Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=\frac{x^{2}+1}{x^{2}}=1+\frac{1}{x^{2}}\) (a) Graph \(f\). (b) Find the following. i. \(\lim _{x \rightarrow \infty} f(x) \quad\) ii. \(\lim _{x \rightarrow 0} f(x) \quad\) iii. \(\lim _{x \rightarrow \infty} f^{\prime}(x)\) iv. \(\lim _{x \rightarrow 0^{+}} f^{\prime}(x) \quad\) v. \(\lim _{x \rightarrow 0^{-}} f^{\prime}(x)\) (c) Graph \(f^{\prime}(x)\). (d) Find \(f^{\prime \prime}(x)\). (e) Are your answers to all parts of this problem consistent? (If not, find your errors.)

Step-by-Step Solution

Answer

a) Graph of function presents a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 1\). b) i. \(\lim _{x \rightarrow \infty} f(x) = 1\), ii. \(\lim _{x \rightarrow 0} f(x) = \infty\), iii. \(\lim _{x \rightarrow \infty} f^{\prime}(x) = 0\), iv. \(\lim _{x \rightarrow 0^{+}} f^{\prime}(x) = -\infty\), v. \(\lim _{x \rightarrow 0^{-}} f^{\prime}(x) = \infty\). c) Derivative graph corresponds to the equation \(f^{\prime}(x)=-\frac{2}{x^3}\). d) The second derivative is \(f^{\prime \prime}(x)=\frac{6}{x^{4}}\). e) All the answers are found to be consistent with each other.

Step 1 - Function Graph

To graph \(f(x)=\frac{x^{2}+1}{x^{2}}=1+\frac{1}{x^{2}}\), one should be aware that \(f(x)\) has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 1\). Plot this along with several points for \(x\) to form the image of the function graph.

Step 2 - Determine the limit

i. As \(x\) approaches infinity, the \(\frac{1}{x^{2}}\) term diminishes and approaches zero, leaving \(1\). Hence, \(\lim _{x \rightarrow \infty} f(x) = 1\). ii. As \(x\) approaches 0, \(\frac{1}{x^2}\) tends to infinity. Therefore, \(\lim _{x \rightarrow 0} f(x) = \infty\).

Step 3 - Differentiation

To find \(f^{\prime}(x)\), differentiate \(f(x)\) with respect to \(x\) to obtain \(f^{\prime}(x)=-\frac{2}{x^3}\). Next, calculate the limits for \(f^{\prime}(x)\). iii. As \(x\) approaches infinity, \(\lim _{x \rightarrow \infty} f^{\prime}(x) = 0\). iv. As \(x\) approaches \(0^{+}\), \(\lim _{x \rightarrow 0^{+}} f^{\prime}(x) = -\infty\). v. As \(x\) approaches \(0^{-}\), \(\lim _{x \rightarrow0^{-}} f^{\prime}(x) = \infty\).

Step 4 - Graph \(f^{\prime}(x)\)

Graph the derivative \(f^{\prime}(x)=-\frac{2}{x^3}\). This can be done by plotting several points for \(x\) based on the function of the derivative.

Step 5 - Find Second Derivative

To find \(f^{\prime \prime}(x)\), differentiate \(f^{\prime}(x)\) with respect to \(x\) to obtain \(f^{\prime \prime}(x)=\frac{6}{x^{4}}\).

Step 6 - Check for Consistency

Review the obtained results from Steps 2 to 5. It is observed that the results are consistent with the properties of the function \(f(x)\), its derivative \(f^{\prime}(x)\) and second derivative \(f^{\prime \prime}(x)\) as evaluated graphically and through limit determination. Hence, no errors are found.

Key Concepts

Asymptotes

An asymptote of a function is a line that the graph of the function approaches but never actually reaches. Asymptotes can be vertical, horizontal, or oblique. In the discussed exercise, the function graph of
\( f(x)=1+\frac{1}{x^2} \)
has a vertical asymptote at
\( x = 0 \)
and a horizontal asymptote at
\( y = 1 \).
Understanding asymptotes is crucial for graphing functions and analyzing their behavior as
\( x \)
approaches infinity or a specific value. For instance, a vertical asymptote often indicates points where the function goes off to infinity, which corresponds to discontinuities or undefined areas in the function.

Function Graph

The graph of a function provides a visual representation of all the points \((x, f(x))\) that make up the function. In our case, the function
\( f(x)=1+\frac{1}{x^2} \)
is graphed by plotting points and observing the behavior indicated by the asymptotes. The graph helps us visualize how the function behaves and allows us to predict values, understand the domain and range, as well as identify continuity, intervals of increase or decrease, and other properties. When graphing \(f(x)\), it's essential to note points where the function has dramatic changes, such as near vertical asymptotes or where the slope of the graph approaches zero, which could indicate a horizontal asymptote or a local extremum.

First Derivative

The first derivative of a function, expressed as \(f'(x)\), represents the rate at which the function's value is changing at any given point. It can inform us about the slope of the function graph at any given point and is used to determine where a function is increasing or decreasing. Calculating the first derivative of the given function
\( f(x) \)
results in
\( f'(x)=-\frac{2}{x^3} \).
By analyzing \(f'(x)\), we can also determine the behavior of the function as \(x\) approaches specific values. For instance, the negative sign in \(f'(x)\) indicates that the graph of \(f(x)\) is decreasing as \(x\) moves away from the asymptote at \(x = 0\).

Second Derivative

The second derivative of a function, denoted by \(f''(x)\), describes the curvature or concavity of the function's graph. It tells us how the rate of change of the function's rate of change (the first derivative) is behaving. For the studied expression
\( f(x)=1+\frac{1}{x^2} \),
the second derivative is
\( f''(x)=\frac{6}{x^4} \).
This positive second derivative indicates the graph is concave up everywhere. In other words, as you move along the graph of \(f(x)\), the slope is always increasing. The concavity can provide information about inflection points, which are points where the graph changes from being concave up to concave down, or vice versa, although no such points exist for this particular function.