Problem 80 - Calculus Single Variable

Question

A function \(f\) and a point \(c\) are given. Graph the function $$ \phi(x)=\frac{f(x)-f(c)}{x-c} $$ in an appropriate viewing window centered about the line \(x=c .\) Use the graph of \(\phi\) to decide whether or not \(f^{\prime}(c)\) exists. Explain the reason for your answer, and, if your answer is that \(f^{\prime}(c)\) exists, use the graph of \(\phi\) to approximate the value of \(f^{\prime}(c)\). $$ f(x)=\sqrt{x^{4}-2 x^{3}+2 x-1}, c=1 $$

Step-by-Step Solution

Answer

The graph of \(\phi(x)\) does not exhibit a finite limit as \(x\) approaches 1, so \(f'(1)\) does not exist.

Step 1: Substitute the Function and Point

Firstly, substitute the given function \(f(x) = \sqrt{x^4 - 2x^3 + 2x - 1}\) and point \(c = 1\) into the expression for \(\phi(x)\). This yields the function \(\phi(x) = \frac{\sqrt{x^4 - 2x^3 + 2x - 1} - \sqrt{0}}{x - 1}\), simplifying to \(\phi(x) = \frac{\sqrt{x^4 - 2x^3 + 2x - 1}}{x - 1}\).

Step 2: Determine the Appropriate Viewing Window

To analyze the behavior of \(\phi(x)\) as \(x\) approaches 1, choose a viewing window centered around \(x = 1\) such as \(x \in [0.8, 1.2]\) and \(y \) range based on trial, for example, \(y \in [-5, 5]\).

Step 3: Graph the Function \(\phi(x)\)

Use graphing software or a graphing calculator to plot \(\phi(x)\) over the chosen viewing window. Examine the graph around \(x = 1\) to observe the behavior of \(\phi(x)\).

Step 4: Analyze the Graph to Determine Existence of \(f'(c)\)

Look at the graph of \(\phi(x)\) near \(x = 1\). If the graph approaches a finite limit as \(x\) approaches 1, then \(f'(1)\) exists. If the graph does not approach a finite limit, then \(f'(1)\) does not exist.

Step 5: Approximate \(f'(c)\) if It Exists

If the graph indicates that \(f'(1)\) exists, estimate the value of \(f'(1)\) by observing the value that \(\phi(x)\) approaches as \(x\) approaches 1 on the graph.

Key Concepts

Limits

Understanding limits is essential to exploring the behavior of functions as we approach a certain point, such as the point where a derivative is calculated. In the given exercise, analyzing the limit of the function \( \phi(x) = \frac{\sqrt{x^4 - 2x^3 + 2x - 1}}{x - 1} \) as \( x \) approaches 1 is crucial.

Limits help us comprehend what value a function approaches, even if it doesn't actually reach that value. This is particularly useful in calculus, where we often need to know the behavior of a function as it gets close to a specific point.

In simpler terms:

• Limits allow for understanding near-behavior without direct substitution.
• They're foundational for calculus concepts like derivatives and integrals.

To examine if a limit exists at a point, check if the values from the left and right sides of the point converge to the same number. If they do, the limit exists.

Continuity

Continuity is the property of a function to be unbroken or uninterrupted; in other words, no gaps or holes in the graph. A function is continuous at a point \( c \) if the following three conditions are met:

• The function \( f(x) \) is defined at \( x = c \).
• The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

In this exercise, understanding the continuity of \( \phi(x) \) around \( x = 1 \) is crucial in determining if the derivative \( f'(1) \) exists.

If any of the above conditions are not met, the function is not continuous at that point. Continuity plays an important role because, for a function to have a derivative at a point, it must be continuous there. If \( \phi(x) \) has no abrupt changes or breaks at \( x = 1 \), it suggests continuity.

Differentiability

Differentiability is the ability of a function to have a derivative at a particular point. If a function is differentiable at a point, it means the slope of the tangent line to the function at that point can be determined. For a function to be differentiable at a point \( c \), it must be continuous there. However, a function can be continuous but not differentiable at some points (e.g., corners or cusps).

Differentiability in this exercise refers to determining if \( f'(1) \) exists. This involves:

• Checking if \( \phi(x) \) settles into a specific path approaching a finite number as \( x \to 1 \).
• Ensuring the absence of jumps, infinite slopes, or vertical tangents in the plotted graph of \( \phi(x) \).

Recognizing differentiability results in understanding rates of change more precisely. If the preconditions aligned with limits and continuity are satisfied, differentiability follows.

Graphing Functions

Graphing functions is a powerful tool for visualizing and understanding the mathematical behavior of expressions. In our exercise, graphing \( \phi(x) \) helps in observing how the function behaves near \( x = 1 \). A clear graph enables us to see if \( \phi(x) \) approaches a finite value, implying the existence of the derivative at that point.

Key elements of graphing:

• Choose an appropriate window that captures behavior around points of interest.
• Look for trends that indicate limits and ensure no abrupt irregularities are present.

By plotting \( \phi(x) \) and focusing on its nature at \( x = 1 \), we can deduce analytically whether \( f'(1) \) exists. If the graph stabilizes around a certain value, it is a strong indication that the derivative exists. Graphs convey critical information visually and can reveal patterns not always obvious through algebraic methods alone.