Let 01 (x) = [* 0(t) dt, for x > 1, where 0 is Chebyshev's function. Let = lim 01(x)/x². 1+00 By considering (t) dt, prove that T-ET 01(x) 01(x Ex) Ex-(x), where = (> 0) is small. - Assuming that 0(x)/x →1 as x → ∞, deduce that (1-1) ≤ 1. By similarly considering (t) dt, prove that (1+½)1 ≥1. 2 Deduce that 01(x) 1½². 2

If f(x) = 5x² , then f' (x) = lim-0 5(x+h)2 –52? h→0 h True False

Suppose f is a function that satisfies the equation f(x + y) = f(x) + f(y) + x2y + xy2 for all real numbers x and y. Suppose also that. Can you please find f(0) , f'(0) , f'(x)

(5) Use the limit definition of derivative to prove the following: (a) For any a e R, (f(ax))' = af'(ax), (b) For any positive integer n, (f(x)")' = nf(x)"-1 f"(x).

3. Use a limit to find f'(4) for f(x) = x²-x.

Q1) Check the continuity of the given function at the given c (or c's). 2 if z+3 (a) f(=) = { -3 c= 3 5 if 1= 3 1+z if zS -2 (b) f(x) = { 2-z if -2 2 Q2) Find the derivative of each function. f). f(e} = In(sec8 + tan 8) 8) f(z) = In () „a) f(z) = %3D b) f(z) = sin %3D %3D 1+cosz c) f(z) = tsinz d) f(z) = (1+z)*(1 – 2)² h) y = (In(ax))² 1+conz g(z) = In(r³ – 3z + 2) i) ) iy = %3D e) f(z) = 19 – 16 +-4 Q3) 27-30 Differentiate f and find the domain of f. 7. f(x) = 28. F(x) = /2 + In x (I – x)u – 1 22. f(x) = In(x² – 2x) 30. (x) = In In In x

Let f(x)=1/x. When using the definition of the derivative to compute f′(x), one of the limits below appears in the computation. Which one? limh→01/x(x+h) limh→0−1/(x2−h) limh→0−1/x(x−h) limh→01/x(x+h) limh→0−1/x(x+h) limh→0−1/(x2+h)

Find the formula for the derivative of l(x) = 1 by using the power p2 rule. Confirm it using difference quotier Choose one l (x) 1 1 - (x+h)? lim h→0 Select the difference quotient for 1(x): h 1 (x+h)? lim h (x + h) – x lim h (x + h) – lim h→0 х h

Let P(x0, y0) be an arbitrary point on the graph of f that f′(x0) ≠ 0, as shown in the figure. Verify each statement.

Could you show step by step how they got from the top part to the second highlighted part?