Question About Average Value of a Function on an Interval

Find the numbers b such that the average value of f(x)=2+6x-3x^2 on the interval [0,b] is equal to 3.

Solution Preview

This is actually an evaluation of the integral
∫f(x)= ∫2+6x-3x^2 dx in the [0, b] ...

Solution Summary

The bound of an interval that allows for a certain average value of a function is found. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

Please see the attached file for the fully formatted problems.
Question 7: For the function find:
a) its averagevalue, (Average f), on the interval [0,9]
b) the value of c in the interval [0,9] guaranteed by the Mean Value Theorem for Integrals.
Question9: Evaluate the following limits:
a.
b.

Find the averagevalue of the function over the given interval and all values of x in the interval for which the function equals its averagevalue. Round your answers to four decimal places.
f(x)=16-x^2, [-4,4]
(x,y)=(_______)(smaller x-value)
(x,y)=(_______)(larger x-value)
Thanks

Use a finite sum to find the averagevalue of the function on the given interval by partitioning the interval and evaluating the function at the midpoints of the subintervals. Show all work!
f(x) = 5 + sin (Pi)x on [0, 2] divided into 4 subintervals

See the data in the attached file and answer the following questions.
Question 1
Construct a 95% confidence interval for an averagevalue of y given that x = 4. Remember the format is (x.xx, x.xx)
Question 2
Construct a 95% prediction interval for y given that x = 4.

Please help with the following problem. Provide step by step calculations for each.
The averagevalue of f(x) = 1/x on the interval [4, 16] is
(ln 2)/3
(ln 2)/6
(ln 2)/12
3/2
0
1
none of these
Find the area, in square units, of the region b

Consider the function f(x)= 4-x^2
a) Find the averagevalue of f on the interval (0,2)
b)Determine the number c that satisfies the mean value theorem for integrals for f on the interval (0,2)
c)sketch the graph of f

A ball is released at the top of a ramp (an inclined plane). The distance it travels can be expressed as the function:
s(t)=5t^2, where t is the time in seconds after the ball is released.
The distance function, s(t), is measured in feet. Find the average velocity of the ball for the time interval [1,4].

Let f(x), g(x) be functions defined on a closed bounded interval [a, b] such that the following conditions hold:
g is differentiable on [a, b].
There are positive constants a, b such that g(x) = a*f(x) - b*(dg/dx).
f(x) > 0 for all x in [a, b]
g(x) >= 0 for all x in [a, b]
g(a) > 0
-----------------------------

What information does a confidence interval give?
A sample of 50 employees was taken and the average number of children in each of the employee's family was computed to be 2.1 with a standard deviation of 0.5.
Compute the 95% confidence interval on the average number of children. Interpret the answer.