Problem 22 - Single Variable Calculus

Question

The position function of an object moving along a straight line is given by \(s=f(t) .\) The average velocity of the object over the time interval \([a, b]\) is the average rate of change of f over \([a, b] ;\) its (instantaneous) velocity at \(t=a\) is the rate of change of \(\bar{f}\) at \(a .\) Velocity of a Car Suppose the distance \(s\) (in feet) covered by a car moving along a straight road after \(t\) sec is given by the function \(s=f(t)=2 t^{2}+48 t\). a. Calculate the average velocity of the car over the time intervals \([20,21],[20,20.1]\), and \([20,20.01]\). b. Calculate the (instantaneous) velocity of the car when \(t=20 .\) c. Compare the results of part (a) with those of part (b).

Step-by-Step Solution

Answer

In summary, we found the average velocities of the car over the time intervals [20, 21], [20, 20.1], and [20, 20.01] to be 98 ft/s, 98.2 ft/s, and 98.02 ft/s, respectively. We also found the instantaneous velocity of the car at t=20 to be 128 ft/s. The average velocities are all less than the instantaneous velocity, but they get closer to the value of the instantaneous velocity as the time intervals become smaller. This demonstrates that the instantaneous velocity is the limit of the average velocities as the time interval approaches zero.

Step 1: Part a: Calculate average velocities over the time intervals

To calculate the average velocity over a given interval, we can use the formula: \[Average\ Velocity = \frac{Change\ in\ position}{Change\ in\ time}\] For the given function s(t) = 2t² + 48t, we will find the average velocities over the given time intervals. 1. Time interval [20, 21]: \[Average\ Velocity = \frac{s(21)-s(20)}{21-20}\] \[Average\ Velocity = \frac{(2(21)^2 + 48(21))-(2(20)^2 + 48(20))}{1}\] \[Average\ Velocity = 98\ ft/s\] 2. Time interval [20, 20.1]: \[Average\ Velocity = \frac{s(20.1)-s(20)}{20.1-20}\] \[Average\ Velocity = \frac{(2(20.1)^2 + 48(20.1))-(2(20)^2 + 48(20))}{0.1}\] \[Average\ Velocity = 98.2\ ft/s\] 3. Time interval [20, 20.01]: \[Average\ Velocity = \frac{s(20.01)-s(20)}{20.01-20}\] \[Average\ Velocity = \frac{(2(20.01)^2 + 48(20.01))-(2(20)^2 + 48(20))}{0.01}\] \[Average\ Velocity = 98.02\ ft/s\]

Step 2: Part b: Calculate instantaneous velocity at t=20

To find the instantaneous velocity at t=20, we need to compute the derivative of the given position function, s(t), with respect to time t and then evaluate it at t=20: \[s(t) = 2t^2 + 48t\] \[s'(t) = \frac{d(2t^2 + 48t)}{dt}\] Applying the power rule and the constant rule for differentiation, we get: \[s'(t) = 4t + 48\] Now, we find the instantaneous velocity at t=20: \[s'(20) = 4(20) + 48\] \[s'(20) = 80 + 48\] \[s'(20) = 128\ ft/s\]

Step 3: Part c: Compare results of part a and part b

In part a, we found the average velocities over the time intervals [20,21], [20,20.1], and [20,20.01] to be 98 ft/s, 98.2 ft/s, and 98.02 ft/s, respectively. In part b, we found the instantaneous velocity at t=20 to be 128 ft/s. As we can see, the average velocities over the time intervals are all less than the instantaneous velocity when t=20. As the time intervals get smaller and closer to the point t=20, the average velocities also get closer to the value of the instantaneous velocity. This demonstrates the concept that the instantaneous velocity is the limit of the average velocities as the time interval approaches zero.

Key Concepts

Average Rate of Change

The average rate of change of a function can be viewed as the slope of the straight line connecting two points on the graph of that function. In the context of motion, it represents the average velocity of an object moving along a straight line within a specified time interval. It's calculated by dividing the change in position by the change in time for that period.

For example, to find the average velocity of a car over the interval \[20, 21\], we look at the positions at \(t=20\) and \(t=21\) and compute \[Average\ Velocity = \frac{s(21)-s(20)}{21-20}\]. This represents the constant velocity at which the car would need to travel to cover the same distance in the same time as it actually did with its varying speed.

Instantaneous Velocity

Instantaneous velocity is the rate of change of position at a specific moment in time. It's the velocity of an object at an exact point in time and is what speedometers in vehicles display. Unlike average velocity, it does not depend on a time interval but is concerned with a precise instant.

In calculus, we determine instantaneous velocity by taking the derivative of the position function with respect to time and evaluating it at a specific time. The step by step solution for finding the instantaneous velocity of the car at \(t=20\) involves taking the derivative \[s'(t) = \frac{d(2t^2 + 48t)}{dt}\] and evaluating it at \(t=20\), which gives us \[s'(20) = 128\ ft/s\]. This value is the car's exact velocity at that moment.

Position Function

A position function describes the location of an object along a line as a function of time. In the scenario with the car, \(s=f(t)=2t^2+48t\) defines the distance covered by the car after \(t\) seconds. This function provides us a way to model the car's motion and calculate essential measures like velocity at any point in time.

The position function is a fundamental concept in kinematics, allowing us to visualize an object's path and determine other motion-related attributes through differentiation and integration.

Derivative

The derivative of a function at a point is the instantaneous rate of change of the function with respect to one of its variables. In relation to motion, it represents instantaneous velocity if the function models distance over time.

For the given function \(s(t)=2t^2+48t\), we use calculus to find the derivative \(s'(t)\) to be \(4t+48\). This formula then allows us to calculate the instantaneous velocity at any point in time by substituting the time value into \(s'(t)\). The derivation of this formula is rooted in the need to understand how a quantity changes moment-to-moment, and it's what ties together the instantaneous velocities with the position function.