If f' is continuous on [a, b], show that 2 [* f(x)f'(x) dx = [f(b)J* – [f(@}J°

Q: Use this definition to find f'(a) at the given number a. f(x) = V5x + 1, a = 3

A: First we find out f'(x) then put x=a=3 in f'(x).

Q: Let f(x) = (a) f(1) = (b) f'(6) = 2x³ - 5x² x2 Evaluate f'(x) at the following points:

A: f(x)=2x3-5x2x23xba=xbad dx(x)m=mxm-1

Q: Let f:A-B and X,Y be subsets of B. Then f-¹(XnY) = f'(X) nf-¹(Y). True False

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Q: what is f'of x

A: Find f’(x).

Q: If f is concave up, then f' is (choose one): (a) increasing. (b) decreasing.

A:

Q: Assume that f"(x) is continuous on [0, 1] an ƒ(0)=1,ƒ(1) = 3, ƒ'(1) = 7. Find [xf"(x) dx.

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Q: Assume that f : (0, 1) → R is a differentiable function with the derivative f' increasing on (0,…

A: Given f:0,1→R is a differentiable function and derivative of f is increasing in 0,1 that is f'x>0…

Q: If f' is continuous, f'(2) = 7, and f(2) = 0, then

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Q: Let f(æ) = – 3x* In æ %3D f'(x) = f'(e²) =

A:

Q: Determine whether the statement is true or false. '8 If f'is continuous on [7, 8], then f '(v) dv =…

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Q: ·b f(b) − ƒ(a) = f'(a)(b − a) – ſå ƒ"(x)(x − b)dx - a

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Q: Assume ƒ is a continuous on [2, 4],, and ƒ(2)= 4. Evaluate ƒ(4).

A: we have been given that f is continuous on [2,4] and f(2)=4. we have to evaluate f(4). the given…

Q: Let f(x) — sin(nx) n³ n=1 Show that f is differentiable on R and that f' is continuous on R.

A: Within its interval of convergence, the derivative of a power series is the sum of derivatives of…

Q: Let f(x) = x?sin (-) for x + 0 and f(0) = 0. c) Show that f'is not continuous at x = 0.

A: Given function  f(x)=x2 sin1x      for x≠00                  for x=0 we have to show that f' is not…

Q: Suppose that S(m)3(4#-4)*(32+ 1). Find f'(#), and then evaluate f' at # = 1 and # = -1 (1)= 1(-1)=

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Q: if f'(a) exists, then f is continuous at x = a. True False

A: Function f differentiable at x=a

Q: Suppose that f(1) = 4,| f(4) = –6, f' (1) = 6, f'(4) = 3, and f" is continuous. Find the value of 1…

A: Given: I=∫14xf"xdx, f1=4, f4=-6, f'1=6, f'4=3 For evaluating given integral, we will use integral by…

Q: A function f: R → R has a third order continuous derivative. Show that f(a) f'(a). f"(a). f"" (a) ≥…

A: Given f:ℝ→ℝ has a third order continuous derivative Case 1: Consider fx is even function f-x=fx…