Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

In Exercises 69-80, determine whether or not fis continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

f(x)=x+4,ifx<23x,ifx2,x=2

Short Answer

Expert verified

The function is continuous atx=2but not differentiable

Step by step solution

01

Step 1. Given information

Given functionf(x)=x+4,ifx<23x,ifx2,x=2

02

Calculate the LHL and RHL and calculate

Calculating, we get

limx2-f(x)=limx2-(x+4)limx2-f(x)=2+4=6limx2+f(x)=limx2+3x=3×2=6f(2)=3xx=2=3×2=6

So the function is continuous atx=2

03

Checking for differentiability

Checking, we get

limx2-f(x)-f(2)x-2=limx2-x+4-(2+4)x-2limx2-f(x)-f(2)x-2=limx2-x+4-6x-2=limx2-x-2x-2=limx2-1=1limx2+f(x)-f(2)x-2=limx2+3x-6x-2limx2+f(x)-f(2)x-2=limx2+3(x-2)x-2=limx2+3=3

So the function is not differentiable atx=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 Vaia!

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

Stuart left his house at noon and walked north on Pine Street for 20minutes. At that point he realized he was late for an appointment at the dentist, whose office was located south of Stuart’s house on Pine Street; fearing he would be late, Stuart sprinted south on Pine Street, past his house, and on to the dentist’s office. When he got there, he found the office closed for lunch; he was 10minutes early for his 12:40appointment. Stuart waited at the office for 10minutes and then found out that his appointment was actually for the next day, so he walked back to his house. Sketch a graph that describes Stuart’s position over time. Then sketch a graph that describes Stuart’s velocity over time.

Each graph in Exercises 31–34 can be thought of as the associated slope function f' for some unknown function f. In each case sketch a possible graph of f.

For each function f(x)and interval localid="1648297458718" a,bin Exercises localid="1648297462718" 81-86, use the Intermediate Value Theorem to argue that the function must have at least one real root on localid="1648297466951" a,b. Then apply Newton’s method to approximate that root.

localid="1648297471865" f(x)=x3-3x+1,a,b=1,2

A bowling ball dropped from a height of 400feet will be s(t)=400-16t2feet from the ground after tseconds. Use a sequence of average velocities to estimate the instantaneous velocities described below:

After t=2 seconds, with h=0.1,h=0.01h=-0.1andh=-0.01

Find the derivative of the absolute value function and piecewise defined function

f(x)=3x+1ifx1x3ifx>1

See all solutions

Recommended explanations on Math Textbooks

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.

Sign-up for free