Help! I really do not understand solving functions and derivatines with the Mean Value Theorum. Can anyone help explain it? (my textbook is not helping at all!)
The MVT states that
If f is continuous on [a,b] and differentiable on (a,b), then there exists a number c on (a,b) such that
f'(c) = [f(b) - f(a)] / b - a
In other words, the instantaneous slope must equal the average slope at some point c. (Your slope from two points will be the same as the slope of the curve at some point.)
The method I use for applying the MVT to find c on an interval is this...
(1) differentiate the function
(2) note differentiability, continuity
(3) find the regular slope (m) of the secant line
(4) set the derivative equal to m
(5) solve for c
For example, "use the MVT to find c on (-2,2) for f(x)=x^3."
(1) f'(x) = 3x^2
(2) Because f(x) is a polynomial, it is differentiable, and therefore, continuous.
(3) m = [f(2) - f(-2)] / 2 - (-2) = (8 + 8) / (2 + 2) = 16 / 4 = 4
(4) f'(c) = 3c^2 from step 1
setting it equal to our slope m=4,
f'(c) = 3c^2 = 4
(5) c^2 = 4/3
c = +-√(4/3) = +- 2/√3
I hope that helps.
Of course, I'd be happy to help explain the Mean Value Theorem and how to apply it when solving functions and derivatives.
The Mean Value Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the open interval (a, b) where the instantaneous rate of change (the derivative) is equal to the average rate of change over the interval [a, b].
To better understand this theorem and its implications, let's break it down into steps:
1. First, ensure that the function you are working with is continuous on the interval [a, b] and differentiable on the interval (a, b).
2. Determine the values of f(a) and f(b), where f(x) represents the function you are studying.
3. Calculate the average rate of change of the function over the interval [a, b]. This is done by finding the slope or gradient of the secant line connecting the points (a, f(a)) and (b, f(b)). The formula for the average rate of change is: average rate of change = (f(b) - f(a))/(b - a).
4. Apply the Mean Value Theorem: If the function meets the conditions of continuity and differentiability on the interval (a, b), there exists at least one point c in (a, b) where the derivative at c is equal to the average rate of change calculated in step 3. Mathematically, this can be represented as: f'(c) = (f(b) - f(a))/(b - a).
5. Solve for c using algebraic or numerical methods. This will allow you to find the specific point on the function where the instantaneous rate of change is equal to the average rate of change over the interval [a, b].
It's important to note that the Mean Value Theorem is a theoretical concept and may not always have practical applications. However, it provides valuable insight into the behavior of functions and can be used in various mathematical proofs.
I hope this explanation helps you understand how to work with the Mean Value Theorem! If you have any further questions, feel free to ask.