Example 2 Find the equation of the tangent line to the curve f (x) = * + 2 for x = 2 at (2,6). (4x - 2 for x > 2 (Hint: need to consider left & right limit at 2) Example 3 Find the equation of the tangent line to the cure f (x) = x3 at (0,0).Definition of Derivatives The derivative of a function fat a number a, denoted by f'(a), is f' (a) = lim ! (x)-f(a) or f' (a) = lim !(ath)-f(a) x-a x-a h-0 h Example 7 Find the derivative of the given function at the number a by definition. a. f(x) = x3 b. f(x) = Vx-1 c. f(x) =Example 4 Find the equation of the tangent line to the curve f (x) = |x| at (0,0).2.1 Derivatives and Rates of Change Definition of Tangents The slope of tangent line to the curve y = f (x) at the point (a, f (a)) is 1. m = lim !(x)-f(a) 2. m = lim !(ath)-f(a) x -a x - a h-0 h Example 1 Find the equation of the tangent line to the curve f (x) = 2 - 3x2 at x = 2.Example 6 Suppose that TTC jumped from the top of a 400-ft skyscraper. The position function s, after t second, is s(t) = 400 - 16t2. a. Find the average velocity during the time interval [1,2]. b. Find the velocity at t = 3.Example 8 Sketch the graph of a function g for which g (0) = g(2) = g(4) = 0, g'(1) = g'(3) = 0, g'(2) = -1, g'(0) = g'(4) = 1, lim g(x) = co, lim g (x) = -co. x-5 x-1+