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

2. 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.
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3. 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)
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4. 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.
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5. 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.
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6. 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.
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7. 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
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8. 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).
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9. 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.
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10. 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.
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11. 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
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12. Trigonometric Functions (4)
Shown before
Lemma
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13. Mika Seppälä: Basic Functions
Examples
Problem 1
Solution
Mika Seppälä: Basic Functions

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

15. Trigonometric Identities 1
Defining Identities
Derived Identities
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16. Trigonometric Identities 2
Derived Identities (cont’d)
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17. Exponential Functions
Mika Seppälä: Basic Functions

18. 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 
Mika Seppälä: Basic Functions