Calculus 1 The Definition of the Derivative Assignment 1. Find the equation of the tangent line to the graph of f (x) = x2 + 2 at x = 3. 2. Find the derivative of f (x) = 3 by using the defintion of the derivative. x 3. Show that f (x) = |x + 3| is not differentiable at x = −3. 4. Is f (x) = |x| differentiable at x = 0? Justify your answer. x Solutions 1. Use the definition of the derivative to arrive at f 0 (x) = 2x. The slope of the tangent line to the graph of f at x = 3 is f 0 (3) = 6. Now use the point-slope equation of a line y−y1 = m(x−x1 ) to arrive at y = 6x − 7. 2. f 0 (x) = − 3 x2 3. We’ll use this version of the definition: f 0 (c) = lim x→c f (x) − f (c) . x−c lim |x + 3| − | − 3 + 3| |x + 3| −(x + 3) = lim = lim = lim (−1) = −1. − − x − (−3) x+3 x+3 x→−3 x→−3 x→−3− lim |x + 3| − | − 3 + 3| |x + 3| x+3 = lim = lim = lim 1 = 1. + + x − (−3) x+3 x→−3 x→−3 x + 3 x→−3+ x→−3− x→−3+ Since the one sided limits are not the same, lim x→−3 |x + 3| − | − 3 + 3| DNE, hence f (x) = |x + 3| x − (−3) is not differentiable at x = −3. 4. This function is not continuous at zero and so it cannot be differentiable there.