Problem 46 - 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.) $$f(x)=3 x+1$$

Step-by-Step Solution

Answer

The limit is 1; graph confirms, interval: \((-0.0033, 0.0033)\).

Step 1: Create a Table of Values

Substitute each value of \(x\) given in part (a) into the function \(f(x) = 3x + 1\). Calculate \(f(x)\) for these specific \(x\) values:- For \(x = 0.1\), \(f(0.1) = 3(0.1) + 1 = 1.3\)- For \(x = 0.01\), \(f(0.01) = 3(0.01) + 1 = 1.03\)- For \(x = 0.001\), \(f(0.001) = 3(0.001) + 1 = 1.003\)- For \(x = 0.0001\), \(f(0.0001) = 3(0.0001) + 1 = 1.0003\)- For \(x = -0.1\), \(f(-0.1) = 3(-0.1) + 1 = 0.7\)- For \(x = -0.01\), \(f(-0.01) = 3(-0.01) + 1 = 0.97\)- For \(x = -0.001\), \(f(-0.001) = 3(-0.001) + 1 = 0.997\)- For \(x = -0.0001\), \(f(-0.0001) = 3(-0.0001) + 1 = 0.9997\)

Step 2: Conjecture the Limit as x Approaches 0

Observe the table values determined in Step 1 as \(x\) approaches 0 from both the left and the right. As \(x\) gets smaller and closer to 0, the function values \(f(x)\) approach 1. Therefore, the conjectured limit is:\[\lim_{x \to 0} f(x) = 1.\]

Step 3: Graph the Function

Graph the function \(f(x) = 3x + 1\). The graph is a straight line with a slope of 3 and a y-intercept at 1. Observe the graph around \(x = 0\) to verify consistency with our conjectured limit. The line passes through \(y = 1\) at \(x = 0\), confirming our conjecture that \(\lim_{x \to 0} f(x) = 1\).

Step 4: Find an Interval Around 0 for Limit Confirmation

Determine an interval around \(x = 0\) where the difference between \(f(x)\) and the limit (1) is less than 0.01. We need \(|3x| < 0.01\), which simplifies to:- \[-0.01 < 3x < 0.01\]- Divide by 3: \[-0.0033 < x < 0.0033\]Thus, the interval is \((-0.0033, 0.0033)\). If \(x\) is within this interval, \(|f(x) - 1| < 0.01\), fitting the described window criteria.

Key Concepts

Limits

Limits help us understand the behavior of functions as the input approaches a particular value. In this problem, we are finding what happens to the function \( f(x) = 3x + 1 \) as \( x \) approaches zero. Calculating limits involves looking at the values of a function very close to a point. If all nearby values point towards a specific number, this is considered the limit of the function at that point.

In the given exercise, as \( x \) gets close to 0, the function values converge towards 1. This is noted in the step-by-step solution as it finds \(\lim_{x \to 0} f(x) = 1\). Limits are foundational in calculus as they lead to the definitions and computations of derivatives and integrals, which describe rates of change and total accumulation.

Functions

A function is a relation between a set of inputs and permissible outputs. In this exercise, the function \( f(x) = 3x + 1 \) relates every input \( x \) to an output that is calculated by multiplying \( x \) by 3 and then adding 1. This particular function is linear, which means it graphs as a straight line.

Linear functions are often the first type of function students learn about in calculus due to their simplicity and predictable behavior. They have constant rates of change, making them easy to graph and analyze. In the context of limits, the predictable pattern of a linear function often makes the calculation straightforward.

Graphing

Graphing is a visual way of understanding functions and their behaviors. By graphing the function \( f(x) = 3x + 1 \), we can visually see that it is a straight line that intersects the \( y \)-axis at 1 and has a slope of 3.

When you graph this line, pay attention to the point where it crosses the \( y \)-axis (y-intercept) and the steepness (slope) of the line. Graphing can help confirm limits and conjectures since it shows the direction and approach of the line around points of interest. In this problem, the graph supports the conclusion that \( \lim_{x \to 0} f(x) = 1 \), as the line passes through this point at \( x = 0 \).

• Make sure your graphing scale is clear to see changes in value.
• Plot several points around the interest area to maintain accuracy.

Conjecture

In calculus and mathematics, a conjecture is an educated guess based on pattern identification or computational evidence. In this task, after calculating values of \( f(x) \) at various inputs approaching zero, one can make a conjecture about the limit value.

The conjecture here states that \( \lim_{x \to 0} f(x) = 1 \). This conjecture is supported by the behavior of the function values as calculated steps showed. Conjectures can be tested by examination, such as graphing, or evaluations, which may provide a stronger validation of correctness in terms of limits or other mathematical concepts. Confirming the conjecture with different methods, such as checking an interval around zero, solidifies the confidence in the hypothesis.