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. a. Compute $f^{\prime}(t)$ b. What is the average speed of a vehicle on that stretch of Route 134 at 6 A.M.? At 7 A.M.? At 8 A.M.? c. How fast is the average speed of a vehicle on that stretch of Route 134 changing at 6: 30 A.M.? At 7 A.M.? At 8 A.M.?

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.
a. Compute $f^{\prime}(t)$
b. What is the average speed of a vehicle on that stretch of Route 134 at 6 A.M.? At 7 A.M.? At 8
A.M.?
c. How fast is the average speed of a vehicle on that stretch of Route 134 changing at 6: 30 A.M.? At 7 A.M.? At 8 A.M.?

Key Concepts

Function Evaluation

Function evaluation involves substituting specific values into a function to determine the corresponding output. In many applied problems, including those involving motion or speed, it is important to evaluate a function at particular points to obtain meaningful real-world measurements.

Instantaneous Rate of Change

The concept of instantaneous rate of change refers to the derivative evaluated at a specific point, providing the rate at which a quantity is changing at that exact moment. It is used to understand the behavior of a function at a precise time, distinguishing it from the average rate of change over an interval.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input variable. In practical applications, derivatives help determine how one quantity changes in response to another and are essential in problems involving rates and motion.

Power Rule

The power rule is a fundamental rule in calculus used to differentiate functions of the form x^n, where n is a real number. It states that the derivative of x^n is n*x^(n-1). This rule simplifies the process of finding derivatives when dealing with monomial terms as part of a larger function.

Chain Rule

The chain rule is a method used to differentiate composite functions, that is, functions formed by the composition of two or more functions. It allows the differentiation of a complex function by taking the derivative of the outer function and then multiplying it by the derivative of the inner function, making it crucial when dealing with functions that contain nested expressions, such as a square root of a function.