Problem 103 - 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 quadratic fit to the data is given by \(G(t)=1.429 t^{2}+0.0857 t-0.1429, \quad\) where \(G\) is the height of the rocket (in meters) and \(t\) is the time elapsed since takeoff. From this equation, determine \(G^{\prime}(t)\). Graph \(G(t)\) with the given data and, on a separate coordinate plane, graph \(G^{\prime}(t)\).

Step-by-Step Solution

Answer

The derivative of the height function is \(G'(t) = 2.858t + 0.0857\).

Step 1: Understand the Problem

You have a table with the height of the Saturn V rocket at various times after launch. You also have a quadratic function \(G(t) = 1.429t^2 + 0.0857t - 0.1429\) that is the best fit for this data. You need to find the derivative \(G'(t)\) and graph both \(G(t)\) and \(G'(t)\).

Step 2: Recall How to Take Derivatives

To find \(G'(t)\), apply basic derivative rules. For a function in the form \(at^2 + bt + c\), the derivative is \(2at + b\). The constant \(c\) becomes zero when differentiated.

Step 3: Differentiate the Quadratic Function

Given \(G(t) = 1.429t^2 + 0.0857t - 0.1429\), use the rule from Step 2 to differentiate term-by-term. The derivative \(G'(t)\) is:- The derivative of \(1.429t^2\) is \(2 \times 1.429 \times t = 2.858t\).- The derivative of \(0.0857t\) is \(0.0857\).- The derivative of \(-0.1429\) is \(0\).Thus, \(G'(t) = 2.858t + 0.0857\).

Step 4: Graph the Original Function G(t)

Plot the quadratic function \(G(t) = 1.429t^2 + 0.0857t - 0.1429\) using a suitable range for \(t\), such as from 0 to 5, aligning with the data table. Mark the given data points on the same graph to visualize the best fit approximation.

Step 5: Graph the Derivative Function G'(t)

Plot the linear function \(G'(t) = 2.858t + 0.0857\) on a separate coordinate plane. This line represents the rocket's rate of change of height over time (velocity). Choose an appropriate range for \(t\) based on the original time values.

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental part of algebra and calculus. A quadratic function is generally represented in the form of a polynomial equation: \[ f(t) = at^2 + bt + c \]where \(a, b,\) and \(c\) are constants, with \(a eq 0\). The highest power of the variable \(t\) is 2, making it quadratic.

In the case of the Saturn V rocket's height, the function \(G(t) = 1.429t^2 + 0.0857t - 0.1429\) is a quadratic equation. This type of function often exhibits a parabolic shape when graphed. The curve can either open upwards (if \(a > 0\)) or downwards (if \(a < 0\)).

The roots of a quadratic function, or the values of \(t\) when the function is equal to zero, can be found using the quadratic formula:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]These roots can provide insights into the realistic applications, such as when a projectile will reach a certain height or return to the ground.

Graphing Functions

Graphing functions is an essential skill to visually interpret relationships between variables. The graph of a function like our quadratic function \(G(t)\) can reveal important characteristics such as symmetry, maximum or minimum points, and intersections with axes.

When graphing a quadratic function, start by identifying key features:

• The vertex: The turning point of the parabola, which can be calculated using \(t = -\frac{b}{2a}\).
• The axis of symmetry: A vertical line passing through the vertex.
• The direction: Determined by the sign of \(a\). If \(a > 0\), the parabola opens upwards.
• The y-intercept: Given by constant \(c\), the point where the graph intersects the y-axis.

To graph \(G(t)\), plot these critical points and use the equation to calculate additional points if needed. Connecting these points with a smooth curve will produce the complete graph of the quadratic function.

Best Fit Polynomial

A best fit polynomial is used to approximate data by finding the polynomial equation that closely follows the trend of a given set of data points. For any set of scattered data, a best fit line, or in this case, a quadratic curve, minimizes the differences between actual and estimated values.

The quadratic function \(G(t) = 1.429t^2 + 0.0857t - 0.1429\) is derived from such methods, often using techniques like least squares regression. This process involves:

• Collecting accurate measurements of the variables and plotting them on a plane.
• Determining the degree of the polynomial that best fits the data.
• Solving for coefficients that minimize the sum of squared differences between observed and estimated values.

This approach is invaluable for predictions and estimations, providing a clear model for real-world data like the Saturn V rocket's height over time.

Differentiation Rules

Differentiation is a key concept in calculus that involves finding the rate at which a function is changing at any point, often referred to as the derivative. For polynomial functions, there are straightforward rules to determine derivatives, making this process simpler.

The basic rule for differentiating a term \(at^n\) is to multiply the coefficient by the exponent and reduce the exponent by one:\[ \frac{d}{dt}(at^n) = nat^{n-1} \]Applying this to the quadratic function \(G(t) = 1.429t^2 + 0.0857t - 0.1429\), the derivative is computed as:

• For \(1.429t^2\), \(\frac{d}{dt}(1.429t^2) = 2 \times 1.429 \times t = 2.858t\).
• For \(0.0857t\), \(\frac{d}{dt}(0.0857t) = 0.0857\).
• The constant \(-0.1429\) becomes zero, as constants have no rate of change.

Thus, the derivative \(G'(t) = 2.858t + 0.0857\) gives the rate of change of the rocket's height at any given time, representing its velocity.