Question about expanding a function to first order

In summary: Interesting. This gives a formal proof that a linear function gives a perfect approximation, since f"(x)==0 for f(x)=mx+b.
  • #1
hgandh
27
2
If we have a function ##f(x+\Delta x)## where ##\Delta x << x##, is it valid to approximate this as:
$$f(x + \Delta x) \approx f(x) + f'(x)\Delta x$$
even if ##\Delta x## is not necessarily small? If not, what is the valid expansion to first order?
 
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  • #2
If ##\Delta x## is big, then ##f'(x)## and ##\dfrac{f(x+\Delta x)-f(x)}{\Delta x}## can be very different, regardless how small ##\Delta x## versus ##x## is. E.g. ##x## could be positive and ##\Delta x## a large negative number, i.e. the function values can be quite different.
 
  • #3
fresh_42 said:
If ##\Delta x## is big, then ##f'(x)## and ##\dfrac{f(x+\Delta x)-f(x)}{\Delta x}## can be very different, regardless how small ##\Delta x## versus ##x## is. E.g. ##x## could be positive and ##\Delta x## a large negative number, i.e. the function values can be quite different.
If I restrict it to positive values only, would this become valid then?
 
  • #4
hgandh said:
If I restrict it to positive values only, would this become valid then?
No. To use a first order approximation, ##\Delta x ## must be "small" in some sense.

You can try some examples, like ##f(x)=x^2##.
 
  • #5
PS another perhaps even more illustrative example would be ##\sin x##.
 
  • #6
Exceptions would be linear, constant functions where expansion would be valid everywhere.
 
  • #7
hgandh said:
If we have a function ##f(x+\Delta x)## where ##\Delta x << x##, is it valid to approximate this as:
$$f(x + \Delta x) \approx f(x) + f'(x)\Delta x$$
even if ##\Delta x## is not necessarily small? If not, what is the valid expansion to first order?

You can answer this yourself. Draw a curve, any curve. That's ##f(x)##. Draw the tangent line at any point ##x_0##. That's the line ##f(x_0) + f'(x_0) \Delta x##. How far is the line from the curve? Anywhere it's "close enough", whatever that means to you, is a place where the approximation is "good enough".

Intuitively you can see that if the curve bends rapidly away from the tangent line, then the error gets big quickly, whereas if the curve is relatively flat and stays close to the line, then the approximation is pretty good over a larger range.

Taylor's Theorem expresses that rigorously with the error term. If you stop at the first derivative, the amount of error depends on the second derivative, i.e. how fast the slope of the curve changes.
 
  • #8
RPinPA said:
You can answer this yourself. Draw a curve, any curve. That's ##f(x)##. Draw the tangent line at any point ##x_0##. That's the line ##f(x_0) + f'(x_0) \Delta x##. How far is the line from the curve? Anywhere it's "close enough", whatever that means to you, is a place where the approximation is "good enough".

Intuitively you can see that if the curve bends rapidly away from the tangent line, then the error gets big quickly, whereas if the curve is relatively flat and stays close to the line, then the approximation is pretty good over a larger range.

Taylor's Theorem expresses that rigorously with the error term. If you stop at the first derivative, the amount of error depends on the second derivative, i.e. how fast the slope of the curve changes.
Interesting. This gives a formal proof that a linear function gives a perfect approximation, since f"(x)==0 for f(x)=mx+b.
 

Related to Question about expanding a function to first order

1. What does it mean to expand a function to first order?

Expanding a function to first order means to approximate the function using a linear function. This involves finding the slope of the function at a specific point and using that slope to create a line that closely follows the behavior of the function near that point.

2. Why would you need to expand a function to first order?

Expanding a function to first order can be useful in situations where you need to estimate the behavior of a function near a specific point. It can also help simplify more complex functions and make them easier to analyze.

3. How do you expand a function to first order?

To expand a function to first order, you will need to use calculus and find the derivative of the function at the point of interest. This derivative will be the slope of the linear function that approximates the original function. You can then use this slope and the point of interest to create the linear function.

4. What is the difference between expanding a function to first order and higher orders?

Expanding a function to first order only involves creating a linear approximation of the function. Expanding to higher orders involves using more terms in the approximation, resulting in a more accurate representation of the original function.

5. Are there any limitations to expanding a function to first order?

Yes, expanding a function to first order is only an approximation and may not accurately represent the behavior of the function outside of the specific point of interest. It also assumes that the function is continuous and differentiable at that point.

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