/*! 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 87 Find all values of \(c\) such th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 87

Find all values of \(c\) such that \(f\) is continuous on \((-\infty, \infty)\). \(f(x)=\left\\{\begin{array}{ll}1-x^{2}, & x \leq c \\ x, & x>c\end{array}\right.\)

Short Answer

Expert verified
The values of \(c\) that make \(f(x)\) continuous over all real numbers are \(c = {-1+\sqrt{5}\over 2}\) and \(c = {-1-\sqrt{5}\over 2}\)

Step by step solution

01

Write down the function

The function \(f(x)\) is given as a piecewise function: \(f(x) = 1 - x^2\) for \(x \leq c\) and \(f(x) = x\) for \(x>c\)
02

Establish the condition for continuity at \(x=c\)

To ensure the function is continuous at \(x=c\), the function values should coincide at \(x=c\) from both sides. In mathematical expression, \(\lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x)\) which simplifies into \(1 - c^2 = c\)
03

Solve the equation

Solving the equation \(1 - c^2 = c\) we have to add \(c^2\) to both sides and subtract \(c\) on both sides which gives \(c^2 + c - 1 = 0\)
04

Solve for \(c\)

To solve for \(c\), use the quadratic formula, \(c={-b\pm\sqrt{b^2-4ac}\over 2a}\), where \(a=1\), \(b=1\), and \(c=-1\). Applying these values, we find \(c = {-1\pm\sqrt{(1)^2-4(1)(-1)}\over 2(1)} = {-1\pm\sqrt{5}\over 2}\)

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!

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

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}-6 x+8, \quad[0,3], \quad f(c)=0 $$

In your own words, describe the meaning of an infinite limit. Is \(\infty\) a real number?

Prove that if \(\lim _{x \rightarrow c} f(x)=0\) and \(|g(x)| \leq M\) for a fixed number \(M\) and all \(x \neq c,\) then \(\lim _{x \rightarrow c} f(x) g(x)=0\).

Rate of Change A 25 -foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate \(r\) of \(r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec}\) where \(x\) is the distance between the ladder base and the house. (a) Find \(r\) when \(x\) is 7 feet. (b) Find \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x \rightarrow 25^{-}\).

In your own words, explain the Squeeze Theorem.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

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.