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Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.
However, if the second derivative is only positive between and +, or only negative (as in the diagram), the curve will increasingly veer away from the tangent, leading to larger errors as increases. The diagram illustrates that the tangent at the midpoint (upper, green line segment) would most likely give a more accurate approximation of the ...
Spivak, Michael (1965), Calculus on manifolds. A modern approach to classical theorems of advanced calculus , W. A. Benjamin Tao, Terence (2006), Analysis II (PDF) , Texts and Readings in Mathematics, vol. 38, Hindustan Book Agency, doi : 10.1007/978-981-10-1804-6 , ISBN 8185931631
Elementary Calculus: An Infinitesimal Approach; Nonstandard calculus; Infinitesimal; Archimedes' use of infinitesimals; For further developments: see list of real analysis topics, list of complex analysis topics, list of multivariable calculus topics
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
This part is sometimes referred to as the second fundamental theorem of calculus [7] or the Newton–Leibniz theorem. Let f {\displaystyle f} be a real-valued function on a closed interval [ a , b ] {\displaystyle [a,b]} and F {\displaystyle F} a continuous function on [ a , b ] {\displaystyle [a,b]} which is an antiderivative of f ...
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [l] is defined as the linear part of the change in the functional, and the second variation [m] is defined as the quadratic part. [22]
It can be thought of as the rate of change of the function in the -direction.. Sometimes, for = (,, …), the partial derivative of with respect to is denoted as . Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:
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