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3.5 Continuity & End Behavior

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Discontinuous – you cannot trace the graph of the function without lifting your pencil. (step and piecewise functions) Infinite discontinuity – the absolute value if f(x) becomes greater and greater as the graph approaches a given x- value. Infinite discontinuity – the absolute value if f(x) becomes greater and greater as the graph approaches a given x- value. Jump discontinuity – the graph stops at a given value of the domain and then begins again at a different range value for the same value of the domain. Jump discontinuity – the graph stops at a given value of the domain and then begins again at a different range value for the same value of the domain. Point discontinuity – when a value in the domain of the function is undefined, but the pieces Point discontinuity – when a value in the domain of the function is undefined, but the pieces of the graph match up of the graph match up Everywhere discontinuous – impossible to graph in the real number system

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Continuity Test - A function is continuous at x = c if it satisfies the following conditions: 1. the function is defined at c; or f(c) exists 2. the function approaches the same y- value on the left and right sides of x = c 3. the y-value that the function approaches from each side is f(c).

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Ex 1 Determine whether the function is continuous at the given x-value. y = 3x 2 + x – 7; x = 1 y = 3x 2 + x – 7; x = 1

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Ex 2 Determine whether the function is continuous at the given x-value.

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Ex 3 Determine whether the function is continuous at the given x-value.

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Continuity on an interval – a function f(x) is continuous on an interval if and only if it is continuous at each number x in the interval Continuity on an interval – a function f(x) is continuous on an interval if and only if it is continuous at each number x in the interval Critical points – points in the domain of the function where the function changes from increasing to decreasing or vice versa. Critical points – points in the domain of the function where the function changes from increasing to decreasing or vice versa.

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End behavior – describes what the y- values do as the absolute value of x becomes greater and greater. When x becomes greater and greater we say that x approaches infinity. (same notation is used for f(x) or y and using real numbers instead of infinity.) End behavior – describes what the y- values do as the absolute value of x becomes greater and greater. When x becomes greater and greater we say that x approaches infinity. (same notation is used for f(x) or y and using real numbers instead of infinity.)

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Ex 4 Describe the end behavior: f(x) = 5x 3 f(x) = 5x 3 g(x) = -5x 3 + 4x 2 – 2x + 4 g(x) = -5x 3 + 4x 2 – 2x + 4

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