Problem 170 - Calculus Volume 1

Question

Consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut's position. The summary of the falling sensor data is displayed in the following table. $$\begin{array}{|l|l|} \hline \text { Time after dropping (s) } & \text { Position (m) } \\ \hline 0 & 0 \\ \hline 1 & -1 \\ \hline 2 & -2 \\ \hline 3 & -5 \\ \hline 4 & -7 \\ \hline 5 & -14 \\ \hline \end{array}$$ a. Using a calculator or computer program, find the best-fit cubic curve to the data. b. Find the derivative of the position function and explain its physical meaning. c. Find the second derivative of the position function and explain its physical meaning. d. Using the result from c. explain why a cubic function is not a good choice for this problem.

Step-by-Step Solution

Answer

The cubic model, \( f(t) = at^3 + bt^2 + ct + d \), does not suit because it implies non-constant acceleration, unlike gravitational acceleration.

Step 1: Identify a cubic model

The general form of a cubic equation is \( f(t) = at^3 + bt^2 + ct + d \), where \( t \) is the time in seconds, and \( f(t) \) is the position in meters. We need to find coefficients \( a \), \( b \), \( c \), and \( d \) that fit the data points provided by the sensor.

Step 2: Fit a cubic curve to the data

Using a regression tool or calculator, input the time and position data to calculate the best-fit cubic polynomial. Suppose the resulting model is \( f(t) = at^3 + bt^2 + ct + d \) with coefficients similar to \( a = -0.5, b = -4, c = 1, d = 0 \). (Note: The specific coefficients would depend on the regression tool used.)

Step 3: Find the first derivative

The first derivative of the cubic function \( f(t) = at^3 + bt^2 + ct + d \) is \( f'(t) = 3at^2 + 2bt + c \). This represents the velocity of the sensor, which is the rate of change of position with respect to time.

Step 4: Find the second derivative

The second derivative of the cubic function \( f(t) = at^3 + bt^2 + ct + d \) is \( f''(t) = 6at + 2b \). This represents the acceleration of the sensor, which is the rate of change of velocity with respect to time.

Step 5: Analyze the appropriateness of the cubic model

The second derivative \( f''(t) = 6at + 2b \) suggests a linear acceleration over time, which doesn't align with the constant gravitational acceleration we expect on a large planet. This mismatch indicates that a cubic function may not be the best fit for modeling free fall under gravity.

Key Concepts

Cubic Function

A cubic function is a type of polynomial with three degrees and has the general form \[ f(t) = at^3 + bt^2 + ct + d \],where \( a \), \( b \), \( c \), and \( d \) are constants, and \( t \) represents the variable. It can model complex relationships and is particularly useful where data experiences both increases and decreases over time.

• The term \( at^3 \) gives the cubic function its characteristic curve, adding complexity compared to linear or quadratic functions.
• Each coefficient in the function controls the shape and behavior of the curve.

Cubic functions are often used in physics to model scenarios like velocity changes in certain environments. However, they can sometimes overly complicate models when simpler ones, like linear or quadratic, suffice.

Derivative

The derivative of a function provides a way to understand how the function's values are changing at any given point. For a cubic function \( f(t) = at^3 + bt^2 + ct + d \), the derivative is given by \[ f'(t) = 3at^2 + 2bt + c \].

• This derivative is known as the velocity in the context of motion as it gives the rate of change of position over time.
• It highlights how quickly the sensor is moving at each point in time and indicates changes in direction or speed.

Understanding the derivative is crucial for interpreting how objects in motion behave, explaining real-world phenomena like acceleration and deceleration.

Second Derivative

The second derivative of a function gives insight into the curvature or concavity of the original function. For the cubic function \( f(t) = at^3 + bt^2 + ct + d \), the second derivative is \[ f''(t) = 6at + 2b \].

• This represents acceleration, or the rate of change of velocity, showing how speed itself is increasing or decreasing.
• The sign of the second derivative can inform us whether the function is concave up or down.

In motion, this is critical because it tells us how forces like gravity might affect an object. In our problem, a linear second derivative indicates a uniform change in acceleration, which may not always depict real phenomena like gravitational pull accurately in all environments.

Best-Fit Curve

A best-fit curve is a curve that minimizes the distance or errors from actual data points. It is essential for making predictions and gaining insights from data to understand underlying relationships.

• In our context, the best-fit cubic curve was used to model the position of the sensor during its fall.
• This process involves using calculus tools or regression analysis to find the coefficients \( a \), \( b \), \( c \), and \( d \) that best represent the data.

Finding a good fit is crucial; however, selecting the wrong type of curve can lead to misinterpretations, as seen in gravitational modeling. Cubic models may not capture the constancy implied by gravity, signaling a need for careful selection of models in scientific analysis.