Q. 7 The function f(x)=4x3-5x+1聽is b... [FREE SOLUTION]

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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,

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

Right hand limit,

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

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,

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

Right hand derivative,

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

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

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91影视

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.

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91影视

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

Mechanics Maths

Decision Maths

Logic and Functions

Applied Mathematics

Pure Maths

Geometry

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.