Given a function f(x) that is continuous on the interval, if d is between f(a) and f(b), then there exists a c between a and b such that f(c) = d.Ħ. This highlights the importance of f(x) being continuous.ĥ. There are numerous ways to draw discontinuous functions that will not reach a height of zero. Much like the function must reach a height of zero (as described in Number 2 above), the function must also reach all other heights between -4 and 3. If the function begins with negative values at x = 1 but eventually has positive values at x = 5, the function had to equal zero (change from positive to negative) somewhere in. A continuous function will have both a minimum and maximum value. If the function is continuous, it must cross the x-axis at some point in the interval. There are many possible answers, but here’s one:Ģ. Note that in my solution for problem 1, fhas 3 roots, although only 1 was guaranteed.ġ. Give one real-life example of the Intermediate Value Theorem’s guarantees.ĪLERT! The Intermediate Value Theorem guarantees that I will hit each intermediate height at least once. Rewrite the Intermediate Value Theorem in your own words to better illustrate its meaning.ħ. Intermediate Value Theorem: Given a function f(x) that is _ on the interval _, if d is between _ and _, then there exists a c between _ and _ such that f(c) = _.Ħ. Based on your work above and the diagram below, complete the Intermediate Value Theorem below.
Draw three different graphs off(x) that are discontinuous and, therefore, do not fulfill the conclusion you drew in Number 2.ĥ. Which of the following height(s) is the function guaranteed to reach, and why?Ĥ.
Decide which of the following must occur between x = 1 and x = 5. You are not given an equation that defines f(x)-only these points.ĭraw one possible graph of f(x) on the axes above.Ģ. Consider a continuous function f(x), which contains points (1,-4) and (5,3). Once again, continuity is a cornerstone of this theorem.ġ. Much like the Extreme Value Theorem guaranteed the existence of a maximum and minimum, the Intermediate Value Theorem guarantees values of a function but in a different fashion. HANDS-ON ACTIVITY 3.3: THE INTERMEDIATE VALUE THEOREM
0 kommentar(er)
