Second Derivative
Math165: Business Calculus
Roy M. Lowman
Spring 2010, Week6 Lec2
Roy M. Lowman Second Derivative
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An in-depth explanation of the concepts of first and second derivatives of a function, f(x). The first derivative, f'(x), represents the slope or rate of change of the function, while the second derivative, f''(x), indicates the rate of change of the slope or concavity of the function. the definition, calculation, and applications of these derivatives, including finding intervals of increase/decrease, critical points, and inflection points.
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Math165: Business Calculus
Roy M. Lowman
Spring 2010, Week6 Lec
definition of f′
Definition (first derivative)
df dx
= f′^ = lim ∆x−> 0
f(x + ∆x) − f(x) ∆x
= lim ∆x−> 0
∆f ∆x
∆f ∆x
average slope of f(x) over delta x, f avg′ (3)
(4)
definition of f′
f′(x) is the slope of the function f(x) f′(x) is the rate of change of the function f(x) w.r.t x f′(x) is (+) where f(x) is increasing. f′(x) is (−) where f(x) is decreasing. f′(x) can be used to find intervals where f(x) is increasing/decreasing f′(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f′(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f′(x) can tell you a lot about a function f(x) f avg′ = ∆f∆x can be used to extimate f′(x)
definition of f′
f′(x) is the slope of the function f(x) f′(x) is the rate of change of the function f(x) w.r.t x f′(x) is (+) where f(x) is increasing. f′(x) is (−) where f(x) is decreasing. f′(x) can be used to find intervals where f(x) is increasing/decreasing f′(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f′(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f′(x) can tell you a lot about a function f(x) f avg′ = ∆f∆x can be used to extimate f′(x)
definition of f′
f′(x) is the slope of the function f(x) f′(x) is the rate of change of the function f(x) w.r.t x f′(x) is (+) where f(x) is increasing. f′(x) is (−) where f(x) is decreasing. f′(x) can be used to find intervals where f(x) is increasing/decreasing f′(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f′(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f′(x) can tell you a lot about a function f(x) f avg′ = ∆f∆x can be used to extimate f′(x)
definition of f′
f′(x) is the slope of the function f(x) f′(x) is the rate of change of the function f(x) w.r.t x f′(x) is (+) where f(x) is increasing. f′(x) is (−) where f(x) is decreasing. f′(x) can be used to find intervals where f(x) is increasing/decreasing f′(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f′(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f′(x) can tell you a lot about a function f(x) f avg′ = ∆f∆x can be used to extimate f′(x)
definition of f′
f′(x) is the slope of the function f(x) f′(x) is the rate of change of the function f(x) w.r.t x f′(x) is (+) where f(x) is increasing. f′(x) is (−) where f(x) is decreasing. f′(x) can be used to find intervals where f(x) is increasing/decreasing f′(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f′(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f′(x) can tell you a lot about a function f(x) f avg′ = ∆f∆x can be used to extimate f′(x)
definition of f′
f′(x) is the slope of the function f(x) f′(x) is the rate of change of the function f(x) w.r.t x f′(x) is (+) where f(x) is increasing. f′(x) is (−) where f(x) is decreasing. f′(x) can be used to find intervals where f(x) is increasing/decreasing f′(x) = 0 or undefined where the sign of the slope can change, i.e. at CNs f′(x) = 0 at critical points The First Derivative Test can be used to determine what kind of critical points. f′(x) can tell you a lot about a function f(x) f avg′ = ∆f∆x can be used to extimate f′(x)
definition of f′′
Definition (second derivative)
d^2 f dx^2
d dx
f′(x) = lim ∆x−> 0
f′(x + ∆x) − f(x)′ ∆x
= lim ∆x−> 0
∆f′ ∆x
∆f′ ∆x
average slope of f′over∆x, f avg′′ (7)
(8)
definition of f′′
f′′(x) gives the concavity of function f(x) f′′(x) is the rate of change of slope w.r.t x f′′(x) is (+) where f(x) is concave up. (holds H 2 O) f′′(x) is (−) where f(x) is concave down. (makes letter A) f′′(x) can be used to find intervals where f(x) is concave up or concave down f′′(x) = 0 or undefined where the concavity can change f′′(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 f′(x) = 0. f′′(x) can tell you a lot about a function f(x) f avg′′ = ∆f ′ ∆x can be used to extimate^ f
′′(x)
definition of f′′
f′′(x) gives the concavity of function f(x) f′′(x) is the rate of change of slope w.r.t x f′′(x) is (+) where f(x) is concave up. (holds H 2 O) f′′(x) is (−) where f(x) is concave down. (makes letter A) f′′(x) can be used to find intervals where f(x) is concave up or concave down f′′(x) = 0 or undefined where the concavity can change f′′(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 f′(x) = 0. f′′(x) can tell you a lot about a function f(x) f avg′′ = ∆f ′ ∆x can be used to extimate^ f
′′(x)
definition of f′′
f′′(x) gives the concavity of function f(x) f′′(x) is the rate of change of slope w.r.t x f′′(x) is (+) where f(x) is concave up. (holds H 2 O) f′′(x) is (−) where f(x) is concave down. (makes letter A) f′′(x) can be used to find intervals where f(x) is concave up or concave down f′′(x) = 0 or undefined where the concavity can change f′′(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 f′(x) = 0. f′′(x) can tell you a lot about a function f(x) f avg′′ = ∆f ′ ∆x can be used to extimate^ f
′′(x)
definition of f′′
f′′(x) gives the concavity of function f(x) f′′(x) is the rate of change of slope w.r.t x f′′(x) is (+) where f(x) is concave up. (holds H 2 O) f′′(x) is (−) where f(x) is concave down. (makes letter A) f′′(x) can be used to find intervals where f(x) is concave up or concave down f′′(x) = 0 or undefined where the concavity can change f′′(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 f′(x) = 0. f′′(x) can tell you a lot about a function f(x) f avg′′ = ∆f ′ ∆x can be used to extimate^ f
′′(x)
definition of f′′
f′′(x) gives the concavity of function f(x) f′′(x) is the rate of change of slope w.r.t x f′′(x) is (+) where f(x) is concave up. (holds H 2 O) f′′(x) is (−) where f(x) is concave down. (makes letter A) f′′(x) can be used to find intervals where f(x) is concave up or concave down f′′(x) = 0 or undefined where the concavity can change f′′(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 f′(x) = 0. f′′(x) can tell you a lot about a function f(x) f avg′′ = ∆f ′ ∆x can be used to extimate^ f
′′(x)