Second DerivativeMath165: Business CalculusRoy M. LowmanSpring 2010, Week6 Lec2Roy M. Lowman Second Derivativesecond derivativedefinition of f0Definition (first derivative)dfdx= f0= lim∆x−>0f(x + ∆x) − f(x)∆x(1)= lim∆x−>0∆f∆x(2)≈∆f∆xaverage slope of f(x) over delta x, f0avg(3)(4)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f0f0(x) is the slope of the function f(x)f0(x) is the rate of change of the function f(x) w.r.t xf0(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) isincreasing/decreasingf0(x) = 0 or undefined where the sign of the slope canchange, i.e. at CNsf0(x) = 0 at critical pointsThe First Derivative Test can be used to determine whatkind of critical points.f0(x) can tell you a lot about a function f(x)f0avg=∆f∆xcan be used to extimate f0(x)Roy M. Lowman Second Derivativesecond derivativedefinition of f00Definition (second derivative)d2fdx2=ddxf0(x) = lim∆x−>0f0(x + ∆x) − f(x)0∆x(5)= lim∆x−>0∆f0∆x(6)≈∆f0∆xaverage slope of f0over∆x, f00avg(7)(8)Roy M. Lowman Second Derivativesecond derivativedefinition of f00f00(x) gives the concavity of function f(x)f00(x) is the rate of change of slope w.r.t xf00(x) is (+) where f(x) is concave up. (holds