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: Suppose that f(x, 0) = 3 for all x ≠ 0 and f(0, y) = 5 for all y ≠ 0. What can you conclude about…

A: The limit of the function, f(x,y) at a point (a,b) does not exist if two paths upon which the…

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

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: x² 1) Let f(x): = (Figure 24). Verify the following: x - 1 (a) f(0) is a local max and f(2) a local…

A:

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

A:

Q: Geometry What is the depth of the pool when it is half filled in terms of volume?   The volume is…

A:

Q: f' (x + h) – f'(x) Use f''(x) = lim to find f''(x). h→0 h f(x) = 1 – 5x f''(x) = Preview

A: Use the given formula to find f'(x).

Q: If f. g and h are differentiable functions at a, then f(g+h) is differentiable at a and ( f(g+…

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: 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: 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: 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: 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: Let 1 f(x) = X - - 1 A) First calculate f'(a) = lim h→0 f'(a)= (i) ƒ' ( − 2) = = (ii) ƒ'(0) = (iii)…

A: To find the derivative.