Differentials
Differentials
Previously, The derivative was defined as:
Calculus
Forum reference:
Book page:
http://course-notes.org/Calculus
End Point Extrema
End Point Extrema:
Relative max. or min. that occur at points other than critical points are end point extrema. The end points must also be part of the domain in order for it to be considered a max. or min.
Point of Inflection
Point of Inflection:
Point of inflection of a graph of a function is the point on the graph such that the concavity of the function changes at that point.
Point of inflection occur when the second derivative of a function is zero or undefined.
Second Derivative Test for Max and Min Points
Second Derivative Test for Max and Min Points
If point A(a,f(a)) is on the graph of function f such that f '(a) = 0 and f ''(a) < 0 , then point A is a relative maximum; if f '(a) = 0 and f ''(a) > 0 , then point A is a relative minimum.
Concavity
Concavity:
If the second derivative of a function f ( f ''(x) ) is positive (or negative) for all x on (a,b) then the graph of f is concave upward (or downward) on (a,b).
Critical Point
Critical Point:
A critical point, (x,f(x)) , of a function f is if f(x) is defined and f '(x) is either zero or undefined. The x-coordinate of the critical point is called a critical value or a critical number.
Absolute Max and Min of a Function
Absolute Max and Min of a Function:
Increasing and Decreasing Functions
Increasing and Decreasing Functions:
A function f is said to be increasing when f '(x) > 0 for every x on (a,b) and decreasing when f '(x) < 0 for every x on (a,b).
Rolle's Theorem
Rolle's Theorem:
If a and b are numbers (a<b) and f is a function, such that f is continuos at [a,b];
f '(x) exist for all x at (a,b) and f(a) = f(b) = 0 then, there is a number c in (a,b) such that f(c) = 0.
Mean-Value Theorem
Mean-Value Theorem and Rolle's Theorem are used to find the relative max. and min. of a function.
Mean-Value Theorem:
If a and b are numbers (a<b) and f is a function, such that f is continuos at [a,b];
f '(x) exist for all x at (a,b) then, there is a number c in (a,b) such that:Â
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