Suppose f ond 9 are continuous functions on [0. b] ond thand gee)>0 for all x E [0,b]. Prove that there exists xin La, b such that X in J fe) gl6) de = f a) Sgct) de gle) %3D

Q: if x 4.

A: The function gx is a piecewise defined function.

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Q: -1+x+ 4x² cos (4) når r + 0 når x = 0. f(x) = %3D %3D

A: We have given function;

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Q: 10) Prove that if fεg are continuous functions R on IR then fog is also continuous on i

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Q: 4. Let f: R→ R be defined by f(x) = Prove that f is continuous at 0. Sx, TEQ, (-x, -x, xER\Q.

A: Definition:A function f(x) is continuous at x=a when,lim x→a– f(x)=lim x→a+ f(x)=f(a)

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Q: Part (b) Define a function g : [0, 1) R by tan(tx) g (1) := x3/2 for all t E [0, 1). (i) Show that g…

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Q: Ex filnd divergenceof f when yj at (o1). 2 e ( + 9 J

A: See the answer below

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Q: Prove thait if y=f(x) f(x)=0 when x is a rotional number, then f(x)=0 for all x € R. is continuous…

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Q: Suppose that *) is continuous, and that Adx = 3, [A*)dr = -2, and rordt - 1. Then redt - O A. 6 О в.…

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Q: Suppose f and g are continuous on [a, b] and f f(x) dx = S% 8(x) dr. Prove that there is cE [a, b]…

A: Suppose f and g are continuous functions on [a, b] such that ∫abf(x) dx=∫abg(x) dx We need to prove…

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Q: Let fa function such that f and f' are continuous in all their domain , If f(-1) = 2 and f'(-1) = 5…

A: limx-0[f(-1+3x)-f(-1-4x) ]/x.

Q: Solve the two items using Integration by parts. 1. [x' v9-x? dx Se sin x dx 2.

A: To find the given integral by integration by parts :-

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A: We will solve the problem

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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.