Problem 49 - Calculus

Question

For the functions in Problems \(46-53,\) do the following: (a) Make a table of values of \(f(x)\) for \(x=0.1,0.01,0.001\) \(0.0001,-0.1,-0.01,-0.001,\) and -0.0001 (b) Make a conjecture about the value of \(\lim _{x \rightarrow 0} f(x)\) (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for \(x\) near 0 such that the difference between your conjectured limit and the value of the function is less than \(0.01 .\) (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.) $$f(x)=\sin 3 x$$

Step-by-Step Solution

Answer

The limit is 0 as \( x \to 0 \).

Step 1: Evaluate Function al Given Values

Compute the values of \(f(x) = \sin(3x)\) for \(x = 0.1, 0.01, 0.001, 0.0001, -0.1, -0.01, -0.001, -0.0001\).1. \( f(0.1) = \sin(0.3) \)2. \( f(0.01) = \sin(0.03) \)3. \( f(0.001) = \sin(0.003) \)4. \( f(0.0001) = \sin(0.0003) \)5. \( f(-0.1) = \sin(-0.3) \)6. \( f(-0.01) = \sin(-0.03) \)7. \( f(-0.001) = \sin(-0.003) \)8. \( f(-0.0001) = \sin(-0.0003) \)Using a calculator, we approximate these values. Since \(\sin(x)\) is an odd function, note that \(\sin(-x) = -\sin(x)\).

Step 2: Conjecture the Limit

Examine the results from the table created in Step 1 to estimate \( \lim_{x \to 0} \sin(3x) \).As \( x \) approaches 0, both positive and negative values get very close to zero. Thus, our conjecture is:\[ \lim_{x \to 0} \sin(3x) = 0 \]

Step 3: Graph the Function

Graph \( f(x) = \sin(3x) \) to visualize its behavior near \( x = 0 \). Notice that the function oscillates, and as \( x \) gets closer to zero from both sides, \( f(x) \) approaches zero.The graph confirms our conjecture from Step 2, showing consistency as the limit approaches zero.

Step 4: Determine Interval Near Zero

Identify an interval around \( x = 0 \) where the difference between \( f(x) \) and the conjectured limit (0) is less than 0.01.We need \( |\sin(3x) - 0| < 0.01 \), which simplifies to \( |\sin(3x)| < 0.01 \).Since \( \sin(3x) \approx 3x \) when \( x \) is close to 0, solve \( |3x| < 0.01 \) to find the interval:\[ -\frac{0.01}{3} < x < \frac{0.01}{3} \]This gives approximately the interval \( -0.00333 < x < 0.00333 \).

Key Concepts

Sine Function

The sine function, often denoted as \( \sin(x) \), is a fundamental trigonometric function that describes a smooth repetitive oscillation. It is essential in many areas of mathematics and physics due its periodic nature. The sine function has a range of values between -1 and 1. As an odd function, it exhibits symmetry with respect to the origin, meaning \( \sin(-x) = -\sin(x) \). Moreover, the sine function is periodic with a period of \( 2\pi \), indicating it repeats its values every \( 2\pi \) units.When transformed with a coefficient, such as \( f(x) = \sin(3x) \), the function oscillates more rapidly compared to the basic \( \sin(x) \). This transformation impacts the frequency or the number of cycles completed within a specific interval. Thus, understanding how coefficients affect the sine function's behavior is crucial for graphing and analysis.

Graphing Functions

Graphing functions is a powerful way to understand the behavior of a function visually. It involves plotting the output value \( f(x) \) for corresponding input values \( x \), creating a curve that can reveal properties like continuity, periodicity, and symmetry.For the example function \( f(x) = \sin(3x) \), graphing lets us observe how the sine function has been altered by the coefficient 3. The graph will exhibit three cycles in the interval \( 0 \) to \( 2\pi \) instead of the usual single cycle of \( \sin(x) \). By observing the plot, students can determine key points where the function achieves maximum and minimum values, crosses the axis (zeroes), and oscillates around a horizontal line. Graphing becomes especially essential when confirming theoretical predictions such as the behavior of \( f(x) \) as \( x \) approaches zero, providing a visual affirmation of calculated limits and conjectures.

Interval Estimation

Interval estimation is a technique used to determine the range within which a function's value falls within a specified accuracy. This method is crucial, particularly in calculus, for evaluating limits and approximating values closely near a point of interest.In solving the exercise, we focus on finding an interval where the difference between \( f(x) = \sin(3x) \) and its limit prediction is less than a tiny value, typically \( 0.01 \). We aim to find a small neighborhood around \( x = 0 \) where this condition holds. To achieve this, observe that as \( x \) nears zero, \( \sin(3x) \) behaves like \( 3x \). Thus, by solving the inequality \( |3x| < 0.01 \), we identify the tight interval \( -0.00333 < x < 0.00333 \). This ensures our limit conjecture stays accurate within a desirable precision.

Conjecture in Calculus

In calculus, a conjecture involves making an educated guess about a mathematical behavior based on observed patterns or computed data. Conjectures can lead to hypotheses or even proofs after sufficient supporting evidence is gathered.When evaluating limits like \( \lim_{x \to 0} \sin(3x) \), conjectures are drawn from trends observed in tabulated function values as \( x \) approaches zero from both the positive and negative sides. If these values trend toward zero, it is reasonable to conjecture that the limit itself is zero. This process, sometimes formalized in coursework, emphasizes understanding rather than computation alone, allowing students to engage deeply with mathematical concepts.Graphical representation can serve as a critical tool for verifying such conjectures, as visual trends alongside numerical ones reinforce the underlying mathematical intuition leading to the conjecture.