Intermediate Value Theorem

1. Intermediate Value Theorem 2.4 cont.

3. Examples • If between 7am and 2pm the temperature went from 55 to 70. • At some time it reached 62. • Time is continuous • If between his 14th and 15th birthday, a boy went from 150 to 165 lbs. • At some point he weighed 155lbs. • It may have occurred more than once.

4. Show that a “c” exists such that f(c)=2 for f(c)=x^2 +2x-3 in the interval [0, 2] f(x) is continuous on the interval f(0)= -3 f(2)= 5

5. Determine if f(x) has any real roots f(x) is continuous on the interval f(1)= - # f(2)= + #

6. Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5. Therefore there must be at least one solution between 1 and 2. Use your calculator to find an approximate solution. Is any real number exactly one less than its cube? (Note that this doesn’t ask what the number is, only if it exists.)

7. Max-Min Theorem for Continuous Functions If f is continuous at every point of the closed interval [a, b], then f takes on a minimum value m and a maximum value M on [a, b]. That is, there are numbers α and β in [a, b] such that f(α) = m, f(β) = M, and m ≤ f(x) ≤ M at all points x in [a, b] Max and mins at interior points Min at interior point and max at endpoint Max and mins at endpoints

8. Why does the IVT fail to hold for f(x) on [-1, 1]? Not Continuous in interval! Point of discontinuity at x = 0

9. Show why a root exists in the given interval Continuous in interval f(-2)= -10 f(-1)= 1

10. Show why a root exists in the given interval Continuous in interval f(1)= 1