4.1 Suppose that f.g and g' are continuous on the interval Show that f(r)dg(z) exists, and furthermore show that | (2)dg(z) = | f(2)g(2)dz.

Q: 3) Differentiation D: F[t] ---> F[t], is linear and satisfies Dt¹ = nt¹-1. It satisfies the usual…

A: (a) We have f(c) = 0 if and only if (t-c) divides f in F[t]. Thus, we can write f(t) = (t-c)g(t) for…

Q: 10 5. Consider a function f (x) that satisfies all the conditions listed. i. fis defined and…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Consider the function f(t, p) 3 (pе' +1—р)". a. Find an expression for f(t, p) and evaluate that…

A: Given that:f(t, p)=(pet+1-p)n

Q: Suppose that f (0) = 2 and f'(x) 0. Show that f(4) 0.

A: f(0)=2   and f'(x)≤3  for all x>0 ⋯(i)integrate (i) w.r.to x ,we getf(x)≤3x+kput…

Q: 3. Find a value of b so that the function x 1 f(x) = is continuous on (-0, 0).

A: A function is said to be continuous if Left Hand limit = Right Hand limit = f(a) In this question,…

Q: Provide concise explanations for the answers to the following: c.) Which is greater, f (2) or f…

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Q: Maximise the function f(x, y) = 4xy² subject to r + y? = 2.

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Q: pose that the function f is continuous everywhere. Suppose that F is any antiderivative of f, and…

A: In this question, the concept of Integral is applied. Integration Integration is a method of…

Q: Suppose that a continuous function g satisfies: g(x)dr = 30 and g(2x)dx = 3. 0. 2 Find r9(3x²)dx.

A: Suppose that a continuous function g satisfies: ∫03gxdx=30 and ∫06g2xdx=3. The, we have to find the…

Q: 4. Find f"(2). (a) f(u) = 2u²5 – u).

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Q: For the function Hlo>x'tle, fnd a)flxth) 5bf(xth)- flx) c) f(x th)-f(x)

A: Given: f(x)=x2+6 To find: a) f(x+h) b)f(x+h)-f(x) c)f(x+h)-f(x)h

Q: Use the values given in the table for differentiable functions f and g to answer the following ques-…

A: a) Q(x) = 2g(x) + f(x) No differentiate above w.r.t. x  We get  Q'(x) = 2 g'(x) + f'(x).   Now for…

Q: for If f (2) is a regular function of z, prove that f (2) 2 = 4 | f'(2) 1?. %3D ve I f12) 1? =4 |…

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Q: 1. (Bx2 |x- 61-8 for x> 2 (a) For what value of Bis the function f(x) = }", for x 2 continuous atx=…

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Q: Si s(2x)dx = 23. If f is a continuous fiunction and if F'(x) = ƒ(x) for all real numbers x, then (A)…

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Q: If f is a positive function and f"(x) < 0 for a < xs b, show that T, < f(x) dx < M,

A: f is a positive function and f''(x)&lt;0 for a≤x≤b This tells us that f is concave down over the…

Q: . What is F(/3) if C = 0?

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Q: 3. Suppose that f(1) = 2, f(4) 7, f'(1) 5, f'(4) = 3, and f" is continu- %3D | af" (x) da. ous. Find…

A: The given data is: f1=2f4=7f'1=5f'4=3  f'' is continuous  To find the value of ∫14xf''xdx.

Q: Which of the following functions is continuous on the interval (-0, +)? …* f(x) = Vx² – 1 (A f(x) =…

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Q: 3. Find f. (g) f'(x) = 2x – e

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Q: Find a point c satisfying the conclusion of the MVT for the function y(x) = 16/x on the interval [0,…

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Q: Let f(x) = ²³ d. Find the value of ƒ¹ (2) af(x) 0 24 4 -8 -4 -24 12 -12 8

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Q: if r>1 if r< 1. 6) Suppose f(r) = Determine whether f'(1) exists.

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Q: 1(b)   Answer with "true or false". In addition, if the answer is wrong, correct the statement.…

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Q: 3. Let there be some a, b R such that f(a)ƒ'(a) = f(b)ƒ'(b) + 1. Show that f f(x) f"(x)dx ≤ 1.

A: This is a problem related to integration. Based on the general formula and concept we will prove the…

Q: S, f(x)dx = 20! ve F(0) = 18! ise F(7) =? %3D A) 20! – 18! B) 20! + 18! C) 0 D) 20! E) 8!

A: It is a application of fundamental theorem of calculus. So by using the theorem we cansolve the…