Problem 7 - Precalculus

Question

Two model rockets are launched at a gathering of the National Association of Rocketry (NAR: www.nar.org). Frank's Apollo II motor burns out at a height of \(500 \mathrm{m},\) at which point the rocket has a velocity of 88.2 meters per second (m/sec). His rocket's height in meters, \(t\) sec after engine burnout, is given by \(f(t)=500+88.2 t-4.9 t^{2}\) Gwen's Icarus Alpha motor burns out at a height of \(600 \mathrm{m}\) at which point the rocket has a velocity of \(78.4 \mathrm{m} / \mathrm{sec} .\) Her rocket's height in meters, \(t\) sec after burnout, is given by \(g(t)=600+78.4 t-4.9 t^{2}\). Find the limit of the difference quotient for \(f,\) to obtain a function \(f(t)\) that represents the instantaneous velocity at time \(t\).

Step-by-Step Solution

Answer

The instantaneous velocity function for Frank's rocket is \(f'(t) = 88.2 - 9.8t\).

Step 1: Understand the Difference Quotient

The difference quotient for any function \(f(t)\) is given as \(\frac{f(t+h)-f(t)}{h}\). It represents the average rate of change of \(f\) over the interval \([t, t+h]\). To find the instantaneous velocity, we need to compute the limit of this quotient as \(h\) approaches zero.

Step 2: Write the Expressions for \(f(t+h)\) and \(f(t)\)

Given \(f(t)=500+88.2t-4.9t^{2}\), we find \(f(t+h)\) by substituting \(t+h\) into the function: \[f(t+h) = 500 + 88.2(t+h) - 4.9(t+h)^2\]. This expands to: \[f(t+h) = 500 + 88.2t + 88.2h - 4.9(t^2 + 2th + h^2)\]. Substitute the expression for \(f(t)\) and simplify.

Step 3: Simplify the Difference Quotient

Subtract \(f(t)\) from \(f(t+h)\):\[f(t+h) - f(t) = 500 + 88.2t + 88.2h - 4.9(t^2 + 2th + h^2) - (500 + 88.2t - 4.9t^{2})\].This simplifies to:\[88.2h - 9.8th - 4.9h^2\].Now, we form the difference quotient:\[\frac{88.2h - 9.8th - 4.9h^2}{h}\] and divide through by \(h\).

Step 4: Compute the Limit for Instantaneous Velocity

Divide each term by \(h\):\[\frac{88.2h}{h} - \frac{9.8th}{h} - \frac{4.9h^2}{h} = 88.2 - 9.8t - 4.9h\].Take the limit as \(h\) approaches 0:\[\lim_{{h \to 0}} (88.2 - 9.8t - 4.9h) = 88.2 - 9.8t\].

Step 5: Resultant Function for Instantaneous Velocity

The resultant function for the instantaneous velocity of Frank's rocket at any time \(t\) after burnout is:\[f'(t) = 88.2 - 9.8t\]. This represents the instantaneous velocity (rate of change of height) of the rocket at time \(t\).

Key Concepts

Difference Quotient

The difference quotient is a powerful mathematical tool used to understand how a function changes. In simple terms, it measures the average rate of change of the function over a small interval. For a given function \( f(t) \), the difference quotient is expressed as \( \frac{f(t+h) - f(t)}{h} \). Here, \( h \) represents a small increment of time.
When you calculate the difference quotient, you are essentially taking the slope of the secant line that passes through the points \( (t, f(t)) \) and \( (t+h, f(t+h)) \). This can be applied to various functions to get a sense of their behavior.

To find the instantaneous rate of change or velocity, we take the limit of this difference quotient as \( h \) approaches zero. This transition from average rate to instantaneous rate is crucial—not just in math, but in understanding real-world phenomena like the speed of a moving object.

Average Rate of Change

The average rate of change is a key concept that tells us how much a quantity changes over a specific interval. In the context of rocket height, it means how much the height changes as time passes. Simply put, it's the slope of the line connecting two points on the function's graph.

To calculate it, you use the formula: \( \text{Average Rate of Change} = \frac{f(b) - f(a)}{b-a} \), where \( a \) and \( b \) are two distinct points in time. This formula shows how much the function’s output changes per unit input.
The idea here is to understand how the function behaves between any two points, which gives valuable insight into overall trends—even if it's not precise at a single moment.

Instantaneous Rate of Change

Much like a snapshot of motion, the instantaneous rate of change provides a precise measure of change at a single moment. Mathematically, it's the derivative of a function, offering the slope of the tangent line at any given point.
This is what we mean by instantaneous velocity: the speed at which an object is moving at an exact moment. In our rocket example, we derived the formula \( f'(t) = 88.2 - 9.8t \) which gives the velocity at any time \( t \) post-burnout.
Calculating instantaneous velocity from the difference quotient requires taking the limit, leading to an exact result that isn't just an average. This nuanced value captures the true velocity of the rocket at a particular time.

Polynomials

A polynomial is essentially a mathematical expression formed by summing powers of a variable, each multiplied by coefficients. It's one of the most common types of functions used due to their simplicity and versatility.
In the exercise, the rocket's height function, \( f(t)=500+88.2t-4.9t^2 \), is a polynomial because it involves terms with \( t \) raised to integer powers with coefficients.

Polynomials are neat because they are continuous and differentiable everywhere, making them easy to work with in calculus. They represent real-world situations effectively, like the height of a rocket over time, which can be graphed as smooth and predictable curves. Furthermore, they allow us to use tools like the difference quotient and limits to find rates of change—essential for problems involving motion and speed.