Find the constant value of the difference

In summary, the two given functions, arctan(x-1) and 2arctan(x-1+sqrt((x-1)^2+1)), have the same derivative. To find the constant value at which they differ, you can evaluate the functions at a good value of x. By setting x=1, we can find that the constant is -pi/2. Additionally, the formula D_x arctan(x+sqrt(1+x^2)) = 1/(2(1+x^2)) can also be used to show their equality.
  • #1
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I know that the following two functions have the same derivative: ##\arctan (x-1)## and ##2 \arctan (x-1 + \sqrt{(x-1)^2+1})##. Out of curiosity, how can I find the constant value at which they differ? I tried to add ##\pi / 2## to arctan(x-1) but I'm not sure if that works or not...
 
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  • #2
Don't know if this works, but if they are equal as functions, they are equal in every point on the domain. So try to find a good x to evaluate the functions in.
 
  • #3
Math_QED said:
Don't know if this works, but if they are equal as functions, they are equal in every point on the domain. So try to find a good x to evaluate the functions in.
So in that case does ##\arctan (x-1) + \pi / 2 =2 \arctan (x-1 + \sqrt{(x-1)^2+1})##? I'm not sure how to check it.
 
  • #4
Let me be a little more specific. You know ##f'(x) = g'(x)##

Hence there exists a constant c such that ##f(x) = c + g(x)##

Now, you can start to find that constant by plugging in values of x.

In your example, put for example ##x = 1##.

Then ##0 = \arctan(0) = c + 2\arctan(1)##

Hence, ##c = -2 \arctan(1) = -2 \pi/4 = - \pi/2##

and indeed, ##\pi/2## works if you add it on the left.
 
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  • #5
The input [itex]x-1[/itex] is an unnecessary complication for the actual result, and we could also state that the formula

[tex]
D_x \arctan\big(x + \sqrt{1 + x^2}\big) = \frac{1}{2}\frac{1}{1 + x^2}
[/tex]

is true.
 

1. What is the constant value of the difference?

The constant value of the difference is the numerical value that remains the same when subtracting two numbers. It is the fixed amount by which one number is larger or smaller than the other.

2. How do you find the constant value of the difference?

To find the constant value of the difference, subtract the smaller number from the larger number. The result will be the constant value of the difference between the two numbers.

3. Why is it important to find the constant value of the difference?

Finding the constant value of the difference is important because it helps determine the relationship between two numbers. It also allows for easier comparison and analysis of data.

4. Can the constant value of the difference be negative?

Yes, the constant value of the difference can be negative. This indicates that the smaller number is actually larger than the larger number by the absolute value of the constant difference.

5. How is the constant value of the difference used in real-life situations?

The constant value of the difference is used in various fields such as mathematics, science, finance, and statistics. It can help calculate change, determine rates of change, and analyze data trends.

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