuous over an interval [a,b], then there is atleast one…

Q: Only one option can be true

A:

Q: Suppose that a < b <c and f is continuous. Select all true statements. (a) f(x) dx = f(x) dx – f(x)…

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Q: If fCxl is continuous on [2,7],then Si f(u) dvz f (7)-(2) True e or false?

A: we have given that f(x) is continuous on[2,7].