posted by jane .
i got answers for this one, but i feel like i did something wrong.
2x+1 when x is less than or equal to 2
(1/2)x^2 + k when x is greater than 2
1) what value of k will make f continuous at x = 2?
my answer: i got k = 3 because it would make the two parts of the function match up at the point (2,5).
2) using that value of k, determine whether f is differentiable at x = 2.
my answer: i said yes because the derivatives of each part of the function are equal.
3) if k = 4, is f differentiable at x = 2?
my answer: no because it is not continuous at x =2 when k = 4.
2) correct. At x=2, the 2x+1 curve has a slope of 2, and the (1/2)x^2 + 3 curve has a slope of x = x = 2 also. Therefore it is differentiable there.
3) correct also. You cannot differentiate at a discontinuity, even if the slopes of the lines on each side are the same.
thank you sososososo much! i was pretty sure of these, but i wanted to double-check.