Course Navigator
Back To Course
Math 104: Calculus13 chapters | 104 video lessons
Erin has taught math and science from grade school up to the post-graduate level. She holds a Ph.D. in Chemical Engineering.
The intermediate value theorem says that if you have some function f(x) and that function is a continuous function, then if you're going from a to b along that function, you're going to hit every value somewhere in that region (a to b). Well, why is this useful? This helps you learn a lot about functions without having to graph them.
For example, if you have the function f(x)= x^3 + x^2, you can start to take a look at what f(x) equals for various values of x, like if x=0, then f(x)= 0 ^3 + 0^2, or just zero. When x=1, f(x) = 1^3 + 1^2, or 2. When x=2, f(x) = 2^3 + 2^2, which is 12.
x | f(x) | |
---|---|---|
0 | 0 | |
1 | 2 | |
2 | 12 |
Okay, great, so you've got this table here of x values and f(x) values. Well, we know that f(x) is a continuous function, so we can use this data to determine that f(x) is going to equal 1 somewhere between 0 and 1. How do we know this? Well, f(0)=0, and f(1)=2, so some value between 0 and 1 will give me f(x)=1. Another example is f(x)=10. Well, for what value of x does f(x)=10? I don't know, it's not actually on my chart, but I know that f(1)=2, and f(2)=12, so some value between 1 and 2 will give me f(x)=10. I can graph this to verify that f(x)=1 between 0 and 1, and f(x)=10 between 1 and 2.
Let's look at another example. In this example we're going to be finding roots of an equation. So let's say we have f(x)= 4x - x^2 - 3. We want to know when f(x)=0. This is called finding the roots of f(x). Similar to the last example, we're going to make a table with x and f(x). When x=0, f(x)=-3, because we have (4)(0) - 0^2 - 3. When x=2, f(x)=1. When x=4, f(x)=-3.
x | f(x) | |
---|---|---|
0 | -3 | |
2 | 1 | |
4 | -3 |
So does this tell us when f(x) will equal zero? Well let's take a look at the three values we calculated and put them on a graph. So I've got f(x) and x. When x=0, f(x)=-3. When x=2, f(x)=1. When x=4, f(x)=-3. Now because I know 4x - x^2 - 3 is a continuous function, I know that to get from -3 to 1, my function has to travel through f(x)=0. Similarly, to go from 1 to -3, f(x) has to pass through zero. So at some point, between 0 and 2, I have a root - there is some place where f(x)=0 for an x value between 0 and 2. Similarly, between 2 and 4, I know that f(x) will equal zero at some point, so there's an x value between 2 and 4. I know for this particular equation, I have at least two roots: one between 0 and 2, and one between 2 and 4. We still could have solved that problem by factoring, so what about another example?
Continue reading... Create an account to read entire transcript...What about trying to find a solution to x^2=cos(x)? There's no f(x) in here, so where do we start? Well let's first subtract x^2 from both sides, so we get 0=cos(x) - x^2. If instead of saying 0, I called this f(x), then I'm just trying to find the roots of the equation cos(x) - x^2. And this I know how to do, so let's make a table. When x=0, f(x)= cos(0) - 0^2, or 1. When x=pi, f(x)=cos(pi), which is -1, - pi^2. So I'm going to leave this as -1 - pi^2. Let's plot these two points. First, I have f(x)=1 when x=0, then I have at x=pi, f(x)= -1 - pi^2. Again this is a continuous function, so somewhere between 0 and pi, this has to have at least one solution. The answer to x^2 = cos(x) is going to be some value of x between 0 and pi.
Let's review. The intermediate value theorem says that if you're going between a and b along some continuous function f(x), then for every value of f(x) between f(a) and f(b), there is some solution. If I'm going between a and b, I'm going to hit every value between f(a) and f(b).
Did you know… We have over 100 college courses that prepare you to earn credit by exam that is accepted by over 2,900 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.
To learn more, visit our Earning Credit Page
Not sure what college you want to attend yet? Education Portal has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.