Problem 44 - Essential Calculus

Question

Use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and points of inflection. (c) Graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) on the same set of coordinate axes and state the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) \(f(x)=\sqrt{2 x} \sin x,[0,2 \pi]\)

Step-by-Step Solution

Answer

The first derivative \(f'(x)\) and the second derivative \(f''(x)\) of the function \(f(x)=\sqrt{2 x} \sin x\) need to be determined first. Then, identify the critical points, classify them into local maximums, minimums or neither using the second derivative test. Plot \(f\), \(f'\), and \(f''\) over the interval [0,\(2\pi\)]. Lastly, analyze the behavior of \(f(x)\), \(f'(x)\), and \(f''(x)\) by observing the signs.

Step 1: Differentiation

The first step is to differentiate the function \(f(x)=\sqrt{2 x} \sin x\). Using the product rule for differentiation \(u'v + uv'\) where \(u=\sqrt{2x}\) and \(v=\sin x\), compute the first derivative, \(f'(x)\). Proceed to calculate the second derivative \(f''(x)\) by repeating the process.

Step 2: Determine Relative Extrema and Inflection Points

To find the relative extrema, set the first derivative equal to zero and solve for \(x\). These points are the critical points. To classify them as local maximums, minimums or neither, use the second derivative test. Evaluate \(f''\) at the critical points. If the value is positive, it’s a local minimum, if negative, it’s a local maximum, and if zero, the test is inconclusive.

Step 3: Graphing the Functions

Using a computer algebra system, plot \(f\), \(f'\) and \(f''\) over the interval [0, \(2\pi\)]. Identify the behavior of the function \(f\) and the signs of \(f'\) and \(f''\).

Step 4: Analyze the Graphs

Observe the points where \(f'(x)\) (the slope of \(f(x)\)) changes its sign, as these are the local maximum and minimum of \(f(x)\). When \(f'(x)\) rises from negative to positive, \(f(x)\) gets a minimum, and when \(f'(x)\) drops from positive to negative, \(f(x)\) gets a maximum. Similarly, points where \(f''(x)\) (the slope of the \(f'(x)\)) changes its sign are the inflection points of \(f(x)\).

Key Concepts

Product Rule

In calculus, the product rule is a critical tool for differentiating functions that are products of two more simple functions. For example, if we have two functions, denoted as u(x) and v(x), the product rule states that the derivative of their product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x).

So, for the function f(x) = \(\sqrt{2x}\) \(\sin x\), we identify u(x) = \(\sqrt{2x}\) and v(x) = \(\sin x\). We compute their derivatives separately and then apply the product rule to find f'(x).

First Derivative

The first derivative of a function represents the rate at which the function's value is changing at any given point, or in more familiar terms, it describes the slope of the tangent line to the function's curve at any point. It's notated as f'(x) or \(\frac{df}{dx}\), and it tells us where the function is increasing or decreasing. When f'(x) is positive, the function is ascending, and when f'(x) is negative, the function is descending.

For our function f(x), we already used the product rule to calculate f'(x). As part of the analysis, we would set f'(x) to zero to find critical points which could be potential relative extrema.

Second Derivative

The second derivative of a function, denoted as f''(x), provides information about the curvature of the function's graph. It tells us whether the graph is concave up or down at a given point, which is crucial for understanding the behavior of the function around its critical points.

If f''(x) is positive, the graph is concave up, and if f''(x) is negative, the graph is concave down. This information is especially useful when the first derivative test is inclusive or when we are seeking to identify points of inflection — where the concavity changes.

Relative Extrema

Relative extrema are the peaks and troughs in the graph of a function, also referred to as local maxima and minima. These are the points where a function changes direction from increasing to decreasing, or vice versa, and they occur at critical points where the first derivative is zero or not defined.

To determine the nature of these extrema, we employ the first or second derivative tests. For the given function f(x), after finding the critical points, we test these points with the second derivative, f''(x), to identify them as local minimums or maximums.

Points of Inflection

A point of inflection is a point on the graph of a function where the curvature changes direction or 'inflects'. This usually corresponds to where the second derivative changes sign, but can also occur where the second derivative is zero or undefined, provided there's an actual change in concavity.

Points of inflection provide valuable information about the geometry of the function's graph. For our exercise, identifying these points involves computing the second derivative of f(x) and analyzing its sign changes over the given interval.

Graphing Functions

Graphing the functions f, f', and f'' gives us a visual representation of their relationships. The graph of f(x) shows the actual function, while the graph of f'(x) illustrates where the function f is increasing or decreasing, and the zeros of f'(x) correspond to the potential extrema of f.

Similarly, f''(x)'s graph provides insight into the function's concavity. By graphing all three on the same axes, one can easily correlate the slopes and curvatures between them, leading to a deeper understanding of function behavior.

Computer Algebra System

A computer algebra system (CAS) is software that facilitates symbolic mathematics. It allows for the manipulation of mathematical expressions in a way that is similar to the traditional manual computations performed by mathematicians. Such systems are extremely useful for differentiating complex functions, solving equations, and graphing functions, helping to visualize the relationships between them.

In the context of our problem, a CAS can be used to quickly compute derivatives, find extrema and inflection points, and graph the function f(x) and its derivatives. This visual aid supports the analytical process and can greatly enhance the learning experience.