Answered

Discover new knowledge and insights with IDNLearn.com's extensive Q&A platform. Get accurate and detailed answers to your questions from our knowledgeable and dedicated community members.

Suppose that [tex]\( f \)[/tex] is a function given as [tex]\( f(x)=\sqrt{3x+1} \)[/tex].

a. Write and simplify the difference quotient, [tex]\(\frac{f(x+h)-f(x)}{h}\)[/tex]. Note that you may need to "rationalize the numerator" as part of the simplification process.

[tex]\[
\frac{f(x+h)-f(x)}{h}=
\][/tex]
[tex]\(\square\)[/tex]

b. The derivative of the function, [tex]\( f'(x) \)[/tex], is the limit of the difference quotient as [tex]\( h \)[/tex] approaches zero.

[tex]\[
f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}=
\][/tex]
[tex]\(\square\)[/tex]

Sagot :

GinnyAnsweridn
Let's analyze and solve the given problem step-by-step.

Given the function [tex]\( f(x) = \sqrt{3x + 1} \)[/tex]:

### Part (a)

We need to write and simplify the difference quotient for the given function.

The difference quotient is given by:

[tex]\[ \frac{f(x + h) - f(x)}{h} \][/tex]

Substituting [tex]\( f(x) = \sqrt{3x + 1} \)[/tex] and [tex]\( f(x + h) = \sqrt{3(x + h) + 1} \)[/tex] into the difference quotient, we get:

[tex]\[ \frac{\sqrt{3(x + h) + 1} - \sqrt{3x + 1}}{h} \][/tex]

To simplify this expression, we need to rationalize the numerator:

[tex]\[ \frac{\sqrt{3(x + h) + 1} - \sqrt{3x + 1}}{h} \times \frac{\sqrt{3(x + h) + 1} + \sqrt{3x + 1}}{\sqrt{3(x + h) + 1} + \sqrt{3x + 1}} \][/tex]

This multiplication gives us:

[tex]\[ \frac{\left(\sqrt{3(x + h) + 1}\right)^2 - \left(\sqrt{3x + 1}\right)^2}{h \left(\sqrt{3(x + h) + 1} + \sqrt{3x + 1}\right)} \][/tex]

Simplifying the numerators:

[tex]\[ \left(\sqrt{3(x + h) + 1}\right)^2 = 3(x + h) + 1 \][/tex]

[tex]\[ \left(\sqrt{3x + 1}\right)^2 = 3x + 1 \][/tex]

Subtracting these, we get:

[tex]\[ 3(x + h) + 1 - (3x + 1) = 3x + 3h + 1 - 3x - 1 = 3h \][/tex]

Thus, the simplified difference quotient becomes:

[tex]\[ \frac{3h}{h \left(\sqrt{3(x + h) + 1} + \sqrt{3x + 1}\right)} = \frac{3}{\sqrt{3(x + h) + 1} + \sqrt{3x + 1}} \][/tex]

So, the simplified difference quotient is:

[tex]\[ \frac{f(x+h)-f(x)}{h}=\frac{3}{\sqrt{3(x + h) + 1} + \sqrt{3x + 1}} \][/tex]

### Part (c)

The derivative [tex]\( f'(x) \)[/tex] is the limit of the difference quotient as [tex]\( h \)[/tex] approaches 0:

[tex]\[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \][/tex]

Using the difference quotient we have simplified, we get:

[tex]\[ f'(x) = \lim_{h \to 0} \frac{3}{\sqrt{3(x + h) + 1} + \sqrt{3x + 1}} \][/tex]

As [tex]\( h \)[/tex] approaches 0, [tex]\( \sqrt{3(x + h) + 1} \)[/tex] approaches [tex]\( \sqrt{3x + 1} \)[/tex]. Therefore, the expression becomes:

[tex]\[ f'(x) = \frac{3}{\sqrt{3x + 1} + \sqrt{3x + 1}} = \frac{3}{2 \sqrt{3x + 1}} \][/tex]

Thus, the derivative of the function is:

[tex]\[ f'(x)= \frac{3}{2\sqrt{3x + 1}} \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.