Suppose that f is twice differentiable at a. (a) Prove that there exist a 0ħ € (0, 1) such that f(x + h) = f(x) + hf'(x + 0µh). (b) Use part (a) to prove that lim h→0 - - f(a+h)-f(a) – f'(a)h h² (c) Evaluate limħ→o Oh for f(x) = x². = =f" (a) lim 0h. h→0

Q: is it true that ∇f(limx)=lim∇f(x) and further that ∇2f(limx)=lim∇2f(x)?

A: We know,  ∇f(xi)=f(xi)-f(xi-h), h=xi-x0i,i=0,1,2...And…

Q: If f (x) = 1 and (a, ƒ(a)) and (a + h, f(a+h)) are points on f(x), where h = 0, then 2x -1 f'(a) =…

A:

Q: If f is continuous from the right of x1 and lim f (x) exists, then lim f'(x) = f'_ (x1). O True…

A: it is true

Q: 1) Use the limit definition of derivative f'(a) = lim h→0 f(a+h)-f(a) h

A:

Q: lim 3 2y ソ→o ソーSiny -sin.

A:

Q: Let f(x) be a function defined in an open interval containing 'a'. The derivatrive of th on f(x) at…

A:

Q: Consider the function 1– cos(x) f(x): (a) (T) Evaluate lim, →0 f(x) = L. (b) (T) As x → 0, at what…

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

Q: Let f(x) be differentiable and lim f(x) = ∞ If lim = 0 then lim,→∞f'(x) : f(x) = 0 or lim,+∞ f'(x) =…

A: We have to check wheather it is true or not

Q: Use the limit laws to derive the product rule of derivatives (fg)' = f'g+ fg' by filling in the…

A:

Q: 16. If f is differentiable at xo, prove that lim h→0 f(xo + αh) - f(xo - ßh) h = (x + ß)ƒ'(xo).

A:

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

A:

Q: X'+2X Lim メーロ 0 @ True eflase laşe 8.

A:

Q: Using the limits sin h lim h→0 h - 1 and cos h lim h→0 h - 1 as facts (which do not need to be…

A:

Q: Given the information below, what can you conclude about the function? f (2) = -4, lim f (x) = -4,…

A: 1.) The given function is fx=3x-22x2+7x-4. The function is not defined at the points where the…

Q: Consider the function h with criterion differentiable functions in R. Which of the following…

A: Using derivatives to find answer

Q: Let f(x) = x²sin () for x + 0 and f (0) = 0. b) Use definition 6.1.1 to show that fis differentiable…

A: Solving by definition of 6.1.1 we have

Q: Let f: RR be a function that is differentiable on (a, o), where a is any real constant. Let g: RR be…

A:

Q: Suppose that a function h satisfies −x^2 ≤ h(x) ≤ x^2 for all x. Prove that h is differentiable at x…

A:

Q: (x + h) – x .3 - 31. lim

A: From the given data:

Q: a) f(x) = 2x + 1, c = 1 [b) f(x)=3√√x+1, c = 3 4 (c) f(x) = [d) f(x) = x+1 X 1- x , C = 2 c = -3…

A: As per the guidelines I am supposed to solve only three subparts of the given problems.

Q: 1) Let fi R²R and suppose that f: lim схіуз эс fcx, y) = 5. What can you say about the value f(1,3)?…

A:

Q: 8. Show lim tanh x =1 X→∞

A:

Q: f(xn) f(xn-1) xna, prove that lim |C| = 1. n→∞ 5. Consider Cn = ƒ'(ïn-1). Under the assumption that…

A: The given sequence Cnn=1∞ is defined as ; Cn = fxn f'xn-1fxn-1 f'xn  and  f'(α) ≠ 0  and xn → α…

Q: If 1/(x^2 + 1) ≤ f(x) ≤ 1 + x^2 for all x in the interval (−1, 1), what is lim x→0 f(x)?

A: Please refer the attached image for complete solution.

Q: Q4: Suppose that f satisfies the equation f (x + y) = f(x) + f(y) + x²y + xy² for all real numbers x…

A:

Q: Determine the Laplace transform for c (0³-0-²} s+2 1- F(s) = In +3 S-3 الخيار 1 0 3- F(s) = √√s…

A:

Q: If f' is continuous, f(6) = 0, and f'(6) = 11, evaluate f(6+ 3x) + f(6 + 5x) lim Limit:

A: The objective is to evaluate the limit.

Q: |2x – 4 lim x→2- 2 — х

A:

Q: (A) Evaluate the limit lim >a+ (x – a)ln(2x – 2a), where a is a positive constant. (B) Determine the…

A: (A) We have to evaluate limx→a+x−aln2x−2a…