Problem 14 - Calculus Early Transcendentals

Question

Let \(g(x)=2 x^{3}-12 x^{2}+26 x+4\) and note that \(\lim _{x \rightarrow 2} g(x)=24\) For each value of \(\varepsilon,\) use a graphing utility to find all values of \(\delta>0\) such that \(|g(x)-24|<\varepsilon\) whenever \(0<|x-2|<\delta\) Sketch graphs illustrating your work. a. \(\varepsilon=1\) b. \(\varepsilon=0.5\)

Step-by-Step Solution

Answer

Question: For the given function g(x) = 2x³ - 12x² + 26x + 4, find the values of delta (δ) for each epsilon (ε) such that 0 < |x - 2| < δ ensures |g(x)-24| < ε, when ε = 1 and ε = 0.5.

Step 1: a. ε = 1

Given the limit of g(x) is 24 as x tends to 2, the inequality we should consider is: $$|g(x)-24|<1$$ This inequality should hold whenever 0 < |x - 2| < delta. To find delta, we can use a graphing utility and analyze the values of x such that the inequality is satisfied.

Step 1 : Plot the graph

First, plot the given function g(x) = 2x^3 - 12x^2 + 26x + 4 and graphically find the points where the function is close to 24. Make sure to focus on the x-values around 2 since x is approaching 2.

Step 2 : Analyze the graph

Using the graphing utility, find the intervals of x such that | g(x) - 24 | < ε = 1 and 0 < |x - 2| < δ. Notice that there will be two intervals because g(x) - 24 can be both positive and negative as long as it's less in absolute value than 1.

Step 3 : Find δ

Now, find the distance between the x-values found for the intervals and 2. Taking the minimum of these distances as delta, you ensure that delta is small enough for the inequality to hold. Record the δ value you found.

Step 5: b. ε = 0.5

Similarly for ε = 0.5, the inequality to consider is: $$|g(x)-24|<0.5$$

Step 1 : Plot the graph

As in part a, first plot the given function g(x) = 2x^3 - 12x^2 + 26x + 4 and graphically find the points where the function is close to 24, with a focus on the x-values near 2.

Step 2 : Analyze the graph

Using the graphing utility we can find the intervals of x such that | g(x) - 24 | < ε = 0.5 and 0 < |x - 2| < δ. Observe that there will be two intervals due to the inequality being in absolute value.

Step 3 : Find δ

Find the distance between the x-values found for the intervals and 2. Take the minimum of these distances as delta, so that delta is small enough to hold the inequality true. Record the δ value found. Conclusion: For each value of epsilon, the corresponding delta value found using the graphing utility ensures that the inequality is satisfied whenever 0 < |x - 2| < δ. Sketch the graphs of the function for each epsilon value and mark the intervals for the delta values found to visualize your work better.

Key Concepts

Delta Epsilon Definition

When dealing with limits, the delta-epsilon (\(\varepsilon-\delta\)) definition is a crucial concept. It provides a rigorous mathematical way of defining the limit of a function. Simply put, it ensures that as the input of a function approaches a particular value, the output gets arbitrarily close to a desired function limit.

To explore this, consider the function \(g(x)=2x^3-12x^2+26x+4\) and its limit as \(x\) approaches 2 is 24. According to the \(\varepsilon-\delta\) definition:

• For every \(\varepsilon>0\), there exists a \(\delta>0\) such that when \(0<|x-2|<\delta\), the inequality \(|g(x)-24|<\varepsilon\) holds.
• Choosing specific \(\varepsilon\) values (like 1 and 0.5 in this exercise) involves finding corresponding \(\delta\) values ensuring this closeness to 24.

Using this approach, mathematicians can make precise claims about how function values behave near specific points.

Understanding this foundational concept helps in grasping calculus ideas more clearly.

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions and their behaviors. In this exercise, it illuminates how \(|g(x)-24|<\varepsilon\) behaves as \(x\) approaches 2. Here's how a graphing utility aids this:

• **Visual Insight**: By plotting \(g(x)=2x^3-12x^2+26x+4\), one can directly see where the function's graph gets close to the limit value of 24.
• **Identifying Intervals**: It helps find intervals around \(x=2\) where the condition \(|g(x)-24|<\varepsilon\) holds true. These intervals show the regions in which the function is within the desired proximity to 24.
**Delta Detection**: From the graph, one can discern the maximum allowable \(\delta\) by observing the x-values where these conditions start and stop holding.

Using graphing utilities allows for an intuitive grasp of the concepts often explored analytically in calculus. They provide immediate feedback and visual clarity, which is invaluable in learning and confirming mathematical theories.

Polynomial Functions

Polynomial functions, like \(g(x)=2x^3-12x^2+26x+4\), are expressions made up of variables raised to non-negative integer powers multiplied by coefficients. These functions are a staple in algebra due to their straightforward nature and ease of manipulation.

Polynomials have several characteristics which make them interesting in calculus:

• **Smooth Curves**: Unlike other functions, polynomials are continuously differentiable, meaning they have no abrupt changes, breaks, or corners in their graphs. This makes them easier to work with when finding limits or derivatives.
• **Predictable Behavior**: At each end of the graph, polynomial functions rise or fall depending on the leading term's degree and coefficient, providing a predictable behavior useful in limit assessments.
• **Analytical Solutions**: Finding limits of polynomial functions as x approaches any real number, like in this problem where \(x\rightarrow2\), tends to be straightforward. They can often be solved with direct substitution, making them ideal for illustrating limit concepts.

Recognizing the role of these functions in calculus helps anchor more complex topics around familiar, simpler cases.