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 13 and \(14,\) find the slope of the curve at the indicated point. $$f(x)=|x| \quad\( at \)\quad\( (a) \)x=2 \quad\( (b) \)x=-3$$

Short Answer

Expert verified
The slope of the function \( f(x) = |x| \) at point \( x = 2 \) is \( 1 \) and at point \( x = -3 \) is \( -1 \).

Step by step solution

01

Understanding \( f(x) = |x| \)

The function \( f(x) = |x| \) can be decomposed into two cases: \( f(x) = x \) when \( x \geq 0 \), and \( f(x) = -x \) when \( x < 0 \) because the absolute value of a number is its non-negative value.
02

Take derivative of \( f(x) = |x| \)

The derivative of the function, \( f'(x) \), will also have two cases: \( f'(x) = 1 \) when \( x > 0 \) and \( f'(x) = -1 \) when \( x < 0 \). The derivative does not exist when \( x = 0 \) as the function is not differentiable at this point.
03

Find slope at \( x = 2 \)

To find the slope when \( x = 2 \), substitute \( 2 \) into the equation of \( f'(x) \). Because \( 2 > 0 \), the case of \( f'(x) = 1 \) applies here. So, the slope at \( x = 2 \) is \( 1 \).
04

Find slope at \( x = -3 \)

To find the slope when \( x = -3 \), substitute \( -3 \) into the equation of \( f'(x) \). Here, \( -3 < 0 \), so the case of \( f'(x) = -1 \) applies. Therefore, the slope at \( x = -3 \) is \( -1 \).

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

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