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

• Sep 20th 2009, 02:58 PM
s3a
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!
• Sep 20th 2009, 03:04 PM
mr fantastic
Quote:

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 $f(x) = \cos x - x^3$. Note that $f(0) > 0$ and $f(\pi) < 0$. 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.
• Sep 20th 2009, 05:23 PM
s3a

f(0) < 0 < f(0.85) ?
• Sep 20th 2009, 05:34 PM
skeeter
Quote:

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!

$[.86,.87]$
• Sep 20th 2009, 05:49 PM
s3a
Do they expect us to plug in a million numbers manually until we find that restriction??
• Sep 20th 2009, 05:56 PM
skeeter
Quote:

Originally Posted by s3a
Do they expect us to plug in a million numbers manually until we find that restriction??

Quote:

Use your calculator to find an interval of length 0.01 that contains a root.
$\cos{x} = x^3$ at $x = 0.86547403...$

what's an interval of length 0.01 that includes that solution?
• Sep 20th 2009, 06:06 PM
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?
• Sep 20th 2009, 06:12 PM
skeeter
Quote:

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 $y = \cos{x} - x^3$ and looked for the zero.
• Sep 20th 2009, 06:15 PM
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.
• Sep 20th 2009, 06:25 PM
skeeter
Quote:

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?