Question

Problem about derivatives
(pdf included)

Answer 1

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-4-limits-and-continuity/MIT18_01SCF10_ex04sol.pdf

Answer 2

I understood the first part. I also understand that the derivatives need to be the same and that 2cos2x = a in x=0, but I don't understand when it says:

"We conclude that f(x) is continuous but not differentiable for b = 0 and any
a ≤ 0. In fact, for any value of a other than 2 the function will be continuous
but not differentiable."
Why a<0, I just see a=0.

Answer 3

Becuase of this part:

"The function is differentiable if it has a well defined tangent line at x = 0. This will be true when the slope of the line y= ax equals the slope of the curve y = sin2x at (0,0)"

The next sentence tells you the slope of sin2x is positive. For the slope of y = ax to be positive as well, 'a' must be positive.

Answer 4

Ok, and why the "In fact, for any value of a other than 2 the function will be continuous
but not differentiable"? Because the slope of sin2x will be 2 at (0,0)?

Answer 5

yes. the only value that would make both their (theoretical) slopes equal at 0

Answer 6

Thanks for your help :)