Arctan absolute value problem, calculus 1

• timon
In summary, the author tries to solve the homework statement prove using the Mean Value Theorem and differentiating, but gets lost and is unsure if he has the right idea.
timon

Homework Statement

prove

$$|arctan(x)-arctan(y)| \leq |x-y|$$

The Attempt at a Solution

I tried separating cases:
1. both x and y are in the same quadrant;
2. x and y are not in the same quadrant;
3. x and y are zero (trivial);
4. either x or y is zero.

Now, i can see why each of these is true from drawing arctan (the asymptotic behavior and |arctan(x)| =< |x| being important), but I'm having trouble producing a mathematical argument.. I tried doing something with the derivative but i just get confused. Also, proving the equality |arctan(x)| =< |x| is the _next_ question, so i doubt I'm supposed to use it here.

Try applying the Mean Value Theorem to arctan(x).

i can get $$arctan(x) - arctan(y) \leq x-y$$, but how do i get that to say something about the absolute value?

is this correct?
$$x-y > 0 \Rightarrow x - y = |x-y|, arctan(x)-arctan(y) = |arctan(x)-arctan(y)|$$
$$\Rightarrow |x-y| \geq |arctan(x)-arctan(y)|$$

$$x-y < 0 \Rightarrow x - y = -|x-y|, arctan(x)-arctan(y) = -|arctan(x)-arctan(y)|$$
$$\Rightarrow -|x-y| \geq -|arctan(x)-arctan(y)| \rightarrow |x-y| \geq |arctan(x)-arctan(y)|$$

$$x-y=0 \Rightarrow x=y=0$$

Last edited:
timon said:
is this correct?
$$x-y > 0 \Rightarrow x - y = |x-y|, arctan(x)-arctan(y) = |arctan(x)-arctan(y)|$$
$$\Rightarrow |x-y| \geq |arctan(x)-arctan(y)|$$
True, but how do you know $$|x-y| \geq |arctan(x)-arctan(y)|$$? You have to justify it.

$$x-y < 0 \Rightarrow x - y = -|x-y|, arctan(x)-arctan(y) = -|arctan(x)-arctan(y)|$$
$$\Rightarrow -|x-y| \geq -|arctan(x)-arctan(y)| \rightarrow |x-y| \geq |arctan(x)-arctan(y)|$$
The step
$$\Rightarrow -|x-y| \geq -|arctan(x)-arctan(y)| \rightarrow |x-y| \geq |arctan(x)-arctan(y)|$$
is wrong.

$$x-y=0 \Rightarrow x=y=0$$
No, you can't conclude x=y=0.

Have you considered the original suggestion? What does the Mean Value Theorem tell you about arctan(x)? (I'm assuming you have studied the theorem.)

You have the right idea about breaking the proof into three cases, though: consider x > y, x < y, and x = y.

awkward said:
True, but how do you know $$|x-y| \geq |arctan(x)-arctan(y)|$$? You have to justify it.
i used the mean value theorem, as you suggested:

if

$$f(x) = arctan(x)$$

then

$$\exists$$ $$c$$

such that

$$f(c)' = \frac{f(y)-f(x)}{y-x}$$

$$\Leftrightarrow f(c)'(y-x)=f(y)-f(x)$$

for any $$x, y \in \Re$$ where x < y, since arctan is continuous and differentiable everywhere. also,

$$f(x)'= \frac{d}{dx}[arctan(x)] = \frac{1}{1+x^2} \leq 1$$

thus

$$y-x \geq arctan(y)-arctan(x).$$

the same process for x > y gives

$$x-y \geq arctan(x)-arctan(y).$$

I think this should give me the wanted result, using the method i used for the first case in my previous post, and realizing $$|x-y| = |y-x|$$. I think i didn't split the cases soon enough when i did it before. I find all this splitting of cases rather confusing..
awkward said:
No, you can't conclude x=y=0.
Why not? could there be any other place where x=y?

Last edited:
You are OK up to the point x=y=0.

Can you think of a number other than 0?

o wait i see what I've been doing. x and y are both on the x axis, so x=y doesn't imply x=y=0, in fact x and y could be any real number.
but the equality is still true, and trivially so. I guess I'm done? if so, thanks a lot for the assistance!

Last edited:
Right!

1. What is the Arctan absolute value problem in calculus 1?

The Arctan absolute value problem in calculus 1 involves finding the inverse tangent of a value with a given absolute value. This problem is used to solve for unknown angles in right triangles.

2. How do you solve the Arctan absolute value problem?

To solve the Arctan absolute value problem, you can use the inverse tangent function on a calculator or manually using the formula arctan(y/x). It is important to note the quadrant of the angle to determine the correct solution.

3. What are the key concepts to understand for the Arctan absolute value problem?

The key concepts for the Arctan absolute value problem include understanding the concept of inverse trigonometric functions, knowing how to use the inverse tangent function, and being able to identify the quadrant of the angle to determine the correct solution.

4. What are some common mistakes when solving the Arctan absolute value problem?

Some common mistakes when solving the Arctan absolute value problem include forgetting to consider the quadrant of the angle, using the inverse tangent function incorrectly, and not simplifying the final answer.

5. How is the Arctan absolute value problem used in real-life applications?

The Arctan absolute value problem is commonly used in real-life applications involving angles and right triangles, such as in engineering, architecture, and navigation. It can also be used to solve for unknown angles in trigonometric equations and to find the slope of a line.

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