Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(h(t)\) denote the height of a rocketship \(t\) seconds after takeoff. (a) Express the average rate of change of height of the rocket betweeen 2 and \(2.01\) seconds after takeoff in terms of the function \(h\). (b) Express the average rate of change of height (average vertical velocity) of the rocket on the time interval \([a, a+0.001]\) in terms of \(h\). (c) Express the average vertical velocity of the rocket on the time interval \([a, a+k]\).

Step-by-Step Solution

Answer

The average rate of change between 2 and 2.01 seconds is \(\frac{h(2.01)-h(2)}{0.01}\), between 'a' and 'a+0.001' seconds is \(\frac{h(a+0.001)-h(a)}{0.001}\) and on the time interval '[a, a+k]' is \(\frac{h(a+k)-h(a)}{k}\)

Step 1: Calculate average rate of change between 2 and 2.01 seconds

The average rate of change formula is given as: \[AR = \frac{f(b)-f(a)}{b-a}\] Here \(a = 2\) and \(b = 2.01\), so replacing in the formula we get \[AR = \frac{h(2.01)-h(2)}{2.01-2}\]

Step 2: Calculate average vertical velocity between 'a' and 'a+0.001' seconds

Continuing with the average rate of change as in the first step, this time however, \(a\) is not a concrete number. Therefore, we keep 'a' as it is and replace 'b' with \(a+0.001\) to get \[AR = \frac{h(a+0.001)-h(a)}{(a+0.001)-a}\] Simplifying this further we get \[AR = \frac{h(a+0.001)-h(a)}{0.001}\]

Step 3: Express average vertical velocity for the interval '[a, a+k]' seconds

Instead of working with concrete numbers or a very small change in time into the average rate of change definition, a general 'k' is used as the change in time. In this case, we replace 'b' with \(a+k\) to derive the formula: \[AR = \frac{h(a+k)-h(a)}{(a+k)-a}\] This simplifies further to \[AR = \frac{h(a+k)-h(a)}{k}\]

Key Concepts

Average Vertical Velocity

Understanding the concept of average vertical velocity is crucial in studies where motion in a vertical direction is analyzed, such as the movement of a rocket after takeoff. It is the rate at which an object moves vertically over a certain time period. Mathematically, it can be expressed as the change in vertical position divided by the time it takes for that change.

For instance, consider a rocket's height as a function of time, denoted as h(t). If we want to find out how fast the rocket is ascending on average between two points in time, we calculate the difference in height divided by the difference in time:

Average Vertical Velocity Formula

\[\text{Average Vertical Velocity (AVV)} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}\]Where h(t_1) and h(t_2) represents the height of the rocket at times t_1 and t_2 respectively. By calculating the average vertical velocity over intervals such as from 2 to 2.01 seconds or [a, a+0.001], one can gain insights into the rocket's performance and ascent characteristics during its flight.

Rate of Change of a Function

The rate of change tells us how a function's output value changes as its input value changes. In calculus, this concept helps to understand the behavior of different physical quantities as they evolve over time or as a function of other variables.

In the scenario of the rocketship, the function given by h(t) describes the height of the rocket at any time t. The rate of change between any two moments, say t=a and t=a+k, can be interpreted as the average velocity of the rocket within that time span. The equation provided in the book solution helps us determine this by subtracting the function's value at the starting time from its value at the ending time and dividing it by the total time elapsed:

Calculation of Rate of Change

\[\text{Rate of Change (RoC)} = \frac{h(a+k) - h(a)}{k}\]If k is very small, the result can give us an approximation of the instantaneous velocity (which we'll discuss soon). The rate of change is a foundation stone in not just physics but in economics, biology, and practically any field where change over time or another dimension is of interest.

Calculus Applications in Physics

Calculus, particularly calculus involving rates of change, is a powerful toolkit for solving problems in physics. We can describe almost all physical phenomena involving change; from the motion of particles to the behavior of electromagnetic fields, using differential calculus to find instantaneous rates of change and integral calculus for cumulative effects.

In our rocket example, calculus allows us to compute not only how the height changes with time but also to explore concepts like velocity and acceleration. The application of calculus crops up in physics in numerous ways:

• Describing the motion of objects using derivatives and integrals to compute velocity and displacement from acceleration data.
• Analyzing wave functions in quantum mechanics.
• Modeling the flow of fluids in dynamics.
• Calculating the electric and magnetic fields in electromagnetism.

These applications are not just academic exercises but are used to predict real-world behaviors and outcomes that affect various engineering and scientific pursuits.

Instantaneous Rate of Change

The instantaneous rate of change of a function at a point is one of the central ideas in calculus. It gives us the slope of the tangent line to the function at that specific point, which represents how fast the function's value is changing right at that moment. Think of it as an 'instant snapshot' of the rate at which one quantity changes with respect to another.

For our rocket, the instantaneous vertical velocity at a time a is the rate of change of the rocket's height at precisely that moment. Mathematically, it's the derivative of the function h(t) with respect to time at t = a:

Calculating Instantaneous Rate of Change

\[\text{Instantaneous Rate of Change (IRC)} = h'(a)\]The prime notation h'(a) denotes the derivative of h at a. If we compare this to the average rate of change over an interval, the instantaneous rate of change does not require a span of time but rather tells us the velocity at an exact instant, leading to more precise and critical information about the motion's nature at that point in time.