Problem 53 - Calculus Single Variable

Question

When the argument of a function \(f\) changes by an increment \(h\) from \(x-h\) to \(x,\) the value of the function changes by the increment \(F(x, h)=f(x)-f(x-h)\). If \(h\) is small, then it is to be expected that \(F(x, h)\) is also small. In \(1615,\) Kepler noticed that when \(f\left(x_{0}\right)\) is a maximum value of \(f,\) the increment \(F\left(x_{0}, h\right)\) is even smaller than one might expect. In this exercise, we use the function \(f(x)=5-24 x^{2}+20 x^{3}-3 x^{4}\) to investigate Kepler's observation. a. Graph \(f\) in the window \([2,5] \times[20,135] .\) Using the graph, determine the value \(x_{0}\) of \(x\) for which \(f(x)\) is maximized. b. Calculate \(F\left(x_{0}, h\right)\) and \(F(3.99, h)\) for \(h=10^{-3}, 10^{-4}\) and \(10^{-5}\). You will notice that for each of these values of \(h, F\left(x_{0}, h\right)\) is much smaller than \(F(3.99, h)\). c. Plot \(F(3.99, h)\) for \(10^{-5} \leq h \leq 10^{-3} .\) You will see that \(F\) \((3.99, h) \approx m h\) for some constant \(m\) d. Now plot \(F\left(x_{0}, h\right)\) for \(10^{-5} \leq h \leq 10^{-3}\). You will see that \(F\left(x_{0}, h\right) \approx A h^{2}\) for some constant \(A .\) The reason for the behavior Kepler observed is that \(h^{2}\) is negligible compared to \(h\) when \(h\) is small.

Step-by-Step Solution

Answer

At \(x \approx 3.98\), the function reaches its maximum; \(F(x_0, h)\) is smaller than \(F(3.99, h)\); \(F(x_0, h) \approx A h^2\).

Step 1: Graph the function

Using graphing software or a calculator, plot the function \(f(x) = 5 - 24x^2 + 20x^3 - 3x^4\) within the window \([2, 5]\times[20, 135]\). From the graph, determine the value of \(x\) where the function reaches its maximum. Observing the graph, the maximum value appears around \(x = 3.98\).

Step 2: Calculate function increments

Calculate the function increment \(F(x, h) = f(x) - f(x-h)\). Start by determining \(F(x_0, h)\) using the approximate \(x_0 = 3.98\) and \(h\) values \(10^{-3}, 10^{-4}, 10^{-5}\). Also calculate \(F(3.99, h)\) for the same \(h\) values.

Step 3: Calculate specific increments

For \(x_0 = 3.98\), calculate \(F(3.98, 10^{-3})\), \(F(3.98, 10^{-4})\), and \(F(3.98, 10^{-5})\). Do likewise for \(x = 3.99\). You'll find that \(F(x_0, h)\) is significantly smaller compared to \(F(3.99, h)\) for each \(h\).

Step 4: Analyze and plot \(F(3.99, h)\)

Plot \(F(3.99, h)\) for \(10^{-5} \leq h \leq 10^{-3}\). You should find that \(F(3.99, h)\) behaves linearly with \(h\), following approximately \(F(3.99, h) \approx m h\) for some constant \(m\).

Step 5: Analyze and plot \(F(x_0, h)\)

Plot \(F(3.98, h)\) over the same range. Here, \(F(x_0, h)\) should show a relationship closer to \(F(x_0, h) \approx A h^2\) for some constant \(A\), illustrating why the increment is much smaller at the maximum point.

Key Concepts

Kepler's Observation

Kepler's insights in 1615 provided a pivotal understanding of calculus before the development of formal calculus regardless. Kepler noticed a unique phenomenon; when a function has a maximum value at a point, the change in the function's value as its input shifts by a small amount is particularly tiny.
This observation comes from the study of incremental changes, or derivatives, which play a crucial role in the fundamental principles of calculus. Kepler's observation leads us to think about function behavior around maximum and minimum points.

• When the function is at its maximum or minimum, small changes in the input result in particularly small changes in the output.
• This signifies the points on the curve where the slope — or rate of change — is zero, hence the function value doesn't change much.

This analysis of function behavior at critical points later contributed to developing more detailed techniques in calculus.

Maximum Value

Finding the maximum or minimum values of a function is a common task in calculus. The maximum value occurs at a point known as a 'vertex' on the curve shown by the function.
For our given function, analyzing its graph reveals the highest point in a certain domain.

• The maximum of the function is identified by finding where the function's derivative equals zero.
• In graph terms, this is where the curve peaks or dips.

In our example with the function \(f(x) = 5 - 24x^2 + 20x^3 - 3x^4\), graphing it with tools can help us locate the maximum point using the window \([2, 5]\times[20, 135]\).
From the graph, this occurs around \(x = 3.98\). This point is crucial as the function behaves differently there compared to other points.

Function Increment

The notion of a function increment, \(F(x, h) = f(x) - f(x-h)\), is a fundamental concept to understand how functions change with little shifts in their input. This increment helps in comparing how dramatically a function's value changes between close points.
For small values of \(h\), especially when we consider points around a maximum value of a function, this increment tells us something profound.

• If \(f\) reaches its maximum value at \(x_0\), then \(F(x_0,h)\) will be smaller than expected.
• This is because the slope of the function at a maximum is zero; less change in function values occurs there.

In practical calculations using \(x_0 = 3.98\) in our example, you calculate \(F(x_0, h)\) and contrast it with increments at close values like \(3.99\). The significant difference in increment values illustrates Kepler's initial observation.

Graph Analysis

Graphing functions is an excellent way to visualize and understand their behavior. A graph displays the curve of a function over a particular interval, showing peaks, valleys, and slopes at a glance.
For the exercise at hand, the graph of the polynomial \(f(x) = 5 - 24x^2 + 20x^3 - 3x^4\) helps pinpoint where the function reaches its maximum value. Visual inspection reveals the function's behavior, making it a powerful tool for comprehension.

• Graphing over the specified window ensures you focus on the critical area where the function's key characteristics emerge.
• The appearance of the graph can reveal immediately where zero slopes occur, signaling maximum or minimum points.

Plotting specific increments, like \(F(3.99, h)\), shows linear behavior, typical for values not at maximum, contrasting with parabolic behavior at maximum as seen with \(F(x_0, h)\). This difference underscores why visual analysis is often essential in calculus.