Mini math wars
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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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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

Mini Math Wars

#1)


Mini math wars

Mini Math Wars

#2)


Mini math wars

Mini Math Wars

#3)


Mini math wars

Mini Math Wars

#4)


Mini math wars

Mini Math Wars

#5)


Mini math wars

Mini Math Wars

#6)


Mini math wars

Mini Math Wars

#7)


Mini math wars

Mini Math Wars

#8)


Section 1 4

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


Definition of continuity

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).


Mini math wars

  • 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.


Example 1

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!


Example 2

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!


Types of discontinuities

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)


Ex 3 examples of discontinuities

Ex 3) Examples of Discontinuities

IF the function is continuous, then the limit must exist!


Ex 4 where does the following function have

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


Intermediate value theorem

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


Ex 5 example applying ivt

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


Target goals

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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