Average Value of a Function and the Mean Value Theorem of Integrals
Concept: Average value
By this point, you have learned quite a bit about functions in your mathematical career. In fact, given a function, \(f(x)\), you can evaluate it, graph it, find the exact/approximate values of the roots, discuss inflection points, investigate the extrema, and a whole lot more. The list of what you can do with a function...
By this point, you have learned quite a bit about functions in your mathematical career. In fact, given a function, \(f(x)\), you can evaluate it, graph it, find the exact/approximate values of the roots, discuss inflection points, investigate the extrema, and a whole lot more. The list of what you can do with a function...
Theorem (Average Value of a Continuous Function)
The average value of a continuous function \(f(x)\) over a closed interval \([a,b]\) is \[f_{avg} = \frac{1}{b  a} \int_{a}^{b}{f(x) dx}.\]
The average value of a continuous function \(f(x)\) over a closed interval \([a,b]\) is \[f_{avg} = \frac{1}{b  a} \int_{a}^{b}{f(x) dx}.\]
Example Group: Computing the average value of functions
1. Find the average value of the function \[h(x) = \cos^{4}{(x)} \sin{(x)}\] on the interval \([0,\pi]\).
1. Find the average value of the function \[h(x) = \cos^{4}{(x)} \sin{(x)}\] on the interval \([0,\pi]\).
2. The following uses Newton's Law of Cooling.
A. A cup of coffee has temperature \(95^{\circ}C\) and takes 30 minutes to cool to \(61^{\circ}C\) in a room with temperature \(20^{\circ}C\). Use Newton’s Law of Cooling to show that the temperature of the coffee after \(t\) minutes is \[T(t) = 20 + 75e^{kt},\] where \(k \approx 0.02\).
B. What is the average temperature of the coffee during the first half hour?
A. A cup of coffee has temperature \(95^{\circ}C\) and takes 30 minutes to cool to \(61^{\circ}C\) in a room with temperature \(20^{\circ}C\). Use Newton’s Law of Cooling to show that the temperature of the coffee after \(t\) minutes is \[T(t) = 20 + 75e^{kt},\] where \(k \approx 0.02\).
B. What is the average temperature of the coffee during the first half hour?
Concept: Mean Value Theorem for Integrals
Recall from differential calculus that the Mean Value Theorem for Derivatives stated that a continuous function on a given closed interval must contain a point such that the derivative of the function at that point is the average derivative on that interval.
Theorem (Mean Value Theorem for Integrals)
If \(f\) is continuous on the closed interval \([a,b]\), then \(\exists c \in [a,b]\) such that \[f(c) \equiv f_{avg} = \frac{1}{b  a}\int_{a}^{b}{f(x)dx}.\] That is, \[\int_{a}^{b}{f(x)dx} = f(c)(b  a).\]
Example Group: Using the Mean Value Theorem for Integrals
3. Consider \(f(x) = 16\sin{(x)}  8\sin{(2x)}\) on the interval \([0,\pi]\).
A. Find \(f_{avg}\) on the given interval.
B. Find \(c\) so that \(f(c) = f_{avg}\). (round to three decimal places)
C. Sketch \(f\) and, on the same graph, a rectangle whose area is the same as the area under \(f\).
A. Find \(f_{avg}\) on the given interval.
B. Find \(c\) so that \(f(c) = f_{avg}\). (round to three decimal places)
C. Sketch \(f\) and, on the same graph, a rectangle whose area is the same as the area under \(f\).
4. If \(f\) is continuous and \[\int_{1}^{3}{f(x)dx} = 8,\] show that \(f\) takes on the value of 4 at least once in the interval \([1,3]\).

