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The top grade, A, is given here for performance that exceeds the mean by more than 1.5 standard deviations, a B for performance between 0.5 and 1.5 standard deviations above the mean, and so on. [17] Regardless of the absolute performance of the students, the best score in the group receives a top grade and the worst score receives a failing grade.
Grading in education is the application of standardized measurements to evaluate different levels of student achievement in a course. Grades can be expressed as letters (usually A to F), as a range (for example, 1 to 6), percentages, or as numbers out of a possible total (often out of 100).
In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
The Taylor polynomial of degree d is the polynomial of degree d which best approximates f, and its coefficients can be found by a generalization of the above formulas. Taylor's theorem gives a precise bound on how good the approximation is. If f is a polynomial of degree less than or equal to d, then the Taylor polynomial of degree d equals f.
This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The second line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a + h.
Marston Morse applied calculus of variations in what is now called Morse theory. [6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. [6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations. [7] [8] [9] [c]
Sometimes the − is used to indicate a better grade if it stands after the grade and a lower grade if it stands before the grade (in which case − is a symbol for "bis", e.g. 'to', rather than 'minus'), for example −5 (4.75) is lower than 5 which is lower than 5− (5.25) in that system. In some regions, decimal grades are used: 5.5, 4.5, etc.
Difference quotients may also find relevance in applications involving Time discretization, where the width of the time step is used for the value of h. The difference quotient is sometimes also called the Newton quotient [10] [12] [13] [14] (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat). [15]