Is the Mean Value Theorem Applicable to Prove sin x < x for x > 0?

In summary, the conversation discusses proving that sin x < x for all x > 0 using the mean value theorem. The person initially struggles with the problem but eventually looks at the solution in the back of the book and notices the significance of using 2pi instead of 1. They then question how this proves the general case in the given interval and ask for clarification on how to show that the slope of the tangent of sin(x) is less than one for 0 < x ≤ 1. They also suggest using the mean value theorem to determine what happens when x > 0 and x = sin(x).
  • #1
dontdisturbmycircles
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3

Homework Statement


Show that sin x < x for all x > 0


The Attempt at a Solution


I thought I was pretty good at calculus so I have kinda been shifting my calc class onto the bottom of my todo list, but this mean value theorem problem is giving me some problems.

For x > 1, sin x [tex]\leq[/tex] 1 < x

This was my start... after about 25 minutes of thinking about how the mean value theorem could be applied I looked in the back.. (with much reluctance, trust me). They chose 2pi instead of 1 for the first part, that is likely significant but I am not getting thepoint. They then proceed to prove that (sinx)/(x) = cos(c) for some c in (0,2pi) and I can see that at that point c sin x< x since cos(c)[tex]\leq 1[/tex] but ONLY at that point c... I don't see how it proves it for the general case in that interval...
 
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  • #2
Can you show that the slope of the tangent of [itex]\sin(x)[/itex] is less than one for [itex]0<x \leq 1[/itex]?

Then, if you assume that there is a point with [itex]x>0,x=\sin(x)[/itex] what happens when you apply the mean value theorem?
 

What is the Mean Value Theorem Problem?

The Mean Value Theorem Problem is a fundamental concept in calculus that states that for any continuous and differentiable function, there exists at least one point in the interval where the slope of the tangent line is equal to the average slope of the secant line.

Why is the Mean Value Theorem important?

The Mean Value Theorem is important because it allows us to make connections between the geometric and algebraic properties of a function. It also provides a way to find the exact value of a function at a specific point using the average rate of change over an interval.

Can you give an example of the Mean Value Theorem Problem?

For example, let's say we have a function f(x) = x^2 on the interval [0,3]. We can use the Mean Value Theorem to show that there exists at least one point c in the interval where the slope of the tangent line is equal to the average slope of the secant line.

How do you solve a Mean Value Theorem Problem?

To solve a Mean Value Theorem Problem, you first need to check if the function is continuous and differentiable on the given interval. Then, you can find the average slope of the secant line using the formula (f(b) - f(a))/(b-a), where a and b are the endpoints of the interval. Finally, you can find the derivative of the function and set it equal to the average slope to solve for the value of c.

What are the real-world applications of the Mean Value Theorem?

The Mean Value Theorem has many real-world applications, such as in economics, physics, and engineering. For example, it can be used to calculate average velocity and acceleration over a given time interval, or to find the optimal production level for a company. It is also used in optimization problems to find the maximum or minimum value of a function.

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