Chapter 2: Q. 7 (page 183)

The function $f (x) = 4 x^{3} - 5 x + 1$ is both continuous and differentiable at $x = 2$ . Write these facts as limit statements.

Short Answer

Expert verified

The function is continuous and differentiable at $x = 2$ .

Step by step solution

Step 1. Given information

We have to prove that the function $f (x) = 4 x^{3} - 5 x + 1$ is both continuous and differentiable at $x = 2$ .

Step 2. Check the function is continuous

Left hand limit,

$\lim_{x \to 2^{-}} 4 x^{3} - 5 x + 1 = 4 {(2)}^{3} - 5 (2) + 1 = 23$

Right hand limit,

$\lim_{x \to 2^{+}} 4 x^{3} - 5 x + 1 = 4 {(2)}^{3} - 5 (2) + 1 = 23$

At point $x = 2$ , the value of function is

$f (2) = 4 {(2)}^{3} - 5 (2) + 1 = 23$

Therefore, the function is continuous at $x = 2$ .

Step 3. Check the function is differentiable.

Left hand derivative,

$\lim_{h \to 0^{-}} \frac{f (2 + h) - f (2)}{h} = \lim_{h \to 0^{-}} \frac{4 {(2 + h)}^{3} - 5 (2 + h) + 1 - [4 {(2)}^{3} - 5 (2) + 1]}{h} = \lim_{h \to 0^{-}} \frac{48 h + 24 h^{2} + 4 h^{3} - 5 h}{h} = \lim_{h \to 0^{-}} 43 + 24 h + 4 h^{2} = 43$

Right hand derivative,

$\lim_{h \to 0^{+}} \frac{f (2 + h) - f (2)}{h} = \lim_{h \to 0^{+}} \frac{4 {(2 + h)}^{3} - 5 (2 + h) + 1 - [4 {(2)}^{3} - 5 (2) + 1]}{h} = \lim_{h \to 0^{+}} \frac{48 h + 24 h^{2} + 4 h^{3} - 5 h}{h} = \lim_{h \to 0^{+}} 43 + 24 h + 4 h^{2} = 43$

Therefore, the function is differentiable at $x = 2$ .

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.

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.

The function $f (x) = 4 x^{3} - 5 x + 1$ is both continuous and differentiable at $x = 2$ . Write these facts as limit statements.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Check the function is continuous

Step 3. Check the function is differentiable.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Calculus

Geometry

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

The function f(x)=4x3-5x+1is both continuous and differentiable at x=2. Write these facts as limit statements.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Check the function is continuous

Step 3. Check the function is differentiable.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Calculus

Geometry

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

The function $f (x) = 4 x^{3} - 5 x + 1$ is both continuous and differentiable at $x = 2$ . Write these facts as limit statements.