Math 16a worksheet 5 let fx x3 - 3x find a linear function

Question

MATH 16A WORKSHEET 5-

(1) Let f(t) = t³(t² - 1). Find f''(3), df/dt, d²f/dt₂|t=1, and d/dt(df/dt).

(2) Let f(x) = x³ - 3x + 2. The slope of the tangent line to f at (2, f(2)) is the same as the slope of the tangent line to f at (a, f(a)), where a ≠ 2. Find a.

(3) Let f(x) = x³ - 3x. Find a linear function (i.e. g(x) = ax + b for some a and b) which passes through (0, f(0)) and has the same slope as the slope of the tangent line to f at (0, f(0)).

(4) Using limits, compute f'(0) where f(x) = √(x² + 1) at x = 0.

(5) Carefully justify why the function

is differentiable at x = 0.