[T] The number of hamburgers sold at a restaurant throughout...

[T] The number of hamburgers sold at a restaurant throughout the day is given in the following table, with the accompanying graph plotting the best cubic fit to the data, $b(t)=0.12 t^{3}-2.13 t^{3}+12.13 t+3.91, \quad$ with $t=0$ corresponding to 9 a.m. and $t=12$ corresponding to 9 p.m. Compute the average value of $b(t)$ to estimate the average number of hamburgers sold per hour.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Average Value of a Function

This concept involves calculating the mean value of a function over a closed interval. It is computed by taking the integral of the function over the interval and dividing by the length of the interval. This average gives a single representative value that summarizes the behavior of the function over that range, which can be used to approximate quantities like the average number of items sold per time period.

Definite Integral

A definite integral is used to calculate the net area under a curve between two limits and provides a measure of the accumulation of a quantity over an interval. It plays a critical role in finding the average value of a function because it aggregates all the individual values over the interval, which can then be normalized by dividing by the interval's length.

Polynomial (Cubic) Function

A cubic function is a third-degree polynomial that provides a smooth curve fitting a set of data points. In many cases, such as estimating trends from experimental data, a cubic function is used as an approximation tool. It allows for the application of calculus techniques to analyze and predict the behavior of the data, including determining average values over a given range.

Transcript

Please give Ace some feedback