Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0…

Question

Chapter 2.3, Problem 47E

To determine

To find: The value of a such that the function f(x)={x2−1, x<32ax, x≥3 is continuous.

Expert Solution & Answer

Answer to Problem 47E

The value of a is 43 such that the function f(x)={x2−1, x<32ax, x≥3 is continuous.

Explanation of Solution

Given information: The function f(x)={x2−1, x<32ax, x≥3 is continuous.

Calculation:

It is given that the function f(x)={x2−1, x<32ax, x≥3 is continuous. So, it is continuous at all the points of x.

The function is also continuous at x=3 . So, limx→3f(x)=f(3) . The function f(x)=x2−1 corresponds to all the values of x<3.

Calculate the left hand limit of function at x=3.

limx→3−f(x)=limx→3−(x2−1)=32−1=9−1=8

The function f(x)=2ax corresponds to all the values of x≥3.

Calculate the right hand limit of function at x=3.

limx→3+f(x)=limx→3+(2ax)=2alimx→3+(x)=2a×3=6a

It is given that the function is continuous at x=3 . So, left hand limit will be equal to right hand limit of the function as:

limx→3−f(x)=limx→3+f(x)8=6aa=86a=43

Therefore, the value of a is 43 such that the function f(x)={x2−1, x<32ax, x≥3 is continuous.

Chapter 2.3, Problem 46E

Chapter 2.3, Problem 48E

Chapter 2 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 2.1 - Prob. 1QR Ch. 2.1 - Prob. 2QR Ch. 2.1 - Prob. 3QR Ch. 2.1 - Prob. 4QR Ch. 2.1 - Prob. 5QR Ch. 2.1 - Prob. 6QR Ch. 2.1 - Prob. 7QR Ch. 2.1 - Prob. 8QR Ch. 2.1 - Prob. 9QR Ch. 2.1 - Prob. 10QR

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