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WARM UP Graph the piecewise function then use it to evaluate each expression.

WARM UP Graph the piecewise function then use it to evaluate each expression. Continuity & Intermediate Value Theorem. Continuity. What does it mean for a graph to be continuous? Informal definition: A function is continuous if it can be drawn without picking up your pencil.

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WARM UP Graph the piecewise function then use it to evaluate each expression.

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  1. WARM UPGraph the piecewise function then use it to evaluate each expression.

  2. Continuity & Intermediate Value Theorem

  3. Continuity What does it mean for a graph to be continuous? • Informal definition: • A function is continuous if it can be drawn without picking up your pencil. • Formal definition: • A function is continuous at x = c if the following three conditions are met. • f(c) exists

  4. Removable vs. non-removable discontinuity Removable = hole in the graph Non-removable = asymptote or jump

  5. Discuss the continuity of each function…

  6. Discuss the continuity of the function on the closed interval.

  7. Determine the value of a so that g(x) is continuous.

  8. Determine the value of b so that f(x) is continuous.

  9. If f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists at least one number c in the interval [a,b] such that f(c) = k. Intermediate Value Theorem (IVT)

  10. Explain why the function f(x) = x3 + 2x - 1 has a zero in the interval [0,1].

  11. Let f(x) = x2 + 3x - 6 Use the Intermediate Value Theorem to show that there is at least one value for c in [0, 4] such that f(c) = 12. Then find any value(s) of c guaranteed by the theorem.

  12. The functions f and g are continuous for all real numbers. The table shows values of the functions at selected values of x. The function h is given by h(x) = f(g(x)) - 6. Explain why there must be a value r, 1 < r < 3, such that h(r) = -5.

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