The average speed of a vehicle on a stretch of Route 134...

The average speed of a vehicle on a stretch of Route 134 between 6 A.M. and 10 A.M. on a typical weekday is approximated by the function $$ f(t)=20 t-40 \sqrt{t}+50 \quad(0 \leq t \leq 4) $$ where $f(t)$ is measured in miles per hour and $t$ is measured in hours, with $t=0$ corresponding to 6 A.M. At what time of the morning commute is the traffic moving at the slowest rate? What is the average speed of a vehicle at that time?

The average speed of a vehicle on a stretch of Route 134 between 6 A.M. and 10 A.M. on a typical weekday is approximated by the function
$$
f(t)=20 t-40 \sqrt{t}+50 \quad(0 \leq t \leq 4)
$$
where $f(t)$ is measured in miles per hour and $t$ is measured in hours, with $t=0$ corresponding to 6
A.M. At what time of the morning commute is the traffic moving at the slowest rate? What is the average speed of a vehicle at that time?

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Key Concepts

Function Modeling

Function modeling involves creating mathematical functions to represent real-world situations. In the context of calculating average speed, a function is used to represent how speed changes over time, allowing for analysis and predictions about traffic behavior during the commute.

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the derivative of a function. The derivative represents the rate of change of the function with respect to the variable, which, in optimization problems, helps identify where the function is increasing or decreasing.

Optimization

Optimization in calculus refers to the process of finding maximum or minimum values of a function. By setting the derivative equal to zero and analyzing the resulting critical points, one can determine the points at which the function attains its minimum or maximum values within a certain interval.

Critical Points

Critical points are values in the domain of a function where its derivative is zero or undefined. Evaluating these points, along with the endpoints of the interval, is essential in determining where the function reaches an optimal (minimum or maximum) value, as required in problems involving speed or other rates.

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