/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 In each part determine whether t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 10

In each part determine whether the function is continuous or not, and explain your reasoning. (a) The Earth's population as a function of time (b) Your exact height as a function of time (c) The cost of a taxi ride in your city as a function of the distance traveled (d) The volume of a melting ice cube as a function of time

Short Answer

Expert verified
(a) Continuous; (b) Discontinuous; (c) Discontinuous; (d) Continuous.

Step by step solution

01

Understanding Continuity

A function is continuous at a point if there are no breaks, jumps, or gaps. Mathematically, for a function \( f(x) \) to be continuous at \( x = c \), it must satisfy three conditions: 1) \( f(c) \) is defined, 2) \( \lim_{{x \to c}} f(x) \) exists, and 3) \( \lim_{{x \to c}} f(x) = f(c) \).
02

Part (a): Earth's Population as a Function of Time

Earth's population increases with events such as births and decreases with deaths. Generally, it's modeled as a smooth, increasing function, barring sudden, large-scale events (e.g., pandemics). Normally, this makes it continuous over short periods, but could be discontinuous over centuries given events.
03

Part (b): Exact Height as a Function of Time

An individual's exact height changes over time in discrete steps, not a smooth process. Growth spurts happen at specific times, creating jumps in height, meaning the function is inherently discontinuous as it isn't smooth over small intervals.
04

Part (c): Taxi Ride Cost as Distance Function

Taxi fares often have a step-wise pattern: a base fare, plus fixed increments per distance unit. As a taxi travels, the cost increases at set intervals (e.g., every mile), creating jumps or steps in the cost function, making it discontinuous.
05

Part (d): Volume of a Melting Ice Cube as a Function of Time

The volume of a melting ice cube decreases smoothly with time, as melting is a continuous physical process. Assuming a consistent environment and no sudden removal or addition of ice, the function is continuous as there are no abrupt changes.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Limits in Calculus
Limits in calculus play a crucial role in understanding the behavior of functions as they approach certain points or infinity. The concept of limits helps in determining the continuity of a function. For a function \( f(x) \) to be continuous at a point \( x = c \), the limit of \( f(x) \) as \( x \) approaches \( c \) must equal the function's value at that point. This can be written mathematically as:\[\lim_{{x \to c}} f(x) = f(c).\]
  • If a limit exists and matches the function's value, we can confidently say the function is continuous at \( x = c \).
  • Conversely, the absence of such a limit usually indicates discontinuity.
Understanding limits allows us to analyze and define functions more precisely, especially where values might not be immediately apparent. They are foundational in calculus for solving real-world problems involving change and approximation.
Discontinuous Functions
Discontinuous functions are those where there are jumps, gaps, or breaks in the function's graph. This can occur for various reasons:
  • Sudden changes in value, like with taxi fares increasing per mile, illustrate discontinuous behavior because the function represents distinct, separate outcomes rather than a smooth progression.
  • In growth patterns, human height changes in distinct intervals rather than smoothly, because the body's growth isn't constant.
Discontinuous functions are common in models where conditions change abruptly or discretely. Recognizing where and why a function is discontinuous helps in accurately representing and understanding phenomena that don't follow a smooth path.
Continuous Functions
Continuous functions allow for smooth transitions without abrupt changes. For a function to be classified as continuous, it needs to meet specific criteria鈥攎ost importantly,\( f(c) \) must be defined, and the limit as \( x \) approaches \( c \) must exist and equal \( f(c) \).
  • A real-life example is the volume of a melting ice cube which reduces consistently without sudden drops or increases, demonstrating a continuous change over time.
  • Earth's population could generally be modeled as continuous over short periods assuming no significant, immediate impacts.
Understanding continuous functions is essential for constructing models that rely on natural processes or incremental changes, capturing a seamless flow in the behavior of variables over time.
Mathematical Modeling
Mathematical modeling involves creating and working with mathematical representations of real-world scenarios to predict, analyze, or understand complex systems. This often involves determining whether a function is continuous or discontinuous to accurately reflect reality.
  • For example, modeling the costs of services like a taxi ride requires understanding the step-wise variations in pricing, revealing discontinuity.
  • Alternatively, processes like the melting of ice or growth patterns can often be modeled continuously, assuming a steady progression without interruptions.
Modeling is essential in virtually all fields of science and engineering, offering a framework for testing hypotheses and making predictions. By assessing the continuity or discontinuity of functions involved, one can more accurately devise models to simulate and explain real-world phenomena.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Recall that unless stated otherwise the variable \(x\) in trigonometric functions such as \(\sin x\) and \(\cos x\) are assumed to be in radian measure. The limits in Theorem 2.5 .3 are based on that assumption. Exercises 49 and 50 explore what happens to those limits if degree measure is used for \(x\). (a) Show that if \(x\) is in degrees, then $$\lim _{x \rightarrow 0} \frac{\sin x}{x}=\frac{\pi}{180}$$ (b) Confirm that the limit in part (a) is consistent with the results produced by your calculating utility by setting the utility to degree measure and calculating (sin \(x\) ) / \(x\) for some values of \(x\) that get closer and closer to \(0 .\)

(a) Find the largest open interval, centered at the origin on the \(x\) -axis, such that for each point \(x\) in the interval. other than the center, the values of \(f(x)=1 / x^{2}\) are greater than 100 (b) Find the largest open interval, centered at the point \(x=1,\) such that for each point \(x\) in the interval, other than the center, the values of the function \(f(x)=1 /|x-1|\) are greater than 1000 (c) Find the largest open interval. centered at the point \(x=3,\) such that for each point \(x\) in the interval, other than the center, the values of the function \(f(x)=-1 /(x-3)^{2}\) are less than -1000 (d) Find the largest open interval, centered at the origin on the \(x\) -axis, such that for each point \(x\) in the interval. other than the center, the values of \(f(x)=-1 / x^{4}\) are less than -10.000

(a) Use the Intermediate-Value Theorem to show that the equation \(x=\cos x\) has at least one solution in the interval \([0 . \pi / 2]\) (b) Show graphically that there is exactly one solution in the interval. (c) Approximate the solution to three decimal places.

A positive number \(\epsilon\) and the limit \(L\) of a function \(f\) at \(+\infty\) are given. Find a positive number \(N\) such that \(|f(x)-L|<\epsilon\) if \(x>N\) $$\lim _{x \rightarrow+\infty} \frac{4 x-1}{2 x+5}=2 ; \epsilon=0.1$$

Find the limits. $$\lim _{x \rightarrow 2} \frac{x^{2}-4 x+4}{x^{2}+x-6}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.