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Index FAQ Basic Functions Polynomials Exponential Functions Trigonometric Functions Trigonometric Identities The Number e.

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Presentation on theme: "Index FAQ Basic Functions Polynomials Exponential Functions Trigonometric Functions Trigonometric Identities The Number e."— Presentation transcript:

1 Index FAQ Basic Functions Polynomials Exponential Functions Trigonometric Functions Trigonometric Identities The Number e

2 Index FAQ 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.

3 Index FAQ 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)

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

5 Index FAQ Mika Seppälä: Basic Functions Measuring of Angles (1) 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. 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. 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. Starting side Ending side Length of the arc = the size of the angle in radians. Circle of radius 1

6 Index FAQ 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.

7 Index FAQ 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 πr 2. Disk of radius r Length of the arc = αr. Area of the sector Angle of size α radians.

8 Index FAQ Mika Seppälä: Basic Functions 1 Trigonometric Functions (1) Definition 1 Consider positive angles α as indicated in the picture. 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)). This definition applies to positive angles α. We extend that to negative angles by setting sin(- α) = - sin(α) and cos(- α) = cos(α).

9 Index FAQ Mika Seppälä: Basic Functions Trigonometric Functions (2) Definition This basic identity follows directly from the definition. Graphs of: 1.sin(x), the red curve, and 2.cos(x), the blue curve.

10 Index FAQ Mika Seppälä: Basic Functions Trigonometric Functions (3) Lemma 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. The above inequality is obvious by the above picture. For negative angles α the inequality is reversed.

11 Index FAQ 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. Lemma This means that, for positive angles α, we have:

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

13 Index FAQ Mika Seppälä: Basic Functions Examples Problem 1 Solution

14 Index FAQ Mika Seppälä: Basic Functions Examples Problem 2 Solution

15 Index FAQ Mika Seppälä: Basic Functions Trigonometric Identities 1 Defining Identities Derived Identities

16 Index FAQ Mika Seppälä: Basic Functions Trigonometric Identities 2 Derived Identities (contd)

17 Index FAQ Mika Seppälä: Basic Functions Exponential Functions

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


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