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

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

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Q: Which of these is the limit definition of the derivative of f(x)? O f'(x) f (x + h) – f (x) lim h 00…

A: use the basic definition of derivative to choose the correct option

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A: it is true

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

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

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

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Q: : limx->0(1/x3)

A: We have to simplify the given expression. The given expression is limx→01x3∫0xt2t4+1dt

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Q: tan(1 – x) limx→1 2 2x +x – 3,

A: Limit

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: Evaluate where f(x) = 7 x + 8. If the limit does not exist enter DNE. Limit= Number lim h→0 ƒ(1+h) —…

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

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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)?…

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Q: -5 -4 -3 -2 -1 n y = دیار 2- 7 1 -2- -3- -4 2 3 4 10 5 Write an equation for the function graphed…

A: Given graph

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 y = f(x) is a function that is differentiable everywhere, then f(-e) – f(-e – h) f' (-e) = lim ,h…

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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…

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Q: 16x4+9y12 (x,y)→(0,0) 4x²+3y6 Show that lim does not exist.

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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…

Q: C is the curve y = e® from (2, e²) to (0,1) and F = (x², -y), compute ſ, F-dr.

A: Given that, C is the curve y=ex from 2,e2 to 0,1 and F=x2,-y Then, F=x2,-y=x2,-ex So, F→=x2i^-exj^