Q. 31 Use (a) the h鈫�0 definition of ... [FREE SOLUTION]

91影视

Chapter 2: Q. 31 (page 184)

Use (a) the h鈫�0 definition of the derivative and then (b) the z鈫抍 definition of the derivative to find f'(c) for each function f and value x = c.
$f (x) = \frac{x 鈭� 1}{x + 3}, x = 2$

Short Answer

Expert verified

(a) $f^{'} (c) = \frac{4}{25}$

(b) $f^{'} (c) = \frac{4}{25}$

Step by step solution

Part (a) Step 1. Given information.

Given function is $f (x) = \frac{x 鈭� 1}{x + 3}$

We have to find $f^{'} (c)$ at x=2

Part (a) Step 2. Find the f'(c)

We have to find the derivative of the function using h鈫�0 definition,

Therefore,

$\begin{aligned} lim_{h 鈫� 0} 鈥� \frac{f (2 + h) 鈭� f (2)}{h} = lim_{h 鈫� 0} 鈥� \frac{\frac{2 + h 鈭� 1}{2 + h + 3} 鈭� \frac{2 鈭� 1}{2 + 3}}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{\frac{1 + h}{5 + h} 鈭� \frac{1}{5}}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{\frac{5 (1 + h) 鈭� 1 (5 + h)}{5 (5 + h)}}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{4 h}{5 h (5 + h)} \\ = lim_{h 鈫� 0} 鈥� \frac{4}{5 (5 + h)} \\ = \frac{4}{25} \end{aligned}$

Part (b) Step 1. Find f'(c)

Find the derivate of the function using $x 鈫� 2$ definition,

$\begin{aligned} lim_{x 鈫� 2} 鈥� \frac{f (x) 鈭� f (2)}{x 鈭� 2} = lim_{x 鈫� 2} 鈥� \frac{\frac{x 鈭� 1}{x + 3} 鈭� \frac{2 鈭� 1}{2 + 3}}{x 鈭� 2} \\ = lim_{x 鈫� 2} 鈥� \frac{\frac{x 鈭� 1}{x + 3} 鈭� \frac{1}{5}}{x 鈭� 2} \\ = lim_{x 鈫� 2} 鈥� \frac{\frac{5 (x 鈭� 1) 鈭� (x + 3)}{5 (x + 3)}}{x 鈭� 2} \\ = lim_{x 鈫� 2} 鈥� \frac{4 x 鈭� 8}{5 (x 鈭� 2) (x + 3)} \\ = lim_{x 鈫� 2} 鈥� \frac{4 (x 鈭� 2)}{5 (x 鈭� 2) (x + 3)} \\ = lim_{x 鈫� 2} 鈥� \frac{4}{5 (x + 3)} \\ = \frac{4}{25} \end{aligned}$

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use (a) the h鈫�0 definition of the derivative and then (b) the z鈫抍 definition of the derivative to find f'(c) for each function f and value x = c.
$f (x) = \frac{x 鈭� 1}{x + 3}, x = 2$

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Find the f'(c)

Part (b) Step 1. Find f'(c)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

91影视

Use (a) the h鈫�0 definition of the derivative and then (b) the z鈫抍 definition of the derivative to find f'(c) for each function f and value x = c.f(x)=x鈭�1x+3,x=2

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Find the f'(c)

Part (b) Step 1. Find f'(c)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Use (a) the h鈫�0 definition of the derivative and then (b) the z鈫抍 definition of the derivative to find f'(c) for each function f and value x = c.
$f (x) = \frac{x 鈭� 1}{x + 3}, x = 2$