How Accurate is Differential Approximation for Fourth Roots?

Karol
Messages
1,380
Reaction score
22

Homework Statement


Approximate ##~\sqrt[4]{17}~## by use of differential

Homework Equations


Differential: ##~dy=f(x)~dx##

The Attempt at a Solution


$$y=\sqrt[4]{x},~~dy=\frac{1}{4}x^{-3/4}=\frac{1}{4\sqrt[4]{x^3}}$$
$$\sqrt[4]{16}=2,~~dx=1,~~dy=\frac{1}{4\sqrt[4]{x^2}}\cdot 1=0.149$$
$$\sqrt[4]{17}=2.031$$
The error is too big
 
Physics news on Phys.org
Karol said:

Homework Statement


Approximate ##~\sqrt[4]{17}~## by use of differential

Homework Equations


Differential: ##~dy=f(x)~dx##

The Attempt at a Solution


$$y=\sqrt[4]{x},~~dy=\frac{1}{4}x^{-3/4}=\frac{1}{4\sqrt[4]{x^3}}$$
$$\sqrt[4]{16}=2,~~dx=1,~~dy=\frac{1}{4\sqrt[4]{x^2}}\cdot 1=0.149$$
$$\sqrt[4]{17}=2.031$$
The error is too big

Nevertheless, your result is correct.

If you want better accuracy you need to take additional, higher order terms (2nd derivatives, maybe 3rd derivatives,etc).
 
Thank you Ray
 
Karol said:

Homework Statement


Approximate ##~\sqrt[4]{17}~## by use of differential

Homework Equations


Differential: ##~dy=f(x)~dx##

The Attempt at a Solution


$$y=\sqrt[4]{x},~~dy=\frac{1}{4}x^{-3/4}=\frac{1}{4\sqrt[4]{x^3}}$$
$$\sqrt[4]{16}=2,~~dx=1,~~dy=\frac{1}{4\sqrt[4]{x^2}}\cdot 1=0.149$$
$$\sqrt[4]{17}=2.031$$
The error is too big
Here is how I would write things.
##f(x + \Delta x) \approx f(x) + df \approx f(x) + f'(x) \Delta x = x^{1/4} + \frac 1 4 x^{-3/4} \Delta x##
When ##x = 16## and ##\Delta x = 1##, we have
##\sqrt[4]{16 + 1} \approx 16^{1/4} + \frac 1 4 \frac 1 {16^{3/4}} \cdot 1 = 2 + \frac 1 32 = 2.03125##
By calculator, ##\sqrt[4]{17} \approx 2.03054##. Since ##\Delta x = 1## is relatively large in comparison to x = 17, the approximation using differentials isn't all that accurate. If ##\Delta x## were smaller, the approximation would be better.
 
Thank you Mark44
By the way, how do i copy your names here, i write them again. when i pause the mouse on your name it becomes a pointer and there is no option to copy
 
Karol said:
Thank you Mark44
By the way, how do i copy your names here, i write them again. when i pause the mouse on your name it becomes a pointer and there is no option to copy
Just hit the 'Reply' button on the lower right of the post. You may also hite the 'Quote' button , also on the lower right.
BTW, small mistake: dy=f'(x)dx , not dy=f(x)dx , unless f(x)=f'(x).
 
Thanks, but i mean i want to copy your name, WWGD, to here, instead of looking and typing it. i usually thank every one that answered my question
 
Karol said:
Thanks, but i mean i want to copy your name, WWGD, to here, instead of looking and typing it. i usually thank every one that answered my question
Maybe you can just use the 'Like' button as a means of thanking.
 
Back
Top