Mean Value Theorem
What is the Mean Value Theorem?
The Mean Value Theorem is another important application of derivatives which is useful to understand. Firstly, let f be a function which satisfies the following two conditions:
1) [latex]f[/latex] is differentiable on the interval (a,b)
2) [latex]f[/latex] is continuous on the interval (a,b)
If this is the case, then:
[latex]f'(c) = \frac{f(b) – f(a)}{b – a}[/latex]
Now let us only concern ourselves with the interval (a,b). The slope of the line that connects points and b is determined as:
[latex]m_{ab} = \frac{f(b)-f(a)}{b-a}[/latex]
Furthermore, the slope of the tangent line at point c is denoted by [latex]f'(c)[/latex], which we can visualize is the same as [latex]m_{ab}[/latex]. This essentially summarizes the mean value theorem, an important concept in calculus, which is used as the basis for other rules and proofs such as L’Hopital’s Rule.
Example 4
Find all values of c that satisfy the mean value theorem for the function:
[latex]\large{f(x) = x^3+x^2-2x}[/latex]
Interval: [latex][0,2][/latex]
First, since the given function is a polynomial, we know that it is continuous and differentiable on the given interval. This means that the mean value theorem applies.
Step 1:
First, let’s find the slope of the line which connects the two endpoints of the interval, which are given as a = 0 and b = 2. So,
[latex]\large{m_{ab} = \frac{f(b)-f(a)}{b-a} = \frac{f(2)-f(0)}{2-0}}[/latex]
We can easily see that [latex]f(2) = 8[/latex] and [latex]f(0) = 0[/latex]
Thus, the slope is found to be:
[latex]\large{m_{ab} = \frac{8-0}{2-0} = \frac{8}{2} = 4}[/latex].
Step 2:
Now, we need to find all the points on the interval which have the same slope. We know that the derivative of the function gives the slope, so, we will find the derivative and set it to 4.
Using the power rule, the derivative of the function is:
[latex]\large{f'(x) = 3x^2+2x-2}[/latex]
Setting the derivative equal to 4 we have:
[latex]\large{3x^2 +2x-2 = 4}[/latex]
[latex]\large{3x^2+2x-6 = 0}[/latex]
Now, using the quadratic formula to solve for x gives us:
[latex]\large{x = 1.120 }[/latex] and [latex]\large{x = -1.786}[/latex].
Although both values work, only the first solution falls within the given interval. Thus, x = 1.120 satisfies the mean value theorem for the given function.
Need more explanation? Chat with a tutor now!
QUESTIONS?
If you have any questions about our services or have any feedback, do not hesitate to get in touch with us!
