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The concept of a strictly decreasing function is explained using a specific function y = x2, which is defined for all real numbers. Suppose A in figure a is the point with coordinates (—3, 9) and B is the point (—2, 4). Over the interval from A to B, the x values increase in size from —3 to —2, but the y values decrease in size from 9 to 4. As x increases, y decreases, and we say that the function y = x2 is strictly decreasing over the interval from A to 15. The gradient of the line segment AS is negative, and this also is an indication that the function is strictly decreasing over this interval. In
fact the function y = x2 is strictly decreasing for all values of x that are less than zero.
Consider another two points on the curve, C(l, 1) and D(2, 4). Over this interval from C to D, the x values increase in size from 1 to 2, and the y values also increase in size from 1 to 4. As x increases, y increases, and we say that the function y = x2 is strictly increasing over the interval from C to D. The gradient of the line segment CD is positive, and this also is an indication that the function is strictly increasing over this interval. In fact the function y = x2 is strictly increasing for all values of x that are greater than zero.
At the origin, where the gradient is zero, the function y = x2 is neither strictly decreasing nor strictly increasing. For this reason it is called a stationary point.
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0 x 0 X
If a function is strictly increasing throughout its domain, we say it is a strictly increasing function. For example the function y = 2X (see figure b) is a strictly increasing function, because it is strictly increasing for all values of x. Similarly, the function y = 2~x is a strictly decreasing function.
References: Function, Gradient, Stationary Point.
There are three different meanings for the word degree as generally used in mathematics.
1. A degree is one of the units used for measuring the size of an angle, which is an amount of turning. Other units are radians and gradians. A degree is a very small amount of turning and in size it is 1/360 of a full turn; or complete turn. There are 360 degrees in a full turn; in symbols we write this as 360°. Angles in degrees are measured with a protractor. A person in the army would soon leam that a half turn is the same as an “about turn” and is a rotation of 180°. This division of a full turn into 360 equal parts called degrees was devised by the Babylonians about 3000 years ago and is still in use today. 360 is a useful figure to use, because it has so many factors.
2. A degree is a unit for measuring the temperature of something with a thermometer. Two of the units that may be used are degrees Celsius and degrees Fahrenheit.
3. The degree of a polynomial is the highest power of the variable. For example,
the polynomial 3x4 — 2x2 + 1 is of degree 4, because the highest power of the variable x is 4. A polynomial does not have negative powers.
References: Acute Angle, Angle, Protractor, Radian.
DEPENDENT VARIABLE 153
This is the fourth letter of the Greek alphabet; the lower case letter is <5, and the capital letter is A. The capital letter A is commonly used in mathematics for the discriminant of a quadratic equation or as a very small, but finite, increase of a variable. For example, Ax is a small increase in x.
References: Quadratic Equations, Variable.
Reference: Decimal System.
A fraction is made up of two numbers, one above the other. The top number is called the numerator of the fraction and the bottom number is called the denominator of the fraction. The numerator is sometimes called the dividend and the denominator is sometimes called the divisor. For example, the numerator of the fraction | is 2 and the denominator is 3. The mixed number 2| must first be written as the improper fraction 19/7, which has a numerator of 19 and a denominator of 7. The algebraic fraction 3ab/4x2 has a numerator of 3ab and a denominator of 4x2.
References: Algebraic Fractions, Fraction, Lowest Common Denominator, Mixed Number. DEPENDENT VARIABLE
Suppose William has $30 at present, and decides to save $5 each week. After x weeks suppose he has saved $y. The equation connecting the variables x and y is y = 5x + 30. In this equation the amount of money saved (y) is dependent on the number of weeks (x) he has been saving, and so y is called the dependent variable. The number of weeks x is called the independent variable. When we graph the equation (see figure) the independent variable x goes on the horizontal axis and the dependent variable goes on the vertical axis.
References: Equations, Straight-Line Graph, Variable.
DEPRESSION (ANGLE OF)
Reference: Angle of Depression.
This is the distance from the top to the bottom of an object, whereas the height of an object is the distance from the bottom to the top. So the depth and the height are the same distance, but viewed from different positions. We talk about the depth of a hole or the depth of a swimming pool when viewed from the surface, but we talk about the height of a tower or a power pole when viewed from ground level.