Democritus was a Greek philosopher who actually developed the atomic theory, he was also an excellent geometer. Democritus was a Greek philosopher who.

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Presentation transcript:

Democritus was a Greek philosopher who actually developed the atomic theory, he was also an excellent geometer. Democritus was a Greek philosopher who actually developed the atomic theory, he was also an excellent geometer. Democritus of Abdera 460 – 370 B.C. Democritus of Abdera 460 – 370 B.C.

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. Length of Fish Parking Meter Cost

Example 1

Example 2

Examples: “The composition of two continuous functions is continuous.”

Example 3

It is continuous at x = 0, x = 3 and x = 4. Take x = 0, since Hence, by definition f is continuous at x = 0. For the function below, discuss the integer values where y = f (x) is continuous and explain. Example y = f (x)

It is continuous at x = 0, x = 3 and x = 4. For the function below, discuss the integer values where y = f (x) is continuous and explain. Example y = f (x) Take x = 3, since Hence, by definition f is continuous at x = 3.

It is continuous at x = 0, x = 3 and x = 4. For the function below, discuss the integer values where y = f (x) is continuous and explain. Example y = f (x) Take x = 4, since Hence, by definition f is continuous at x = 4.

For the function below, discuss the integer values where y = f (x) is not continuous and explain. Example y = f (x) This function has discontinuities at x = 1 and x = 2. Take x = 1, since Hence, by definition f is discontinuous at x = 1.

For the function below, discuss the integer values where y = f (x) is not continuous and explain. Example y = f (x) This function has discontinuities at x = 1 and x = 2. Take x = 2, since Hence, by definition f is discontinuous at x = 2. Determine the intervals where y = f (x) is continuous.

Example 6

Jump Infinite Types of Discontinuities: Undefined Removable

From the graph of f, state the numbers at which f is discontinuous and describe the type of discontinuity. Example 7

Consider the function f has discontinuities at. a) What type of discontinuities occur at x = 1 and x =  1. Solution By definition, x =  1 is a vertical asymptote, infinite discontinuity. Example 8

Note: The other discontinuity at x =  1 can not be removed, since it is a vertical asymptote. b) Write a piece-wise function using f (x) that is continuous at x = 1. Take x = 1, it follows Hence, by definition f is continuous at x = 1.

Example 9

Example 10

Example 11

Example 12

Example 13