Mathematics 1 Applied Informatics tefan BEREN MATHEMATICS 1
Mathematics 1 Applied Informatics Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics
2 nd lecture Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics
Contents • The Derivative • Applications of Differentiation Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 3
The Derivative Definition: Let a be a point of the domain of f(x). The derivative of f(x) at x = a is the limit: provided this limit exists. If it does exist, we say f(x) is differentiable at x = a, otherwise f(x) is not differentiable at x = a. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 4
The Derivative There is a useful alternative form of the limit defining the derivative. Replace a + h by x, and note that x a is equivalent to h 0. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 5
The Derivative Differentiation Rules: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 6
The Derivative Differentiation Rules: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 7
The Derivative Differentiation Formulas: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 8
The Derivative Differentiation Formulas: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 9
Applications of Differentiation We sometimes refer to a differentiable function as a smooth function and to its graph as a smooth graph or smooth curve. Let y = f(x) be a smooth function, and let P = a, f(a) be a point on its graph. By the slope of the graph at P we mean simply derivative f (a). Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 10
Applications of Differentiation The tangent to the graph at P is the line passing trough P whose slope equals the slope f (a) of the graph at P. By the pointslope from the equation of this line is: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 11
Applications of Differentiation Mean Value Theorem (Lagrange’s Theorem): Let function f be continuous on the closed interval a, b and let it be differentiable on the open interval (a, b). Then there exists a point (a, b) such that: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 12
Applications of Differentiation Theorem: Let f be a continuous function on interval I = a, b. Then the following implications hold: • f (x) 0 for all x (a, b) f is increasing on interval I. • f (x) 0 for all x (a, b) f is non-decreasing on interval I. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 13
Applications of Differentiation • f (x) 0 for all x (a, b) f is decreasing on interval I. • f (x) 0 for all x (a, b) f is non-increasing on interval I. • f (x) = 0 for all x (a, b) f is a constant function on interval I. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 14
Applications of Differentiation Theorem: If function f has a local extreme value at point x 0 and if f is differentiable at this point then f (x 0) = 0. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 15
Applications of Differentiation Theorem: The only points where function f can have a local extreme value in an interval I = a, b are: • points of interval (a, b) where f is equal to zero, • points of interval (a, b) where f does not exist. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 16
Applications of Differentiation Theorem: The only points where function f can have an absolute extreme value in an interval I = a, b are: • points of interval (a, b) where f is equal to zero, • points of interval (a, b) where f does not exist, • endpoints of interval (a, b) (if interval I is not open). Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 17
Applications of Differentiation Definition: Function f is called strictly concave up on set M if M D(f) and if for each tree points x 1, x 2, x 3 M such that x 1 x 2 x 3, is holds that: The point Q 2 = x 2, f(x 2) is below the straight line Q 1 Q 3, where Q 1 = x 1, f(x 1) and Q 3 = x 3, f(x 3). Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 18
Applications of Differentiation Definition: Function f is called strictly concave down on set M if M D(f) and if for each tree points x 1, x 2, x 3 M such that x 1 x 2 x 3, is holds that: The point Q 2 = x 2, f(x 2) is above the straight line Q 1 Q 3, where Q 1 = x 1, f(x 1) and Q 3 = x 3, f(x 3). Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 19
Applications of Differentiation Definition: Function f is called concave up on set M if M D(f) and if for each tree points x 1, x 2, x 3 M such that x 1 x 2 x 3, is holds that: The point Q 2 = x 2, f(x 2) is below or on the straight line Q 1 Q 3, where Q 1 = x 1, f(x 1) and Q 3 = x 3, f(x 3). Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 20
Applications of Differentiation Definition: Function f is called concave down on set M if M D(f) and if for each tree points x 1, x 2, x 3 M such that x 1 x 2 x 3, is holds that: The point Q 2 = x 2, f(x 2) is above or on the straight line Q 1 Q 3, where Q 1 = x 1, f(x 1) and Q 3 = x 3, f(x 3). Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 21
Applications of Differentiation The condition saying that Q 2 = x 2, f(x 2) finds itself below the straight line Q 1 Q 3, where Q 1 = x 1, f(x 1) and Q 3 = x 3, f(x 3) , can be computatively expressed by the inequality: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 22
Applications of Differentiation Theorem: Let function f be continuous on interval I = a, b. Then the following implications hold: • f (x) 0 for all x (a, b) f is strictly concave up on interval I. • f (x) 0 for all x (a, b) f is strictly concave down on interval I. • f (x) 0 for all x (a, b) f is concave down on interval I. • f (x) = 0 for all x (a, b) f is a linear function on interval I. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 23
Applications of Differentiation Definition: Suppose that function f is differentiable at point x 0 (and, consequently, there exists a tangent to the graph of f at the point x 0, f(x 0) ). The tangent divides the x, y plane into two half-planes. If the tangent passes from one half -plane to the other at the point x 0, f(x 0) then x 0 is called the point of inflection or the inflection point of function f. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 24
Applications of Differentiation Theorem: If f (x 0) = 0 and f (x 0) 0, then function f has a strict local minimum at point x 0. If f (x 0) = 0 and f (x 0) 0, then function f has a strict local maximum at point x 0. Theorem: If x 0 is an inflection point of function f and if the second derivative f (x 0) exists, then f (x 0) = 0. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 25
Thank you for your attention. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 26
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