# Thread: Prove that the equation has at least one real root IN AN INTERVAL OF 0.01

1. ## Prove that the equation has at least one real root IN AN INTERVAL OF 0.01

Question:

(a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that contains a root.

41. cos x = x^3

If I were given the interval, I would know what to do but how do I know which interval of length 0.01 I should use??

Any help would be greatly appreciated!

2. Originally Posted by s3a
Question:

(a) Prove that the equation has at least one real root.
(b) Use your calculator to find an interval of length 0.01 that contains a root.

41. cos x = x^3

If I were given the interval, I would know what to do but how do I know which interval of length 0.01 I should use??

Any help would be greatly appreciated!
Consider the function $\displaystyle f(x) = \cos x - x^3$. Note that $\displaystyle f(0) > 0$ and $\displaystyle f(\pi) < 0$. Therefore .... (why?)

Now start an iterative process of subdividing the above interval and use your calculator to test each subdivision until you get one that satisfies the stated requirements.

3. So is a good answer:

f(0) < 0 < f(0.85) ?

4. Originally Posted by s3a
question:

(a) prove that the equation has at least one real root.
(b) use your calculator to find an interval of length 0.01 that contains a root.

41. Cos x = x^3

if i were given the interval, i would know what to do but how do i know which interval of length 0.01 i should use??

Any help would be greatly appreciated!
$\displaystyle [.86,.87]$

5. Do they expect us to plug in a million numbers manually until we find that restriction??

6. Originally Posted by s3a
Do they expect us to plug in a million numbers manually until we find that restriction??
Use your calculator to find an interval of length 0.01 that contains a root.
$\displaystyle \cos{x} = x^3$ at $\displaystyle x = 0.86547403...$

what's an interval of length 0.01 that includes that solution?

7. Did you plug in y = 0 to find x = 0.8655 ? If so, could you please explain the steps involved in solving for x with that rule?

8. Originally Posted by s3a
Did you plug in y = 0 to find x = 0.8655 ? If so, could you please explain the steps involved in solving for x with that rule?
I graphed the equation $\displaystyle y = \cos{x} - x^3$ and looked for the zero.

9. Do you have some sort of trick to do that fast because the way I do it, it takes so much time I would not be able to answer this on a test in time.

10. Originally Posted by s3a
Do you have some sort of trick to do that fast because the way I do it, it takes so much time I would not be able to answer this on a test in time.
what kind of calculator are you using?

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### solving for at least one real root

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