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: Evaluate the function shown, where ƒ is continuous on [3, 8], ƒ(3) = 4,and ƒ(8) = 20.

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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;

Q: 7. Suppose f is a continuous function on (a, b) with b> a. Stat f (recall this theorem was taught in…

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)3Df(x)+f) x, yER. Show that f(x)3Dcx…

A: Since you have posted a multiple questions. So,  we will solve only first question  for you. To get…

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

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Q: 2.6. Let ƒ be continuous on [0, a]. If x e [0, a], define fo(x) = f(x) and let %3D 1 fa+1(x) = -…

A: We will use the basic knowledge of integral calculus to answer this question. Note that we can only…

Q: 10. Prove the following change of variables formula: Let p be a real-valued differentiable function…

A: Introduction: The department of calculus wherein we look at integrals and their houses is called…

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)

Q: The function f is a composite of two continuous functions: y = cos x and y = x 2, so f is continuous…

A: The function fx is a composition of functions y=cosx and y=x2and it is a continuous function.

Q: Q3 Prove that if f and g are continuous at zo in R, then min(f, g) is continuous at 0.

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Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)-f(x) f(y) ♥x, yER. Show…

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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: Theorem 11. Let f be a continuous function on R such that f(x+y)Df(x)+fy) vx, yER. Show that…

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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…

Q: Let D = R² and f:D → R that f₁ (partial derivative off with respect to the 1st coordinate) and £2…

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Q: -2 x< -1 Q1)a) For what values of a and b is g(x) = ах — b -1<x<1, x21 Continuous at x= -1

A: The suitable formula for continious       LEFT HAND LIMIT = RIGHT HAND LIMIT

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: 15) Prove that if ffx) is continuous at x=0 then so is V f(x) provided that fix) zo for all x in the…

A: Step 1: Step 2: Step 3: Step 4:

Q: Assume that f is continuous on the set of real numbers. Find a and b given that b 3 [ f(x) dx - f₁…

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Q: that f 1s a continuous on IR Creal a nd h ale odif feren tiable funcho ns mberr). I denhfy a for…

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Q: 4. Consider the Poisson's equation: Let u be a continuous and differentiable function in R, and let…

A: Let us start proving the problem step by step... Let us take f, g as two different variables which…

Q: A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1…

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Q: Let f, g to be two continuous functions on [a, b] and f(a) ≥ g(a), f(b) ≤ - Prove that co € [a, b],…

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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.