Explain why or why not? Determine whether the following statements are true and give an explanation or counterexample. a.? For linear functions, the slope of any secant line always equals the slope of any tangent line. b.? The slope of the secant line passing throu?gh the?points P ? and ?Q is less than the slope of the ?tangent line at ?P. c.? Consider the gra?ph ? of t?he para?bola f ? (?x)? ?x2 For ?x > 0 and ?h > 0, the secant line ? ? t? rou? g? ?? f(?x? )) and (?x +? ,? (?x + ?h)) always has a greater slope than the tangent ? ? ? line at (?x, ? ? )). ? d.? If the function f ? is differe?ntiable? for all values of ?x, then f ? is continuous for all ? values of ?x.
SOLUTION STEP 1 (a). Any function of the form f(x) = ax+b,where a is not equal to 0 is called a linear function.The graph of f is a line with slope m and y intercept b. For a linear function ,the graph is a straight line.Therefore the secant line for the line is the line itself which itself is the tangent line.Therefore the slope of the tangent and the secant line is always same for a linear function. STEP 2 (b). The slope of the secant line passing thr ough the points P and Q is less than the slope of th e tangent line at . Consider an example. Let f(x) = 5x+6.consider 2 points P(1,2) and Q(2,10) The slope of the tangent line at P is given by f(x ) = f(1) = 5 1 The slope of the secant line at PQ is given by the equation m = 2 1 = 102 = 8 x21 21 Thus we got the slope of the tangent line is less than the slope of the secant line. Therefore ,the slope o f the secant line passing through the points P and Q is not alw ays less than the slope of the tangent line at P . STEP 3 (c). Consider the graph of the parabola f(x) = x For x > 0 and h > 0, the secant line through (x,f(x)) and (x + h, f(x + h)) always has a greater slope than the tangent line at (x, f(x)). f(x) = x for x > 0 and h > 0. 2 2 The points are P(x,x ) and Q((x+h),(x+h) ) The slope of the secant line PQ is given by y y (x+h) x 2 2 2 h(2x+h) m = x x = x+hx = x +h h2xhx = h = 2x+h 2 1 ie, slope m sec= 2x+h Now the slope of the tangent line is the derivative of f(x) at x 1 ie, (f(x)) = (2x) = 2x 1 x Therefore m tan= 2x Therefore for any positive values of x and h the slope of the secant line at PQ is always greater than the slope of the tangent line at P STEP 4 (d). If the function f is differentiable for all values of x, then f values of x. Let a be any point in f .Then if f is differentiable at a then, f(x)f(a) f(a) = lxa xa exists. To show that f is continuous at athen we have to show that limf(x)f(a) = 0. We can write f(x)f(a) = Taking limit on both sides we get, lim(f(x)f(a)) = lim f(xaf(a) xa xa limf(x)f(a) = lim f(xaf(a) xa xa =f (x)×0 = 0 Thus limxa)f(a) = 0 Therefore,f is continuous at a. Hence, If the function f is differentiable for all values of x all values of x.