× Didn't find what you were looking for? Ask a question
Top Posters
Since Sunday
5
o
5
4
m
4
b
4
x
4
a
4
l
4
t
4
S
4
m
3
s
3
New Topic  
bio_man bio_man
wrote...
Administrator
Valued Member
Educator
Posts: 33224
5 years ago
Set:

\(f\left(x\right)=\sqrt[3]{x^2}\)

Apply into limit expression, do not take the limit yet:

\(\lim _{x\rightarrow 0}\ \frac{\sqrt[3]{\left(x+\Delta x\right)^2}-\sqrt[3]{x^2}}{\Delta x}\)

Set a and b:

\(a=\sqrt[3]{\left(x+\Delta x\right)^2}\)

Solve for a3:

\(a^3=\left(x+\Delta x\right)^2\)

Next, we look at b:

\(b=\sqrt[3]{x^2}\)

Solve for b3:

\(b^3=x^2\)

Recall:

\(a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\)

Rearrange this formula:

\(\frac{a^3−b^3}{(a^2+ab+b^2)}=a−b\)

\(\frac{\left(x+\Delta x\right)^2−x^2}{\left(\sqrt[3]{\left(x+\Delta x\right)^4}\right)+\sqrt[3]{\left(x+\Delta x\right)^2x^2}+\sqrt[3]{x^4}}=a−b\)

We expand the numerator:

\(\left(x+\Delta x\right)^2−x^2=2\Delta x\cdot x+\Delta x^2\)

Common factor it further!

\(2\Delta x\cdot x+\Delta x^2=\Delta x\left(2x+\Delta x\right)\)

Apply into limit expression, do not take the limit yet:

\(\lim _{x\rightarrow 0}\frac{\Delta x\left(2x+\Delta x\right)}{\left[\left(\sqrt[3]{\left(x+\Delta x\right)^4}\right)+\sqrt[3]{\left(x+\Delta x\right)^2x^2}+\sqrt[3]{x^4}\right]\Delta x}\)

The Δx cancel each other out:

\(\lim _{x\rightarrow 0}\frac{2x+\Delta x}{\left[\left(\sqrt[3]{\left(x+\Delta x\right)^4}\right)+\sqrt[3]{\left(x+\Delta x\right)^2x^2}+\sqrt[3]{x^4}\right]}\)

Take the limit now:

\(=\frac{2x}{\left(\sqrt[3]{x^4}\right)+\sqrt[3]{x^4}+\sqrt[3]{x^4}}\)

\(=\frac{2x}{3\left(\sqrt[3]{x^4}\right)}\)

Technically you're done, but you can use the laws of exponents to go further:

\(=\frac{2x}{3\left(\sqrt[3]{x^4}\right)}\rightarrow \frac{2}{3}\cdot \frac{x}{x^{\frac{4}{3}}}\rightarrow \ \frac{2}{3}x^{1-\frac{4}{3}}\rightarrow \ \frac{2}{3}x^{-\frac{1}{3}}=\frac{2}{3\sqrt[3]{x}}\)

Final answer:

\(\frac{2}{3\sqrt[3]{x}}\)
Source  First principles, Calculus
Read 306 times
1 Reply

Related Topics

Replies
wrote...
Staff Member
4 years ago
👍
- Master of Science in Biology
- Bachelor of Science
New Topic      
Explore
Post your homework questions and get free online help from our incredible volunteers
  963 People Browsing
Related Images
  
 86
  
 213
  
 181
Your Opinion
What's your favorite funny biology word?
Votes: 156

Previous poll results: Do you believe in global warming?