Second Derivative Math165: Business Calculus Roy M. Lowman Spring 2010, Week6 Lec2 Roy M. Lowman Second Derivative second derivative definition of f0 Definition (first derivative) df f(x + ∆x) − f(x) = f0 = lim (1) dx ∆x−>0 ∆x ∆f = lim (2) ∆x−>0 ∆x ∆f ≈ average slope of f(x) over delta x, f0 (3) ∆x avg (4) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f0 f0(x) is the slope of the function f(x) f0(x) is the rate of change of the function f(x) w.r.t x f0(x) is (+) where f(x) is increasing. f0(x) is (−) where f(x) is decreasing. f0(x) can be used to find intervals where f(x) is increasing/decreasing f0(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f0(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f0(x) can tell you a lot about a function f(x) 0 ∆f 0 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f00 Definition (second derivative) d2f d f0(x + ∆x) − f(x)0 = f0(x) = lim (5) dx2 dx ∆x−>0 ∆x ∆f0 = lim (6) ∆x−>0 ∆x ∆f0 ≈ average slope of f0over∆x; f00 (7) ∆x avg (8) Roy M. Lowman Second Derivative second derivative definition of f00 f00(x) gives the concavity of function f(x) f00(x) is the rate of change of slope w.r.t x 00 f (x) is (+) where f(x) is concave up. (holds H2O) f00(x) is (−) where f(x) is concave down. (makes letter A) f00(x) can be used to find intervals where f(x) is concave up or concave down f00(x) = 0 or undefined where the concavity can change f00(x) = 0 at inflection points, but must check if actually IP. The Second Derivative Test can be used to determine what type of critical points where f0(x) = 0. f00(x) can tell you a lot about a function f(x) 00 ∆f0 00 favg = ∆x can be used to extimate f (x) Roy M. Lowman Second Derivative second derivative definition of f00 f00(x) gives the concavity of function f(x) f00(x) is the rate of change of slope w.r.t x 00 f (x) is (+) where f(x) is concave up. (holds H2O) f00(x) is (−) where f(x) is concave down. (makes letter A) f00(x) can be used to find intervals where f(x) is concave up or concave down f00(x) = 0 or undefined where the concavity can change f00(x) = 0 at inflection points, but must check if actually IP.