Problem 23 - Essential Calculus Early Transcendentals

Question

A car is moving with speed 20 \(\mathrm{m} / \mathrm{s}\) and acceleration 2\(\mathrm{m} / \mathrm{s}^{2}\) at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?

Step-by-Step Solution

Answer

The car travels approximately 21 meters in the next second. The Taylor polynomial is not suitable for estimating distance over a minute due to potential changes in conditions.

Step 1: Write the Position Function as a Taylor Series

The position of the car can be represented by the function \( s(t) \), which is nearly equal to its Taylor series about \( t = 0 \). The general form for the Taylor polynomial up to the second degree is:\[s(t) \approx s(0) + s'(0)t + \frac{s''(0)}{2}t^2. \]

Step 2: Identify the Initial Conditions

From the problem, we know:- The initial speed \( s'(0) = 20 \, \text{m/s} \)- The acceleration \( s''(0) = 2 \, \text{m/s}^2 \).- The initial position \( s(0) = 0 \), assuming the car starts at the origin.

Step 3: Substitute the Values into the Polynomial

Substitute the known values into the polynomial:\[s(t) \approx 0 + 20t + \frac{2}{2}t^2 = 20t + t^2.\]

Step 4: Calculate the Distance for t = 1 Second

Substitute \( t = 1 \) second into the polynomial:\[s(1) = 20(1) + (1)^2 = 20 + 1 = 21 \, \text{meters}.\]

Step 5: Discuss the Reasonableness for a Longer Time Interval

A Taylor series locally approximates a function, often near zero or an initial point. Given that the acceleration is constant, higher-order effects (jerk, etc.) won't affect a quadratic model over very short times. However, over a whole minute (60 seconds), this polynomial might not account for changes in speed or acceleration that could occur in practical scenarios.

Key Concepts

Position Function

When we're talking about a car's position over time, we often refer to the "position function." The position function, typically denoted as \( s(t) \), gives us the location of the car at any given time \( t \). Think of it as a snapshot of where the car is on a timeline. In the context of the Taylor Series, the position function can be expanded into a polynomial to estimate positions at small time intervals.

The Taylor series expansion helps in estimating the function's value at a particular point when we know its value at another point. For our purpose, a second-degree Taylor polynomial expansion of the position function is expressed as:

• \( s(t) \approx s(0) + s'(0)t + \frac{s''(0)}{2}t^2 \)

Here, \( s(0) \) is the initial position, \( s'(0) \) is the initial velocity or speed, and \( s''(0) \) is the acceleration at the start. By including these terms, we get a practical estimate of the distance the car will cover in a short time after it starts moving.

Acceleration

Acceleration plays a crucial role in predicting the car's future position. Acceleration, signified by \( s''(t) \), refers to how the speed of the car changes over time. Think of it as the push that gets the car moving faster or slower. In our exercise, the acceleration is constant at \( 2 \, \text{m/s}^2 \).

By incorporating this constant acceleration into the Taylor expansion, we can form part of our position function. In the equation \( s(t) \approx s(0) + s'(0)t + \frac{s''(0)}{2}t^2 \), the term \( \frac{s''(0)}{2}t^2 \) is what accounts for the acceleration's effect on the car's position. Essentially, this term tells us how much further the car moves due to its acceleration, compared to how far it would go at a constant speed.

In real-world terms, this means that as long as the car maintains a constant acceleration, the distance covered in small time intervals can be accurately estimated using this quadratic term.

Initial Conditions

Initial conditions give us the necessary starting points to solve motion problems using math tools like the Taylor Series. In this specific context, our initial conditions include knowing the car's initial speed and acceleration.

Let's break it down:

• Initial Speed (Velocity): Denoted \( s'(0) \), is given as \( 20 \, \text{m/s} \). This speed is the velocity of the car at the start of the observation.
• Initial Acceleration: Represented as \( s''(0) \), is \( 2 \, \text{m/s}^2 \). This describes how the speed changes over time, starting from time zero.
• Initial Position: While not always given, in many problems like this, it can be assumed to be \( 0 \) for simplicity, indicating the start point is the origin.

With these conditions, we can plug values into our Taylor polynomial. Starting with these known quantities is key to making accurate predictions about where the car will be after a specific time, like one second as discussed in our exercise.