Understanding Derivatives and Their Applications in Mathematics

Mathematics1 Applied Informatics Štefan BEREŽNÝ

2nd lecture

Contents • The Derivative • Applications of Differentiation MATHEMATICS 1 Applied Informatics

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, wesay f(x) is differentiable at x = a, otherwise f(x) is not differentiable at x = a. MATHEMATICS 1 Applied Informatics

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. MATHEMATICS 1 Applied Informatics

The Derivative Differentiation Rules: MATHEMATICS 1 Applied Informatics

The Derivative Differentiation Formulas: MATHEMATICS 1 Applied Informatics

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). MATHEMATICS 1 Applied Informatics

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 point-slope from the equation of this line is: MATHEMATICS 1 Applied Informatics

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: MATHEMATICS 1 Applied Informatics

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. MATHEMATICS 1 Applied Informatics

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. MATHEMATICS 1 Applied Informatics

Applications of Differentiation Theorem: If function f has a local extreme value at point x0 and if f is differentiable at this point thenf(x0) = 0. MATHEMATICS 1 Applied Informatics

Applications of Differentiation Theorem: The only points where function fcan 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. MATHEMATICS 1 Applied Informatics

Applications of Differentiation Theorem: The only points where function fcan have anabsolute 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). MATHEMATICS 1 Applied Informatics

Applications of Differentiation Definition: Function f is called strictly concave up on set Mif MD(f) and if for each tree points x1, x2, x3M such that x1x2x3, is holds that: The point Q2 = x2, f(x2) is below the straight line Q1Q3, where Q1 = x1, f(x1) and Q3 = x3, f(x3). MATHEMATICS 1 Applied Informatics

Applications of Differentiation Definition: Function f is called strictly concave down on set Mif MD(f) and if for each tree points x1, x2, x3M such that x1x2x3, is holds that: The point Q2 = x2, f(x2) is above the straight line Q1Q3, where Q1 = x1, f(x1) and Q3 = x3, f(x3). MATHEMATICS 1 Applied Informatics

Applications of Differentiation Definition: Function f is called concave up on set Mif MD(f) and if for each tree points x1, x2, x3M such that x1x2x3, is holds that: The point Q2 = x2, f(x2) is below or on the straight line Q1Q3, where Q1 = x1, f(x1) and Q3 = x3, f(x3). MATHEMATICS 1 Applied Informatics

Applications of Differentiation Definition: Function f is called concave down on set Mif MD(f) and if for each tree points x1, x2, x3M such that x1x2x3, is holds that: The point Q2 = x2, f(x2) is above or on the straight line Q1Q3, where Q1 = x1, f(x1) and Q3 = x3, f(x3). MATHEMATICS 1 Applied Informatics

Applications of Differentiation The condition saying that Q2 = x2, f(x2) finds itself below the straight line Q1Q3, where Q1 = x1, f(x1) and Q3 = x3, f(x3), can be computatively expressed by the inequality: MATHEMATICS 1 Applied Informatics

Applications of Differentiation Theorem: Let function f be continuous on interval I = a, b. Then thefollowing 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 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 intervalI. MATHEMATICS 1 Applied Informatics

Applications of Differentiation Definition: Suppose that function f is differentiable at point x0 (and, consequently, there exists a tangent to the graph of f at the point x0, f(x0)). 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 x0, f(x0) then x0 is called the point of inflection or the inflection point of function f. MATHEMATICS 1 Applied Informatics

Applications of Differentiation Theorem: If f(x0) = 0 and f(x0)  0, then function f has a strict local minimum at point x0. If f(x0) = 0 andf(x0)  0, then function f has a strict local maximum at point x0. Theorem: If x0 is an inflection point of function f and if the second derivative f(x0) exists, then f(x0) = 0. MATHEMATICS 1 Applied Informatics

Thank you for your attention. MATHEMATICS 1 Applied Informatics

Understanding Derivatives and Their Applications in Mathematics