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Chapter 10: Problem 44

Describe the method of evaluating limits graphically. Give at least one disadvantage of this method.

Short Answer

Expert verified
To evaluate limits graphically, one must first understand that the limit of a function is the value it approaches as the input value nears a chosen point. For example, consider the function \(f(x) = x^2\). By plotting this function and observing its behavior at the point \(x = 2\), it becomes clear that as x approaches 2, the function's curve gets closer and closer to the value of 4. Thus, the limit of this function as x goes to 2 is 4, written as \(\lim_{x \to 2} x^2 = 4\). However, one significant disadvantage of this method is its potential inaccuracy due to visual approximation, particularly for functions with more complex or unclear behavior near the point of interest.

Step by step solution

01

Understand the concept of limit graphically

The limit of a function is the value the function approaches as the input value approaches a given point. Graphically, we can visualize this by looking at the behavior of the function's curve near that particular point.
02

Choose an example function

For the purpose of explaining the method, let's choose a simple example: \(f(x) = x^2\). We will try to find the limit of this function as x approaches 2.
03

Plot the function

First, plot the given function, \(f(x) = x^2\), on the graph. This will help visualize the behavior of the function around the point of interest.
04

Identify the limit

Now, to find the limit of the function as x approaches 2, we will observe the behavior of the function's curve around the point \(x = 2\). We can see that the function's curve gets closer and closer to 4 (the value of the function at \(x = 2\)), which indicates that the limit of the function as x goes to 2 is 4. So we would write it as: \(\lim_{x \to 2} x^2 = 4\)
05

Mention a disadvantage

One major disadvantage of evaluating limits graphically is the inaccuracy of the results, particularly for functions with complex or unclear behavior at the point of interest. The graphical method relies on visual approximation, which can lead to errors or misinterpretations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits of a Function
When we talk about the limit of a function, we're referring to the value that a function approaches as its input gets closer to a specific point. Think of it like a game of 'hot and cold' where 'hot' is the function's value nearing the target outcome. In mathematics, expressing this notion can look something like \( \lim_{x \to a} f(x) = L \) where 'L' is the limit value as 'x' approaches 'a'.

To grasp this better, imagine you're on a hike, walking towards a mountain peak (the limit). Even if the path doesn't take you to the very top, you can get close enough to touch the sky (the function getting close to the limit value). The path represents the function's curve, and the closer you get to the peak, the clearer you see what the summit looks like.
Visualizing Functions
Visualizing a function means plotting its graph on coordinate axes. This visual representation can reveal a lot about the function's personality鈥攚here it lives, how it behaves during change, and its defining features, like peaks and valleys. In our hiking analogy, visualizing functions is like following the trail on a map before setting foot on the actual path.

When plotting the function \(f(x) = x^2\), the resulting parabola is like a smooth road bending upwards in a predictable pattern. By examining this visual curve as 'x' nears 2, we can see our 'mountain peak' lying at '4', indicating that \( \lim_{x \to 2} x^2 = 4 \). Thus, visually mapping a function is a powerful tool to predict where the function may lead you and to make educated guesses about its limits.
Graphical Approximation Disadvantages
However, relying only on what our eyes can see on a graph can sometimes be misleading, much like mistaking a mirage for a waterfall. Graphically finding limits can be particularly tricky if the function's behavior is more erratic than our serene quadratic function. Imagine trying to follow a twisting, turning trail with sudden drops and unfamiliar terrain鈥攖his is comparable to trying to determine the limit of a complex function visually.

The main pitfall here is the precision, or lack thereof. A graph can look different based on how it's drawn鈥攖hink of the scale, the viewing window, and even one's interpretive skills. This imprecision can lead to approximations that miss the mark, giving you a misguided sense of the landscape. Therefore, while a picture's worth a thousand words, a graph's worth should be understood with a grain of salt.

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Most popular questions from this chapter

In Exercises 39-44, find the equation of the tangent to the graph at the indicated point. HINT [Compute the derivative algebraically; then see Example \(2(\mathrm{~b})\) in Section \(10.5 .]\) $$ f(x)=x^{2}-3 ; a=2 $$

The price per barrel of crude oil in constant 2008 dollars can be approximated by $$ P(t)=0.45 t^{2}-12 t+105 \text { dollars } \quad(0 \leq t \leq 28) $$ where \(t\) is time in years since the start of \(1980.5^{5}\) a. Compute the average rate of change of \(P(t)\) over the interval \([0,28]\), and interpret your answer. HINT [See Section \(10.4\) Example 3.] b. Estimate the instantaneous rate of change of \(P(t)\) at \(t=0\), and interpret your answer. HINT [See Example 2(a).] c. The answers to part (a) and part (b) have opposite signs. What does this indicate about the price of oil?

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=\frac{-1}{x} $$

Company A's profits are given by \(P(0)=\$ 1\) million and \(P^{\prime}(0)=-\$ 1\) million/month. Company B's profits are given by \(P(0)=-\$ 1\) million and \(P^{\prime}(0)=\$ 1\) million/month. In which company would you rather invest? Why?

A few weeks into the deadly SARS (Severe Acute Respiratory Syndrome) epidemic in 2003, the number of reported cases can be approximated by $$ A(t)=1,804(1.04)^{t} \quad(0 \leq t \leq 30) $$ \(t\) days after April \(1,2003 .\) a. What, approximately, was the instantaneous rate of change of \(A(t)\) on April \(21(t=20) ?\) Interpret the result. b. Which of the following is true? During April, the instantaneous rate of change of the number of cases (A) increased (B) decreased (C) increased and then decreased (D) decreased and then increased
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