Derivative, Angle and Integers material

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Derivative

Derivative is a measure of how a function changes as its input changes. A derivative can be thought of as how much one quantity is changing in response to changes in some other quantity.

Let f  be a real valued function. In classical geometry, the tangent line to the graph of the function f at a real number a was the unique line through the point (a, f(a)) that did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at a. The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). These lines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope m of the secant line is the difference between the y values of these points divided by the difference between the x values, that is,

For example a function is f(x)=4x²-8x+3

Find the f’(3) .So its derivative is

We can find the general formula

So the general formula for a function  f(x)=4x²-8x+3 adalah f’(x)=8x-8

Angle

In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

 An angle equal to 1/4 turn (90° or π/2 radians) is called a right angle.

Two lines that form a right angle are said to be perpendicular or orthogonal.

Angles equal to 1/2 turn (180° or two right angles) are called straight angles.

Angles equal to 1 turn (360° or four right angles) are called full angles.

Angles that are not right angles or a multiple of a right angle are called oblique angles.

Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").

Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").

Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles.

 Angles that have the same measure  are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all right angles are congruent).

Two angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. These angles are equal in measure.

 Angles that share a common vertex and edge but do not share any interior points are called adjacent angles.

Two angles that sum to one right angle (90°) are called complementary angles.

Two angles that sum to a straight angle (180°) are called supplementary angles.

Two angles that sum to one turn (360°) are called explementary angles or conjugate angles.

Integers

An integer  is a whole number (not a fractional number) that can be positive, negative, or zero.

Examples of integers are: -5, 1, 5, 8, 97, and 3,043.

Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .09, and 5,643.1.

The set of integers, denoted Z, is formally defined as follows:

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}