Question

Friends please if you can help me with this exercise, this came into my calculus test 1: parameters found for "w" and "n" the equation cos (wx) - x / n has a real solution and calculate how many second solutions has exactly ..... please still do not know how to solve it thanks

Answer 1

The directions are somewhat unclear. Are you quoting word for word?

Answer 2

Is this the equation :
\( \Large \cos (wx) - \frac xn = 0 \)

Answer 3

yes that is the equation

Answer 4

search parameters of "w" and "n" for the equation has a real solution

Answer 5

but also sought to find the exact number of solutions of the equation

Answer 6

That's harder :)
We can try using calculus

Answer 7

That confused me

Answer 8

This exercise was in my calculus test 1

Answer 9

this problem is within the subject of calculus 1

Answer 10

we can see that as n gets large
\(\Large \cos (wx) - \frac xn \to \cos(wx) \)

Answer 11

We could modify the equation
\(
\Large n\cdot \cos (wx) -x = 0
\)
The derivative is
\( \Large -n \sin(wx)w -1 = 0\)
However I don't see how this helps

Answer 12

Again just to be clear you want to know exactly how many real solutions are there to the equation:
$$\Large \cos (wx) - \frac xn = 0$$
Does n have to be an integer? Clearly n=0 won't work.

Answer 13

I don't see anything that pops out.
@Michele_Laino
@ParthKohli

Answer 14

They call these things transcendental equations, don't they? Graphing is the way to go about these.

Answer 15

yes . not an algebraic equation.
i dont see any other way than by checking for sign change, using intermediate value theorem . but this does not give you a nice result

Answer 16

theorem of Bolzano or intermediate value theorem that thought but how to use it with the values of "w" and "n"

Answer 17

but forget to mention
\[w >0,n \ge1\]

Answer 18

we can assume n are integers.
https://www.desmos.com/calculator/0pdovzb6kn

Answer 19

I think it has infinitely many solutions. As n goes larger, the continuity of the cosine (regarding W values) will ensure the existence of number k= cos (c) for some c in R such that k between (-1,1). Now we can vary n in such a way that x/n is between (-1,1) for any value of x (since n is independent).
That is to say, for any x in the domain, there exist some n such that their ratio is between (-1,1). on the other hand, we can change w as we want such that the equality still holds . in short, infinitely many solutions
That's just what I think