Question

Let v(t) denote the velocity of the car (in thousands of feet per minute) at time t (in minutes). Which graph, a-f, is the best representative of the derivative function?

1) a
2) b
3) c
4) d
5) e
6) f

Answer 1

It shows a constant positive slope, indicating a consistent rate of change. Therefore the correct option is Graph d is the best representative of the derivative function.

Explanation

Graph d, depicting a constant positive slope, aligns with the characteristics of a derivative function representing velocity. The derivative of velocity with respect to time yields acceleration, which, if constant, results in a linear graph. This representation indicates a constant rate of change in velocity, signifying steady acceleration or deceleration.

Derivatives in calculus indicate the rate of change of a function at any given point. In the context of velocity, the derivative represents the rate of change of velocity concerning time.

A constant slope in the derivative graph implies a consistent change in velocity, either an unchanging acceleration or deceleration. Graph d's consistent and steady slope reflects this continuous change in velocity, making it the most appropriate depiction of the derivative function.

Therefore the correct option is Graph d is the best representative of the derivative function.

Answer 2

The best representative of the derivative function is graph (2) b.

Graph b aligns with the characteristics of the derivative function, showing constant velocity (derivative = 0), increasing velocity (positive derivative), and decreasing velocity (negative derivative), making it the most suitable representation.

Step-by-step explanation:

The derivative of a function represents its rate of change. In this case, we are looking for the graph that best represents the derivative of the velocity function v(t). The key is to identify the characteristics of the velocity function and its derivative.

Firstly, if the velocity is constant, the derivative is zero since there is no change. Looking at graph b, it shows a constant value of zero, indicating that the car's velocity is constant during that time.

Secondly, if the velocity is increasing, the derivative is positive. Graph b also correctly represents this, showing a positive slope, indicating an increasing velocity.

Thirdly, if the velocity is decreasing, the derivative is negative. Graphs a, c, and d show negative values, but graph b best represents a decreasing velocity with a smooth curve.

In summary, graph b aligns with the expected characteristics of the derivative function based on the given information. It correctly represents constant velocity (derivative = 0), increasing velocity (positive derivative), and decreasing velocity (negative derivative). Therefore, graph b is the most suitable representation of the derivative function for the given velocity function v(t).