Solving equations with derivatives

In summary, the conversation discusses two statements that need to be proved or disproved. The first statement states that if a and b are real constants and 0<a<1, then the equation x - a*cosx = b has only one solution. The second statement states that between every two solutions to arctanx = sinx, there is at least one solution to 1 - cosx = x^2 cosx. The conversation provides explanations and proofs for both statements, concluding that both statements are true.
  • #1
daniel_i_l
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Homework Statement


Prove or disprove:
1)If a and b are real constants and 0<a<1 then the equation x - a*cosx = b has only one solution.
2)Between every two solutions to arctanx = sinx there's atleast one solution to 1 - cosx = x^2 cosx


Homework Equations





The Attempt at a Solution



1)True: The limit of x - a*cosx at +infinity is +infinity and at -infinity is
-infinity. Also, the derivative is 1+a*sinx > 0 for all x and so it's an injective continues function. Which means that there's one and only one solution for every b. (one solution because of the mean value theorem and only one cause it's injective)

2)True: The derivative of f(x) = arctanx - sinx is 1/(1+x^2) - cos x =
(1 - cos x - (x^2)cosx)/(1+x^2). So if for x_1 and x_2 f(x)=0 then there's some x_1<x_3<x_2 where f'(x) = 0. And since (1+x^2) =/= 0 then
1 - cos x - (x^2)cosx which means that
1 - cos x = (x^2)cosx
 
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  • #2
Yeah, all seems ssssooo good to me. They are both perfect. ^.^ Well done, daniel_i_l. :)
 
  • #3
Thanks for your reply.
 

1. What is the purpose of solving equations with derivatives?

The purpose of solving equations with derivatives is to find the rate of change or slope of a function at a specific point. This can help in understanding the behavior of a function and how it changes over time.

2. How do I solve an equation with derivatives?

To solve an equation with derivatives, you will need to use the rules of differentiation to find the derivative of the function. Then, set the derivative equal to 0 and solve for the variable. This will give you the critical points of the function, which can help in finding the minimum or maximum points.

3. Can I use derivatives to find the minimum or maximum points of a function?

Yes, derivatives can be used to find the minimum or maximum points of a function. This is done by finding the critical points of the function, which are the points where the derivative is equal to 0. By analyzing the behavior of the function at these points, you can determine if it is a minimum or maximum point.

4. What are some common applications of solving equations with derivatives?

Solving equations with derivatives has many practical applications in fields such as physics, engineering, economics, and finance. It can be used to determine the velocity and acceleration of a moving object, optimize production processes, and analyze financial data, among many other uses.

5. Are there any limitations to solving equations with derivatives?

While solving equations with derivatives can provide valuable information about the behavior of a function, it does have some limitations. For example, it may not be able to accurately predict the behavior of a function in complex or non-linear systems. Additionally, it may not always provide a complete picture, as it only considers the behavior of the function at a specific point and not the overall behavior.

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