Suggested Exercises: 1. Consider the function f defined by if f(z) = 16 NIHOIL CO if Use Definition 2.11 to find f' Determine if f is differentiable at x = yright @2020 Department of Mathematics and Computer Science University of the Philippines Baguio 12 Math 100 2. Let f be the function defined by ar2 +b if I 1. Find the values of a and b such that f'(1) exists and then sketch the graph of f.erentiation_Part 1... i A Q Search We wish to define the slope of the tangent line to the graph of f at P(x1, f(x1)). Let Ar = $2 - 21. We remark here that the symbol Ar denotes an increment of a and does not mean "delta multiplied by a". Now, the slope of the secant line PQ is mpQ f($2) - f(21) Ax (1) Since 2 - 21 + Ax, then equation (1) yields to mpQ = f(21 + Ax) - f(21) Ax Now think of point P as being fixed and move point Q along the curve towards P, (see Figure 3) that is Q approaches P. This is equivalent to stating that Ar approaches zero. Q (x2, f(x2)) P(X X)) Figure 3: The line PT is the tangent line to the graph of f at point P(21, f(x1)). Our previous discussion leads to the following definition. Definition 2.1: Tangent Line Suppose the function f is continuous at $1. The tangent line to the graph of f at the point P(21, f(21)) is i. the line P having the slope m(21) given by m(x1) = lim f (21 + Ax) - f(21) (2) if this limit exists; ii. the line x = x1 if lim f ( 21 + At) - f(21) is + 00 or - 00 12-+0+ and lim f ( 21 + Ax) - f(31) is + 00 or - 00 Ax