Problem 74 - Thomas Calculus

Question

You will find a graphing calculator useful. Let \(G(t)=(1-\cos t) / t^{2}\) a. Make tables of values of \(G\) at values of \(t\) that approach \(t_{0}=0\) from above and below. Then estimate lim_{t\rightarrow0} G ( t ) . b. Support your conclusion in part (a) by graphing \(G\) near \(\quad t_{0}=0. \)

Step-by-Step Solution

Answer

\( \lim_{t \to 0} G(t) = 0.5 \). The function approaches 0.5 graphically as well.

Step 1: Define the Function

The function given is \( G(t) = \frac{1-\cos t}{t^2} \). We will need to create tables of values for this function as \( t \) approaches 0 from both the positive and negative directions.

Step 2: Create Table of Values for \( t \to 0^+ \)

Choose a set of positive values for \( t \) that get closer to 0, such as \( t = 0.1, 0.01, 0.001, \) and \( 0.0001 \). For each of these values, compute \( G(t) \) using a calculator:- \( t = 0.1 \Rightarrow G(0.1) \approx 0.4950 \)- \( t = 0.01 \Rightarrow G(0.01) \approx 0.4999 \)- \( t = 0.001 \Rightarrow G(0.001) \approx 0.499999 \)- \( t = 0.0001 \Rightarrow G(0.0001) \approx 0.49999999 \)

Step 3: Create Table of Values for \( t \to 0^- \)

Choose a set of negative values for \( t \) to approach 0, such as \( t = -0.1, -0.01, -0.001, \) and \( -0.0001 \). Compute \( G(t) \) for each:- \( t = -0.1 \Rightarrow G(-0.1) \approx 0.4950 \)- \( t = -0.01 \Rightarrow G(-0.01) \approx 0.4999 \)- \( t = -0.001 \Rightarrow G(-0.001) \approx 0.499999 \)- \( t = -0.0001 \Rightarrow G(-0.0001) \approx 0.49999999 \)

Step 4: Estimate \( \lim_{t \to 0} G(t) \)

From the table of values for both \( t \to 0^+ \) and \( t \to 0^- \), we see that \( G(t) \) approaches 0.5 as \( t \) approaches 0. Therefore, we estimate that \( \lim_{t \to 0} G(t) = 0.5 \).

Step 5: Graph the Function

Use a graphing calculator or software to plot \( G(t) = \frac{1-\cos t}{t^2} \) near \( t = 0 \). Observe that as \( t \) gets closer to zero from both sides, the graph approaches a horizontal line at \( y = 0.5 \), supporting our conclusion from the tables.

Key Concepts

Graphing Calculators

Graphing calculators play a crucial role in understanding the behavior of complex functions, especially near points of interest such as asymptotes or discontinuities.
When you graph a function like \(G(t) = \frac{1-\cos t}{t^2}\), it can provide visual confirmation of what tables of values suggest.
A graphing calculator can help by:

• Plotting the function accurately to show its behavior near zero.
• Allowing zoom in and out for a better view of the graph's changes as \(t\) approaches zero.
• Supporting interactive learning, so students can change functions or perspectives.

For this particular function, visually observing that \(G(t)\) approaches 0.5 as \(t\) nears zero gives a more intuitive grasp than numbers alone. Graphing the function solidifies the understanding of limits in calculus.

Tables of Values

Creating tables of values is an approachable way to estimate limits. By calculating \(G(t)\) for values close to zero, both from positive and negative sides, it's easier to perceive how the function behaves.
Significantly, a table of values helps to:

• Identify patterns. You can see that as \(t\) becomes very small, \(G(t)\) consistently approaches 0.5.
• Provide numerical evidence supporting the visual observations from graphing.
• Serve as a reference to detect discrepancies in manual calculations or graph interpretations.

This systematic approach is vital in calculus to back claims with numerical proofs when predicting limits.

Function Behavior Near a Point

Understanding the behavior of a function like \(G(t)\) near a specific point, such as \(t = 0\), is essential in calculus. This behavior helps in predicting limits and understanding continuity or discontinuity in functions.
Near \(t=0\), we observe:

• The function \(G(t)\) is undefined precisely at \(t=0\) due to division by zero.
• Despite this, the behavior around \(t=0\) suggests that the function approaches a stable limit as \(t\) nears zero.
• The limit, in this case, is estimated numerically and visually to be 0.5.

Studying the function's tendency allows students to make predictions and understand the broader concept of limits, which is a foundation of calculus. Explaining how the function behaves near \(t=0\) rather than at \(t=0\) helps to bridge the gap created by the undefined point.