Q. 97 Use the definition of two-sided ... [FREE SOLUTION]

91影视

Chapter 2: Q. 97 (page 186)

Use the definition of two-sided and one-sided derivatives, together with properties of limits, to prove that $f' (c)$ exists if and only if $f'_(c)$ and $f'_{+} (c)$ exist and are equal.

Short Answer

Expert verified

$f_{鈭�}^{鈥�} (c) and f_{+}^{鈥�} (c) are exist and they are equal. Thus f^{鈥�} (c) exist.$

Step by step solution

Step1. Given Information

Consider a function

$f (x)$

$Here the objective is to show that f^{鈥�} (c) exist if and only if f_{鈭�}^{鈥�} (c) and f_{+}^{鈥�} (c) exist and are$ equal.

Step2. Left Continuity

Now,

$\begin{aligned} lim_{x 鈫� c^{鈭�}} 鈥� \frac{f (x) 鈭� f (c)}{x 鈭� c} = f_{鈭�}^{鈥�} (c) \\ \frac{lim_{x 鈫� c^{鈭�}} 鈥� f (x) 鈭� f (c)}{lim_{x 鈫� c^{鈭�}} 鈥� x 鈭� c} = f_{鈭�}^{鈥�} (c) \\ lim_{x 鈫� c^{鈭�}} 鈥� f (x) 鈭� f (c) = f_{鈭�}^{鈥�} (c) [lim_{x 鈫� c^{鈭�}} 鈥� x 鈭� c] \\ lim_{x 鈫� c^{鈭�}} 鈥� f (x) 鈭� lim_{x 鈫� c^{鈭�}} 鈥� f (c) = 0 \\ lim_{x 鈫� c^{鈭�}} 鈥� f (x) = f (c) \end{aligned}$

Therefore the function is left continuous at $x = c$ .

Step3. Right Continuity

Again,

$\begin{aligned} lim_{x 鈫� c^{+}} 鈥� \frac{f (x) 鈭� f (c)}{x 鈭� c} = f_{+}^{鈥�} (c) \\ \frac{lim_{x 鈫� c^{+}} 鈥� f (x) 鈭� f (c)}{lim_{x 鈫� c^{+}} 鈥� x 鈭� c} = f_{+}^{鈥�} (c) \\ lim_{x 鈫� c^{+}} 鈥� f (x) 鈭� f (c) = f_{+}^{鈥�} (c) [lim_{x 鈫� c^{+}} 鈥� x 鈭� c] \\ lim_{x 鈫� c^{+}} 鈥� f (x) 鈭� lim_{x 鈫� c^{+}} 鈥� f (c) = 0 \\ lim_{x 鈫� c^{+}} 鈥� f (x) = f (c) \end{aligned}$

Therefore the function is right continuous at $x = c$ .

$Therefore f_{鈭�}^{鈥�} (c) and f_{+}^{鈥�} (c) are exist and they are equal. Thus f^{鈥�} (c) exist.$

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 the definition of two-sided and one-sided derivatives, together with properties of limits, to prove that $f' (c)$ exists if and only if $f'_(c)$ and $f'_{+} (c)$ exist and are equal.

Short Answer

Step by step solution

Step1. Given Information

Step2. Left Continuity

Step3. Right Continuity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

91影视

Use the definition of two-sided and one-sided derivatives, together with properties of limits, to prove that f'(c)exists if and only if f'_(c) and f'+(c) exist and are equal.

Short Answer

Step by step solution

Step1. Given Information

Step2. Left Continuity

Step3. Right Continuity

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Use the definition of two-sided and one-sided derivatives, together with properties of limits, to prove that $f' (c)$ exists if and only if $f'_(c)$ and $f'_{+} (c)$ exist and are equal.