Derivatives from graphs? ?Use the graph of f to sketch a grap ? h of f?.
SOLUTION The objective is to use the graph of f sketch a graph of f . STEP 1 The graph consists of line segments which are the tangent lines. Now to find the slope of the tangent line ,select any 2 points on the line. The slope of a line never changes,so we are free to choose any 2 points. y y Now substitute the points on the equation of slope m = x x1. 2 1 Thus,since it is a tangent line ,the slope of the line is itself the derivative. Now,let the 2 points on the line x < 1 be (1,5) and (-1,1). Substitute the points in the equation of the slope Then we get , 15 4 slope m = 11 = 2 = 2 Therefore the slope of the curve y=f(x) for x<1 is 2.ie f (x) = 2 for x < 1 Similarly, The slope of the curve y = f(x) for x > 1 is -1. ie f(x) =1 for x > 1 STEP 3 Using the above information sketch a graph of f At x=1 the slope of the line changes.So the derivative is undefined and the graph is discontinuous at the point x=1.