Problem 101 - 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. $$\begin{array}{|l|l|} \hline \text { Time (seconds) } & \text { Height (meters) } \\ \hline 0 & 0 \\ \hline 1 & 2 \\ \hline 2 & 4 \\ \hline 3 & 13 \\ \hline 4 & 25 \\ \hline 5 & 32 \\ \hline \end{array}$$ Construct a table of values for \(h^{\prime}(t)\) and graph both \(h(t)\) and \(h^{\prime}(t)\) on the same graph. (Hint: for interior points, estimate both the left limit and right limit and average them.)

Step-by-Step Solution

Answer

Use finite differences to approximate \(h'(t)\), then plot both \(h(t)\) and \(h'(t)\).

Step 1: Understanding the Problem

We need to find the derivative of the height function with respect to time, \(h'(t)\), which represents the velocity of the rocket. We'll use the given data points and estimate derivatives, then create a table with these estimations.

Step 2: Calculating h'(t) for Interior Points

For the interior points, calculate the average of the left and right differences for \(t = 2, 3, 4\). For instance, at \(t = 2\), use the points (1, 2) and (3, 13) to find the average rate of change: \(h'(2) = \frac{(13-4)}{(3-2)} + \frac{(4-2)}{(2-1)} = 9.5\).

Step 3: Calculating h'(t) for Boundary Points

For boundary points \(t=0\) and \(t=5\), use just one side difference. For example, \(h'(0) = \frac{2-0}{1-0} = 2\). Similarly, \(h'(5) = \frac{32-25}{5-4} = 7\).

Step 4: Construct Table for h'(t)

Construct a table to include \(t\) and the corresponding \(h'(t)\) for each specified time:| Time (Seconds) | h'(t) (m/s) ||---|---|| 0 | 2 || 1 | 2 || 2 | 9.5 || 3 | 10.5 || 4 | 3.5 || 5 | 7 |

Step 5: Graphing h(t) and h'(t)

Graph both sets of values: plot \(h(t)\) using the original height values and \(h'(t)\) using the estimated derivatives. Ensure that both curves are distinguishable (use different colors or styles).

Key Concepts

Rate of Change

The rate of change is a fundamental concept in calculus that describes how a quantity changes relative to another. In this context, we're observing how the height of the Saturn V rocket changes over time. The root idea is to measure how fast or slow the height of the rocket, denoted as \(h(t)\), changes as time progresses.

• The rate of change between two points is calculated as: \(\text{Rate of Change} = \frac{\text{Change in Height}}{\text{Change in Time}}\).
• This gives us an average velocity between two specific times.
• In the dataset provided, this involves simple subtraction and division for pairs of values in the table.

For example, from \(t = 1\) second to \(t = 2\) seconds, the height goes from 2 meters to 4 meters. The rate of change would then be \((4 - 2) / (2 - 1) = 2 \text{ meters per second}\). This means, on average, the rocket was rising 2 meters every second between these times.
Understanding the rate of change is the first step towards grasping more complex concepts like the derivative.

Numerical Differentiation

Numerical differentiation is a method used to approximate derivatives when you have discrete data points instead of a continuous function. Instead of finding an exact analytical derivative, we rely on data points to estimate changes.

• This technique is particularly helpful when dealing with real-world data, like the Apollo 11 launch.
• The key is to compute the difference between successive values to estimate the so-called derivative \(h'(t)\).

For points within our dataset, we find the average of the left and right differences. This approach helps smooth out estimations and provides a more reliable derivative value. For example, to find \(h'(2)\), we use both the left and right neighbors:
- Calculate \(\frac{4 - 2}{2 - 1} = 2\)
- Calculate \(\frac{13 - 4}{3 - 2} = 9\)
- Average these two results to get \(h'(2) = \frac{2 + 9}{2} = 9.5\)
In this way, numerical differentiation provides a viable solution when calculus formulas are non-practical, giving us insight into the dynamic changes of the scenario.

Velocity Estimation

Velocity estimation involves determining the speed and direction of an object, expressed as a derivative of position over time. In the example of the Saturn V rocket, velocity is represented by \(h'(t)\), the rate of change of height with respect to time. Finding this estimation helps understand the rocket's speed at different moments post-launch.

• Boundary values are estimated using a one-sided difference approach.
• Interior points use the more accurate centered difference discussed earlier.

For boundary points, like \(t = 0\) or \(t = 5\), we only use one interval to compute \(h'(t)\). For instance, at \(t = 0\), we calculate \(h'(0) = \frac{2-0}{1-0} = 2 \text{ m/s}\).
By analyzing these estimations, you can plot a velocity graph alongside the height graph. This provides visual insight into how quickly the rocket is speeding up or slowing down over the first few seconds of its flight.