Lecture II. The elements of higher mathematics . The derivative of function. Lecture question s. Function R epresent ation o f a function Function derivative G eometric interpretation of function derivative Some differentiation rule s
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The elements of higher mathematics .
The derivative of function
Definition of function. Independent and dependent variables. The domain and the range
A relationship between two variables, typically x and y, is called a function if there is a rule that assigns to each value of x one and only one value of y.
When that is the case, we say that y is a function of argument of x.
The values that x may assume are called the domain of the function. We say that those are the values for which the function is defined.
There are many ways to represent or visualize functions: a function may be described by a formula, by a plot or graph, by an algorithm that computes it, by arrows between objects, or by a description of its properties. Sometimes, a function is described through its relationship to other functions (for example, inverse functions). In applied disciplines, functions are frequently specified by tables of values or by formulas. The equation y = ƒ(x) is viewed as a functional relationship between dependent and independent variables.
of the second derivative:
of a higher order derivatives:
At B and R points, where a function is increasing, the tangent makes an acute angle with the axis of x, hence the slope is positive. At M and Q points, where a function is decreasing, the tangent makes an obtuse angle with the axis of x, therefore the slope is negative.C and P points are maxima. N point is a minimum.