172k views
2 votes
In the following exercise, find the average value f_ave of f between a and b, and determine a point c where f(c) equals the average value.

1 Answer

2 votes

Final Answer:

The average value
\( f_{\text{ave}} \) of \( f \) between \( a \) and \( b \) is \( f_{\text{ave}} = (1)/(b-a) \int_(a)^(b) f(x) \, dx \). A point \( c \) where \( f(c) \) equals the average value is \( c \) in the interval \([a, b]\) such that \( f(c) = f_{\text{ave}} \).

Step-by-step explanation:

To find the average value
\( f_{\text{ave}} \) of \( f \) between \( a \) and \( b \), we use the formula \( f_{\text{ave}} = (1)/(b-a) \int_(a)^(b) f(x) \, dx \).This formula involves finding the definite integral of \( f \) over the interval \([a, b]\) and then dividing by the length of the interval.

The integral represents the accumulated area under the curve of \( f \) over the given interval. Dividing by the length of the interval (\( b - a \)) normalizes this accumulated area, providing the average value of \( f \) over the interval.

Now, to determine a point \( c \) where \( f(c) \) equals the average value, we look for a specific \( c \) in the interval \([a, b]\). This point \( c \) satisfies
\( f(c) = f_{\text{ave}} \), making the function value at \( c \) equal to the average value of the function over the interval. This can be interpreted as a balance point where the function is equally distributed above and below its average value in the given interval.

User Ricab
by
7.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories