Differentiability

• Nov 4th 2007, 06:44 PM
Truthbetold
Differentiability
Find all values of x for which the function is differentiable.

\$\displaystyle P(x)= sin(|x|)-1\$
• Nov 4th 2007, 07:11 PM
topsquark
Quote:

Originally Posted by Truthbetold
Find all values of x for which the function is differentiable.

\$\displaystyle P(x)= sin(|x|)-1\$

Can you think of any points of the graph of \$\displaystyle y = |x|\$ where the function isn't differentiable?

-Dan
• Nov 4th 2007, 09:46 PM
Truthbetold
Quote:

Originally Posted by topsquark
Can you think of any points of the graph of \$\displaystyle y = |x|\$ where the function isn't differentiable?

-Dan

To shorten the below: no.

If something is not differentiable only when it at the very middle (where the right and left come together, usually in examples as (0,0,) of cusps, corners, vertical tangents, and a non-removable discontinuity, then I cannot think or see, using a graphing calculator, any points.
• Nov 5th 2007, 03:46 AM
topsquark
Quote:

Originally Posted by Truthbetold
To shorten the below: no.

If something is not differentiable only when it at the very middle (where the right and left come together, usually in examples as (0,0,) of cusps, corners, vertical tangents, and a non-removable discontinuity, then I cannot think or see, using a graphing calculator, any points.

\$\displaystyle f(x) = |x|\$

Consider x near 0. When x approaches 0 from the left, the first derivative is -1. When x approaches 0 from the right, the first derivative is 1. So the derivative of |x| does not exist at x = 0.

We can make a similar argument for \$\displaystyle f(x) = sin(|x|) - 1\$. See the graph below. Notice the cusp at x = 0.

-Dan