Problem 104 - Calculus Volume 1

Question

Use the following table, which shows the height \(h\) of the Saturn \(\mathrm{V}\) rocket for the Apollo 11 mission \(t\) seconds after launch. $$\begin{array}{|l|l|} \hline \text { Time (seconds) } & \text { Height (meters) } \\ \hline 0 & 0 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 13 \\ \hline 4 & 25 \\ \hline 5 & 32 \\ \hline \end{array}$$ The best cubic fit to the data is given by \(F(t)=0.2037 t^{3}+2.956 t^{2}-2.705 t+0.4683, \quad\) where \(F\) is the height of the rocket (in \(\mathrm{m}\) ) and \(t\) is the time elapsed since take off. From this equation, determine \(F^{\prime}(t)\). Graph \(F(t)\) with the given data and, on a separate coordinate plane, graph \(F^{\prime}(t)\). Does the linear, quadratic, or cubic function fit the data best?

Step-by-Step Solution

Answer

The cubic function fits the data best.

Step 1: Identify the Function

Given the function for the height of the rocket is \( F(t) = 0.2037 t^3 + 2.956 t^2 - 2.705 t + 0.4683 \).

Step 2: Differentiate the Function

Differentiate the function \( F(t) \) with respect to \( t \) to find its derivative, \( F'(t) \). The derivative will be \( F'(t) = 3 imes 0.2037 t^2 + 2 imes 2.956 t - 2.705 \).

Step 3: Simplify the Derivative

Perform the multiplication to simplify the derivative. Thus, \( F'(t) = 0.6111 t^2 + 5.912 t - 2.705 \).

Step 4: Plot the Given Function

Graph the function \( F(t) = 0.2037 t^3 + 2.956 t^2 - 2.705 t + 0.4683 \) using a graphing tool or coordinate plane against time \( t \) to visualize the height of the rocket.

Step 5: Plot the Derivative

Graph the derivative \( F'(t) = 0.6111 t^2 + 5.912 t - 2.705 \) on a separate coordinate plane. This shows the rate of change of height or the velocity of the rocket.

Step 6: Determine the Best Fit

Compare the shape of the graphs of \( F(t) \) and the derivative \( F'(t) \) with the data points. The cubic equation \( F(t) \) should fit the data best because it is specifically described as the best fit in the problem and represents the mathematical model for the data.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, particularly useful for understanding how a function changes over time or space. In this exercise, we've been given a cubic function, which describes the height of a rocket over time, and we're tasked with finding its derivative.
The process of differentiation involves taking a function and calculating its derivative.

• The function we start with is a polynomial: \( F(t) = 0.2037 t^3 + 2.956 t^2 - 2.705 t + 0.4683 \).
• To differentiate this, apply the power rule, which states that \( n \times a_n t^{n-1} \) for each term, where \( a_n \) is the coefficient.
• For each term, you reduce the exponent of \( t \) by one and multiply by the original exponent.

Thus, the derivative \( F'(t) \) becomes \( 0.6111 t^2 + 5.912 t - 2.705 \).
This derivative function represents the velocity, or the rate of change of the rocket's height over time.

Polynomial Fit

A polynomial fit is an approximation of given data by a polynomial equation.
This type of fitting is particularly useful when data points follow a certain trend that can be approximated by a smooth curve.
In this problem, we're presented with a data table showing the height of the Saturn V rocket over each second after launch.

• To find the best mathematical model for this data, a cubic polynomial was used.
• The specific polynomial, in this case, is \( F(t) = 0.2037 t^3 + 2.956 t^2 - 2.705 t + 0.4683 \).
• The polynomial fit suggests that this cubic function best approximates the data over the observed time period.

Using a cubic function allows for capturing the more complex behavior in the rocket's height changes, providing a more precise fit than linear or quadratic models might offer. The coefficients \( 0.2037 \), \( 2.956 \), and \( -2.705 \) define the curve's shape, while the constant term \( 0.4683 \) adjusts the curve's position vertically.

Graph Interpretation

When it comes to graph interpretation, it involves visually analyzing the graphed functions to derive specific conclusions from the data.
With graphs, it becomes easier to see how a rocket's height and velocity change over time.
The original function \( F(t) \) is graphed to show how the rocket's height varies.

• The graph of \( F(t) \) tends to resemble a curve that explains the height reaching specific points over time.
• On a separate graph, \( F'(t) \) is plotted to visualize the rocket’s velocity or rate of change of height.
• The derivative's graph is typically a parabola that highlights where the growth rate, or velocity, is increasing or decreasing.

These graphical interpretations reveal much about the motion of the rocket. The shape and slope of \( F(t) \) and \( F'(t) \) can help identify acceleration periods and maximum speeds. Observing the graphs helps not just in verifying the polynomial fit visually but in understanding intricate details regarding the rocket's flight that numbers alone might not convey.