Let f (x) be differentiable at x = 1 and f '(1) = 2, then [LIM (H → 0) f (1-h) - f (1)] / h is equal to

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• 1. Let f (x) be differentiable at x = A and lim [f (a + 5H)] - f (a-5h)] / 2H = 1, then f '(a)=
• 2. Why can derivative be written as f '(a) = LIM (x → a) f (x) - f (a) / x-a Shouldn't it be written as f '(x) = LIM (△ x → 0) (x + Δ x) - f (x) / Δ x? I hope you can help me
• 3. It is proved that LIM (1-e ^ 1 / x) / (1 + e ^ 1 / x) does not exist when x tends to 0 LIM (1 / X of 1-e) / (1 / X of 1 + e) does not exist when x tends to zero
• 4. Let f (x) be differentiable on [0,1], and f (1) = 2 ∫ 0 ~ 1 / 2 XF (x) DX. It is proved that there exists ξ belonging to (0,1), such that f (ξ) + ξ f '(ξ) = 1
• 5. Given that f (x) is differentiable at x = 0, f (0) = 0, f '(0) = 2, what is the limit of F (sin3x) / X when x approaches 0
• 6. Let f (x) be differentiable at point X. = 0, and f (0) = 0 and f '(0) = 3, then the value of LIM (x →∞) [f (x) / x] ()
• 7. Let f (x) be differentiable and find Lim [f (x + △ x)] ^ 2 - [f (x)] ^ 2 △ x → 0 lim [f(x+△x)]^2-[f(x)]^2 △x→0 =lim{[f(x+△x)+f(x)]*[f(x+△x)-f(x)]}/△x Why is it equal to ＝2f(x)lim[f(x+△x)-f(x)]/△x Especially why is it equal to 2F (x) Please give specific reasons,
• 8. Let f (x) be differentiable in a neighborhood of x = 0, and lim f '(x) = 1, then f (x) has extremum in x = 0, and the detailed solution is obtained
• 9. X tends to 0) Lim [(x + 4) ^ 1 / 2-2] / sin3x
• 10. Lim x tends to 0 arcsin2x / sin3x
• 11. If the function f (x) is differentiable at x = a, then Lim h → 0 (f (a + 3H) - f (A-H)) △ 2H =?
• 12. Let f (x) be differentiable at x = 2 and f '(2) = 2, then Lim h → 0 [f (2-3H) - f (2)] / h =?
• 13. If f (x) is differentiable at point X and lim f (x-3h) - f (0) / h = 1, then f '(x) =? H-0
• 14. When Lim h tends to zero, (f (x0 + H) - f (x0-h)) / 2H = f '(x0) can't understand
• 15. If f ′ (x0) = - 2, then Lim [f (x0 + H) - f (x0-h)] / h=
• 16. If f ′ (x0) = - 3, then limh → 0f (x0 + H) - f (x0-h) H = () A. -3B. -6C. -9D. -12
• 17. If f ′ (x0) = - 3, then Lim [f (x0 + H) - f (x0-3h)] / h= process
• 18. The existence of partial derivative of binary function, why can we deduce the following first 1. X - > x0 Lim f (x, Y0) = y - > Y0 Lim f (x0, y) 2. F (x, y) is differentiable at P0. If a function of one variable is differentiable, the curve is smooth. Why is it not that a function of two variables is differentiable? What is the geometric meaning of differentiable function of two variables?
• 19. Is the existence of partial derivative the same as the existence of partial derivative? (function of two variables)
• 20. Let the minimum value of F (x) = ax ^ 3 + BX ^ 2 + CX be - 8, and the image of its derivative function y = f '(x) passes through points (- 2,0) and (2 / / 3,0), as shown in the figure (1) Finding the analytic expression of F (x) (2) If f (x) ≥ m ^ 2-14m holds for X ∈ [- 3,3], the value range of real number m is obtained (the picture shows: the opening of the image is downward, the intersection of the image and the X axis is - 2,2 / 3, and the intersection of the image and the Y axis is above.)