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Mini Math Wars

Mini Math Wars. Review of Limits Windows vs. Doors I will pick a person to answer. If correct = +1; If incorrect  other team can steal that point. Round is over after pts awarded; question goes to other side of room. Mini Math Wars. #1) . Mini Math Wars. #2) . Mini Math Wars. #3) .

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Mini Math Wars

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  1. Mini Math Wars Review of Limits Windows vs. Doors I will pick a person to answer. If correct = +1; If incorrect  other team can steal that point. Round is over after pts awarded; question goes to other side of room.

  2. Mini Math Wars #1)

  3. Mini Math Wars #2)

  4. Mini Math Wars #3)

  5. Mini Math Wars #4)

  6. Mini Math Wars #5)

  7. Mini Math Wars #6)

  8. Mini Math Wars #7)

  9. Mini Math Wars #8)

  10. Section 1.4 Target Goals: 1. Be able to evaluate continuity of a function using left and right hand limits 2. Be able to apply the Intermediate Value Theorem to closed interval functions. Continuity and One-Sided Limits

  11. Definition of continuity • A function f(x) is continuous at x=c if all of the following conditions exist: • The function has a value at x=c, i.e., f(c) exists. • The limit exists at c (check left and right hand sides). • The limit at c = f(c).

  12. A function is everywhere continuous if it is continuous at each point in its domain. • A function is continuous on an interval if it is continuous at every point on that interval.

  13. Example 1 • Is the function continuous at x=2? • Does f(2) exist? • Does the limit exist at x=2? • Does the limit equal f(2)? YES! The function is CONTINUOUS!! YES! 3 3 YES! YES!

  14. Example 2 • Is the function continuous at x=2? • Does f(2) exist? • Does the limit exist at x=2? • Does the limit equal f(2)? YES! 3 5 NO! NOT CONTINUOUS at x = 2!

  15. Types of Discontinuities • Jump discontinuity - the curve breaks at a particular place and starts somewhere else (non-removable) • Point discontinuity - curve has a hole in it and a point that’s off of the curve (removable) • Essential discontinuity - vertical asymptote (non-removable)

  16. Ex 3) Examples of Discontinuities IF the function is continuous, then the limit must exist!

  17. Ex 4) Where does the following function have… • An essential discontinuity? At vertical asymptotes! x = -4 • A removable discontinuity? What can you cancel? x = 5

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

  19. Ex 5) Example applying IVT Verify that the IVT applies to the given function on the given interval. Then find the value of c guaranteed by the theorem such that f(c) = 0. 1. Is f (x) continuous on [0, 3]? 2. Is 0 in the interval [f (0), f ( 3)]? Yes Yes

  20. Target Goals • Be able to evaluate continuity of a function using left and right hand limits. • Be able to apply the Intermediate Value Theorem to closed interval functions.

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