Basic Functions Polynomials Exponential Functions Trigonometric Functions Trigonometric Identities The Number e

Mika Seppälä: Basic Functions Polynomials Definition The polynomial P is of degree n. A number x for which P(x)=0 is called a root of the polynomial P. Theorem A polynomial of degree n has at most n real roots. Polynomials may have no real roots, but a polynomial of an odd degree has always at least one real root. Mika Seppälä: Basic Functions

Graphs of Linear Polynomials Graphs of linear polynomials y = ax + b are straight lines. The coefficient “a” determines the angle at which the line intersects the x –axis. Graphs of the linear polynomials: 1. y = 2x+1 (the red line) 2. y = -3x+2 (the black line) 3. y = -3x + 3 (the blue line) Mika Seppälä: Basic Functions

Graphs of Higher Degree Polynomials Problem The picture on the right shows the graphs and all roots of a 4th degree polynomial and of a 5th degree polynomial. Which is which? Solution The blue curve must be the graph of the 4th degree polynomial because of its behavior as x grows or gets smaller. Mika Seppälä: Basic Functions

Mika Seppälä: Basic Functions Measuring of Angles (1) Ending side Angles are formed by two half-lines starting from a common vertex. One of the half-lines is the starting side of the angle, the other one is the ending side. In this picture the starting side of the angle is blue, and the red line is the ending side. Starting side Length of the arc = the size of the angle in radians. Angles are measured by drawing a circle of radius 1 and with center at the vertex of the angle. The size, in radians, of the angle in question is the length of the black arc of this circle as indicated in the picture. Circle of radius 1 In the above we have assumed that the angle is oriented in such a way that when walking along the black arc from the starting side to the ending side, then the vertex is on our left. Such angles are positive. Mika Seppälä: Basic Functions

Mika Seppälä: Basic Functions Measuring of Angles (2) The first picture on the right shows a positive angle. The angle becomes negative if the orientation gets reversed. This is illustrated in the second picture. This definition implies that angles are always between -2 and 2. By allowing angles to rotate more than once around the vertex, one generalizes the concept of angles to angles greater than 2 or smaller than - 2. Mika Seppälä: Basic Functions

Areas of Sectors of Disks By the definition of the mathematical constant π, the circumference or the length of a circle of radius r is 2πr. From this definition it also follows that the area of a disk of radius r is πr2. Length of the arc = αr. Angle of size α radians. Disk of radius r Area of the sector Mika Seppälä: Basic Functions

Trigonometric Functions (1) Consider positive angles α as indicated in the picture. Definition The quantities sin(α) and cos(α) are defined by placing the angle α at the origin with starting side on the positive x-axis. The intersection point of the ending side with the circle of radius 1 and with the center at the origin is (cos(α),sin(α)). 1 This definition applies to positive angles α. We extend that to negative angles by setting sin(- α) = - sin(α) and cos(- α) = cos(α). Mika Seppälä: Basic Functions

Trigonometric Functions (2) This basic identity follows directly from the definition. Definition Graphs of: sin(x), the red curve, and cos(x), the blue curve. Mika Seppälä: Basic Functions

Trigonometric Functions (3) The size of an angle is measured as the length α of the arc, indicated in the picture, on a circle of radius 1 with center at the vertex. On the other hand, sin(α) is the length of the red line segment in the picture. Lemma The above inequality is obvious by the above picture. For negative angles α the inequality is reversed. Mika Seppälä: Basic Functions

Trigonometric Functions (4) We know now: The sector of size α radians of the disk of radius 1 is included in the larger right angle triangle in the picture. Hence the area of the sector ≤ the area of the larger triangle. This means that, for positive angles α, we have: Lemma Mika Seppälä: Basic Functions

Trigonometric Functions (4) Shown before Lemma Mika Seppälä: Basic Functions

Mika Seppälä: Basic Functions Examples Problem 1 Solution Mika Seppälä: Basic Functions

Mika Seppälä: Basic Functions Examples Problem 2 Solution Mika Seppälä: Basic Functions

Trigonometric Identities 1 Defining Identities Derived Identities Mika Seppälä: Basic Functions

Trigonometric Identities 2 Derived Identities (cont’d) Mika Seppälä: Basic Functions

Exponential Functions Mika Seppälä: Basic Functions

Mika Seppälä: Basic Functions The Number e a=1/2 a=1 a=3/2 a=5/2 From the picture it is obvious that, as the parameter a grows, the slope of the tangent line of the graph of the function ax at x = 0 grows. Definition The mathematical constant e is defined as the unique number for which the slope of the tangent line of the graph of the function ex at x = 0 is 1. The slope of a tangent line is the tangent of the angle at which the tangent line intersects the x-axis. e  2.718281828 Mika Seppälä: Basic Functions