Chapter 2: Q. 23 (page 184)

Use (a) the $h \to 0$ definition of the derivative and then
(b) the $z \to c$ definition of the derivative to find $f' (c)$ for each function f and value $x = c$ in Exercises 23–38.
23. $f (x) = x^{2}, x = - 3$

Short Answer

Expert verified

$f' (- 3) = - 6$ .

Step by step solution

Part (a) Step 1. Given information.

A function is given as $f (x) = x^{2}$ and $x = c = - 3$ .

Part (a) Step 2. Find f'(c) using h→0 definition of the derivative.

We have

$f' (- 3) = \lim_{h \to 0} \frac{f (- 3 + h) - f (- 3)}{h} = \lim_{h \to 0} \frac{{(- 3 + h)}^{2} - {(- 3)}^{2}}{h} = \lim_{h \to 0} \frac{9 - 6 h + h^{2} - 9}{h} = \lim_{h \to 0} \frac{- 6 h + h^{2}}{h} = \lim_{h \to 0} \frac{h (h - 6)}{h} = \lim_{h \to 0} (h - 6) = 0 - 6 = - 6$

Part (b) Step 2. Find f'(c) using z→(-3) definition of the derivative.

We have

$f' (c) = \lim_{z \to c} \frac{f (z) - f (c)}{z - c} f' (- 3) = \lim_{z \to - 3} \frac{z^{2} - {(- 3)}^{2}}{z - (- 3)} = \lim_{z \to - 3} \frac{z^{2} - 9}{z + 3} = \lim_{z \to - 3} \frac{(z - 3) (z + 3)}{(z + 3)} = \lim_{z \to - 3} (z - 3) = [(- 3) - 3] = - 6$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use (a) the $h \to 0$ definition of the derivative and then
(b) the $z \to c$ definition of the derivative to find $f' (c)$ for each function f and value $x = c$ in Exercises 23–38.
23. $f (x) = x^{2}, x = - 3$

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Find f'(c) using h→0 definition of the derivative.

Part (b) Step 2. Find f'(c) using z→(-3) definition of the derivative.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use (a) the h→0definition of the derivative and then(b) the z→cdefinition of the derivative to find f'(c)for each function f and value x=c in Exercises 23–38.23.f(x)=x2,x=-3

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Find f'(c) using h→0 definition of the derivative.

Part (b) Step 2. Find f'(c) using z→(-3) definition of the derivative.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use (a) the $h \to 0$ definition of the derivative and then
(b) the $z \to c$ definition of the derivative to find $f' (c)$ for each function f and value $x = c$ in Exercises 23–38.
23. $f (x) = x^{2}, x = - 3$