i got answers for this one, but i feel like i did something wrong.

f(x)=
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.

1) correct

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.

1) To find the value of k that will make f continuous at x = 2, you need to ensure that the two parts of the function (2x+1 and (1/2)x^2 + k) match up at x=2. In other words, the value of the function from both sides of x=2 should be the same.

To check this, we evaluate the function for x = 2 from the left side: f(2-) = 2(2) + 1 = 5
And from the right side: f(2+) = (1/2)(2)^2 + k = 1 + k

For the function to be continuous at x = 2, we need f(2-) = f(2+), which means 5 = 1 + k. Solving this equation, we find k = 4. So, the value of k that will make f continuous at x = 2 is k = 4.

2) After determining the value of k, you need to check whether f is differentiable at x = 2. Differentiability at a point requires two conditions:
a) The function must be continuous at that point.
b) The limit of the slopes (or derivatives) from both sides of the point should be the same.

Since we found k = 4 to make f continuous at x = 2, let's now check if f is differentiable at x = 2. To do this, we need to calculate the derivatives of each part of the function:

The derivative of the first part, 2x+1, is simply 2.
The derivative of the second part, (1/2)x^2 + 4, is x.

Therefore, the derivative from the left side at x = 2 (f'(2-)) is 2, and from the right side at x = 2 (f'(2+)) is 2. Since f'(2-) = f'(2+), f is differentiable at x = 2 using the value k = 4.

3) Now, let's consider the case where k = 4 and check if f is differentiable at x = 2. You already correctly determined that the function is not continuous at x = 2 when k = 4.

Differentiability requires the function to be continuous, so since f is not continuous at x = 2 when k = 4, it cannot be differentiable at that point either. Therefore, for k = 4, f is not differentiable at x = 2.